A Mathematical Model of Protein Translation and

Otto-von-Guericke-University Magdeburg

Faculty of Process and Systems Engineering

Diploma Thesis A Mathematical Model of Protein Translation and the Competition for Rare Cellular Resources in Response to Amino Acid Starvation Author:

Stefan Heldt November 16, 2009

Supervisor:

Dr. Marco Thiel University of Aberdeen Department of Physics Fraser Noble Building, Kings College Aberdeen, AB24 3UE, Scotland

In the middle of difficulty lies opportunity. Albert Einstein

Heldt, Stefan: A Mathematical Model of Protein Translation and the Competition for Rare Cellular Resources in Response to Amino Acid Starvation Diploma Thesis, Otto–von–Guericke–University Magdeburg, 2009.

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OTTO-VON-GUERICKE-UNIVERSITÄT MAGDEBURG

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Aufgabenstellung für die Diplomarbeit von Herrn Stefan Heldt, Matr.-Nr.: 171596 Title:

Stochastic simulation of protein translation: Effects of ribosome competition for rare tRNA species The process of protein synthesis, which is crucial for biological systems, has been the focus of intensive study over the past few decades. The translation of messenger RNAs (mRNA) into corresponding proteins is carried out by ribosomes via transfer RNAs (tRNAs) carrying specific amino acids (aa). Thereby, thousands of ribosomes are engaged in translation in one cell at the same time. It is generally accepted that the translation rate, and therefore the protein production, depends on the availability of cognate aa-tRNAs to these ribosomes. This thesis aims to extend an existing modeling approach for protein translation based on the totally asymmetric exclusion process, by considering the dependence on tRNA abundance. Competition among elongating ribosomes on multiple mRNA strands for rare tRNA species will be one fundamental aspect of the improved model. Furthermore the (re-)charging of used tRNAs with amino acids plays a crucial role. On the basis of biological data of S. cerevisiae the student will develop a stochastic simulation for both processes. Assessing and modeling the dynamic relation between tRNA binding to ribosomes, recharging of tRNAs and ribosome movement represents one major task. To evaluate the impact of competition for rare resources on protein production in principle, the student will first implement and analyse artificial mRNA sequences with respect to their translation rate. Based on these numerical simulations and further biological data, an analysis of real mRNA sequences employing the actual abundance of tRNAs in yeast will follow. Depending on the progress the student will then implement genome wide data of S. cerevisiae to compute protein synthesis. Differences in the translation rate of different protein types caused by competition are of particular interest. The student will employ parallel programming techniques in order to evaluate the model on a computer cluster.

Beginn der Diplomarbeit: 20.05.2009 Ende der Diplomarbeit:

20.11.2009

___________________________ Prof. Dr.-Ing. E. D. Gilles Verantwortlicher Hochschullehrer

___________________________ Dr. M. Thiel University of Aberdeen, Dept. of Physics Betreuer der Arbeit

d

Erklärung zum selbständigen Arbeiten Hiermit versichere ich, Stefan Heldt, dass ich die vorliegende Arbeit selbständig verfasst und keine anderen, als die im Literaturverzeichnis angegebenen, Quellen benutzt habe. Stellen und Zitate, die wörtlich oder sinngemäß aus veröffentlichten oder noch nicht veröffentlichten Quellen entnommen sind, sind als solche kenntlich gemacht. Die Zeichnungen oder Abbildungen in dieser Arbeit sind von mir selbst erstellt worden oder mit einem entsprechenden Quellennachweis versehen. Diese Arbeit ist in gleicher oder ähnlicher Form noch bei keiner anderen Prüfungsbehörde eingereicht worden.

Ort, Datum

Stefan Heldt

e

Acknowledgements I wish to thank all those who supported me during my thesis and my studies. Without them, I would not have completed this part of my life. A special thanks goes to my adviser Marco Thiel for his continuous support. Marco was responsible for involving me in this modelling project in the first place and he was always there to listen and give advice. He showed me different ways to approach a research problem and the need to be persistent to accomplish any goal. I also thank Andreas Kremling for giving me the opportunity to start my diploma thesis at the University of Aberdeen. His help with finding a thesis project and assistance throughout it made this work possible. It were also his lectures that sparked my interest in Systems Biology. Beside my advisers, I would like to thank the following colleagues at the Department of Physics of the University of Aberdeen: Christopher Brackley for reviewing my work on a very short notice and giving valuable feedback. Ian Stansfield who found the time to discuss the biological side of this project; and Mamen Romano and Luca Ciandrini for helping me at any time, and solving any unsolvable problems. Last, but not least I thank my family for their advice, their trust and the unconditional support and encouragement to pursue my interests, even when these interests went beyond practical application. They listened to my complaints and frustrations, while always believe in me. With their immense help, they contributed more to my thesis then they might be aware of.

Abstract Precise regulation of protein synthesis lies at the heart of living systems. However, despite the advances made in genomics and proteomics during the past decades, the exact mechanism by which biased codon usage influences the production of proteins on a translational level remains rather unclear. Mathematical modelling can constitute an important step toward a more comprehensive understanding of such a fundamental biological process. In this context, current approaches to the computational analysis of protein translation can be divided into deterministic and stochastic models. Although both types of model account for the special role of aa–tRNAs, they largely neglect how these molecules are charged with cognate amino acids, and that they form a finite pool of resources. We present a deterministic description of translation which explicitly accounts for tRNA binding, charging and discharging. Application of analytic techniques demonstrates that amino acid starvation significantly decreases the abundance of aa–tRNAs, reducing their availability to ribosomes. As these results are in agreement with in vivo experiments, we use the model to predict the consequences of there being a limited number of charged tRNAs. Numeric simulations suggest that translation is more sensitive to the availability of amino acids than to changes in initiation rate, which is a common method of regulating protein expression. Additionally, the biased usage of codons is shown to determine an mRNA’s translational robustness to starvation conditions and its susceptibility to control mechanisms. We discuss several artificial mRNA configurations with special codon distributions that cause these phenomena. We also develop a stochastic model of protein translation which is based on the totally asymmetric exclusion process. By implementing extended ribosomes and accounting for variation in tRNA charging level, this description is applied to realistic mRNA sequences of S. cerevisiae. Our computational analysis shows a relation between the sensitivity of an mRNA to nutritional stress and the biological function of its respective protein. Beside predicting ribosome densities and tRNA charging levels, the model also highlights the characteristics of flow optimised nucleotide sequences. In this regard, competition of the ribosome population for a finite pool of aa–tRNAs is shown to constitute one form of regulation on a translational level.

Contents

i

Contents List of Figures

iii

List of Tables

vi

List of Abbreviations

vii

1. Introduction

1

2. Biological and Theoretical Background

2

2.1. Protein Bio–synthesis in Eukaryotes . . . . . . . . . . . . . . . . . . . . .

2

2.2. Protein Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

2.2.1. The Genetic Code . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2.2.2. Ribosomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2.2.3. Transfer RNAs . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.2.4. Phases of Protein Translation . . . . . . . . . . . . . . . . . . . .

8

2.3. Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.4. Competition for Rare Resources . . . . . . . . . . . . . . . . . . . . . . .

11

2.5. Michaelis–Menten Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . .

12

3. Deterministic Modelling Approach

14

3.1. Deterministic Model Neglecting Steric Hindrance . . . . . . . . . . . . .

14

3.1.1. Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . .

14

3.1.2. Analytic Solution for the steady–state tRNA charging level . . . .

17

3.1.3. Numerical Simulations of Artificial Sequences . . . . . . . . . . .

20

3.2. Phase Transition in Protein Production Rates . . . . . . . . . . . . . . .

24

3.3. Deterministic Model Accounting for Steric Hindrance . . . . . . . . . . .

26

3.3.1. Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . .

26

3.3.2. Numerical Simulations of Artificial Sequences . . . . . . . . . . .

27

3.3.3. Relative Protein Production . . . . . . . . . . . . . . . . . . . . .

32

3.3.4. Influence of the Charging Enzyme Activity . . . . . . . . . . . . .

33

3.4. Effects of Two Codon Species and Competing Ribosomes . . . . . . . . .

34

Contents

ii

3.4.1. Two Codon Species on a Single mRNA . . . . . . . . . . . . . . .

34

3.4.2. Ribosomal Competition for Rare aa–tRNAs . . . . . . . . . . . .

36

3.5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

4. Stochastic Modelling Approach

43

4.1. The TASEP Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

4.1.1. Improved TASEP Algorithm . . . . . . . . . . . . . . . . . . . . .

44

4.1.2. Analytic Solution for the Ribosome Flow . . . . . . . . . . . . . .

46

4.2. Numerical Simulations of Artificial Sequences . . . . . . . . . . . . . . .

47

4.2.1. Ribosomes Covering One Codon . . . . . . . . . . . . . . . . . . .

47

4.2.2. Ribosomes Covering Multiple Codons . . . . . . . . . . . . . . . .

50

4.3. Numerical Simulations of Realistic Sequences . . . . . . . . . . . . . . . .

50

4.3.1. Derivation of a Realistic Parameter Set . . . . . . . . . . . . . . .

51

4.3.2. Example Proteins and mRNA Configurations . . . . . . . . . . .

52

4.3.3. Charging Level of tRNAs . . . . . . . . . . . . . . . . . . . . . . .

55

4.3.4. Ribosome Density on Realistic mRNAs . . . . . . . . . . . . . . .

57

4.3.5. Protein Production Rate . . . . . . . . . . . . . . . . . . . . . . .

60

4.3.6. Effects of Competing Ribosomes . . . . . . . . . . . . . . . . . . .

62

4.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

5. Outlook

67

Bibliography

69

List of Symbols

75

A. Simulation Parameters

77

A.1. Analytic Solution for the Steady–State tRNA Charging Level . . . . . . .

77

A.2. Numerical Simulation of the Deterministic Model . . . . . . . . . . . . .

77

A.3. Numerical Simulation of the Stochastic Model . . . . . . . . . . . . . . .

79

B. Additional Results

80

B.1. Deterministic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

B.2. Stochastic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

List of Figures

iii

List of Figures 2.1. Scheme of eukaryotic protein bio–synthesis. . . . . . . . . . . . . . . . . .

3

2.2. The three phases of protein translation. . . . . . . . . . . . . . . . . . . .

4

2.3. The genetic code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.4. Ribosome reader density as a function of position measured by Ingolia et al.

6

2.5. Simplified scheme of protein translation and the influence of amino acid starvation on tRNA charging. . . . . . . . . . . . . . . . . . . . . . . . .

11

2.6. Michaelis–Menten like equation with double substrate limitation. . . . . .

13

3.1. Simplified scheme of protein translation employed to derive a deterministic translation model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

3.2. Analytic solution for the steady–state tRNA charging level. . . . . . . . .

20

3.3. Steady–state charging level and protein production rate for the deterministic model neglecting steric hindrance. . . . . . . . . . . . . . . . . . . .

21

3.4. Relative charging level of different tRNA species in response to amino acid starvation measured by Zaborske et al. . . . . . . . . . . . . . . . . . . .

22

3.5. Phase transition in the steady–state protein production of the deterministic model neglecting steric hindrance. . . . . . . . . . . . . . . . . . . .

22

3.6. Steady–state ribosome density for the deterministic model neglecting steric hindrance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

3.7. Protein types according to the dependence of ribosome current on initiation probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

3.8. Different mRNA configurations correspond to distinct protein types. . . .

24

3.9. Steady–state charging level and protein production for the deterministic model accounting for steric hindrance and a slow codon at i = 50. . . . .

28

3.10. Steady–state ribosome density for the deterministic model accounting for steric hindrance and a slow codon at i = 50. . . . . . . . . . . . . . . . .

29

3.11. Phase transition in protein production for the deterministic model accounting for steric hindrance and a slow codon at i = 50. . . . . . . . . .

30

List of Figures

iv

3.12. Steady–state charging level and protein production for the deterministic model accounting for steric hindrance and a slow codon at i = 1. . . . . .

30

3.13. Phase transition in protein production for the deterministic model accounting for steric hindrance and a slow codon at i = 1. . . . . . . . . . .

31

3.14. Relative protein production in response to amino acid starvation . . . . .

32

3.15. Normalised sensitivity of protein production rate to changes in amino acid concentration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

3.16. Different mRNA configurations employed during numerical simulations. .

34

3.17. Steady–state tRNA charging levels for an mRNA strand containing two codon species. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

3.18. Steady–state protein production for an mRNA strand containing two slow codon species. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

3.19. Steady–state protein production for two mRNA strands containing two codon species. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

3.20. Relative protein production in response to kα with and without competition. 37 3.21. Protein production of a type II mRNA in response to competition and changes in amino acid concentration. . . . . . . . . . . . . . . . . . . . .

38

3.22. Relative protein production of simultaneously translated mRNAs is coupled through the pool of available aa–tRNAs. . . . . . . . . . . . . . . .

39

4.1. Scheme of a TASEP resembling protein translation. . . . . . . . . . . . .

44

4.2. Ribosome movement employing a random update. . . . . . . . . . . . . .

46

4.3. Steady–state charging level and protein production for the stochastic TASEP model with ribosomes covering one codon. . . . . . . . . . . . . .

48

4.4. Quantitative differences in steady–state protein production rate and tRNA charging level employing the deterministic model and the stochastic TASEP description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

4.5. Steady–state protein production for the stochastic TASEP model considering different ribosome sizes. . . . . . . . . . . . . . . . . . . . . . . . .

51

4.6. Position of codons decoded by rare tRNAs and of codons representing leucine in an mRNA coding for the leucine–tRNA ligase. . . . . . . . . .

53

4.7. Biased usage of codons representing leucine in mRNAs coding for leucine– tRNA ligase and Efb1. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

List of Figures

v

4.8. Position of codons decoded by rare tRNAs and of codons representing leucine in an mRNA coding for the branched–chain–amino–acid transaminase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9. Relative steady–state charging level of tRNA

Leu

55

isoacceptors in response

to amino acid starvation. . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

4.10. Predicted steady–state ribosome density on an mRNA coding for leucine– tRNA ligase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

4.11. Predicted steady–state ribosome density on an mRNA coding for the branched–chain–amino–acid transaminase. . . . . . . . . . . . . . . . . .

59

4.12. Steady–state protein production rate against initiation probability for three realistic mRNAs. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

4.13. Steady–state protein production rate against amino acid concentration for three realistic mRNAs. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

4.14. Normalised protein production rate of an mRNA coding for leucine–tRNA ligase and transaminase. . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

4.15. Steady–state protein production of two simultaneously translated realistic mRNAs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

4.16. Steady–state protein production of an mRNA coding for leucine–tRNA ligase in response to competition with different realistic sequences. . . . .

63

B.1. Steady–state charging level and protein production against kα and vmax . .

80

B.2. Local sensitivity of the deterministic model accounting for steric hindrance to selected parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

B.3. Different mRNA configurations employed in the appendix. . . . . . . . .

82

B.4. Steady–state charging level and protein production for two mRNA strands containing two codon species. . . . . . . . . . . . . . . . . . . . . . . . .

83

B.5. Difference in steady–state protein production for the stochastic TASEP model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

B.6. Steady–state charging level and protein production for the stochastic TASEP model with ribosomes covering nine codons. . . . . . . . . . . . .

85

B.7. Steady–state protein production against initiation probability for the translation of an mRNA coding for the leucine–tRNA ligase. . . . . . . .

86

B.8. Steady–state ribosome density on an mRNA coding for Efb1. . . . . . . .

87

B.9. Steady–state protein production of all three realistic mRNAs in response to competition and changing initiation probability. . . . . . . . . . . . . .

88

List of Tables

vi

List of Tables 2.1. Abundances of tRNA species in S. cerevisiae. . . . . . . . . . . . . . . . .

6

3.1. System behaviour of type I and II mRNAs in response to changing initiation rate and amino acid concentration. . . . . . . . . . . . . . . . . . .

31

4.1. Parameters employed in stochastic simulations of realistic mRNA sequences with the improved TASEP algorithm. . . . . . . . . . . . . . . .

52

A.1. Parameter set employed to calculate the steady–state tRNA charging level with the analytic solution. . . . . . . . . . . . . . . . . . . . . . . . . . .

77

A.2. Kinetic parameters of the leucine–tRNA ligase according to Brenda. . . .

77

A.3. Parameters used in numerical simulations of artificial sequences with the deterministic model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

A.4. Parameter set employed to simulate artificial sequences with the stochastic TASEP model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

vii

List of Abbreviations aa aaRS aa–tRNA ASEP ATP DNA Efb1 GCN GTP LeuRS MCS mRNA ODE RNA TASEP tRNA

amino acid aminoacyl tRNA synthetase aminoacyl transfer ribonucleic acid asymmetric simple exclusion process adenosine-5’-triphosphate deoxyribonucleic acid translation elongation factor 1 beta gene copy number guanosine-5’-triphosphate leucine–tRNA ligase Monte–Carlo–Step messenger ribonucleic acid ordinary differential equation ribonucleic acid totally asymmetric simple exclusion process transfer ribonucleic acid

Chapter 1. Introduction

1

Chapter 1

Introduction

Elucidating the complex interactions in biological systems and unravelling the mystery of life itself is one of the great challenges of our times. In this context, scientists have to bridge the gap between diverse disciplines to gain an insight into the basic processes that connect all living organisms. As one of these central processes, protein synthesis lies at the heart of a wide variety of relevant phenomena [1, 2]. Although substantial progress has been made in several areas during the past decades, the role of biased codon usage in regulating protein synthesis on a translational level remains largely unclear. However, adaptation of the proteome, as the set of all cellular proteins, to environmental changes is vital [3, 4] and translational regulation constitutes an important step toward it [5, 6]. As a consequence of complexity, mathematical modelling has been the method of choice to study many biological systems [7, 8]. In the field of protein translation, two opposing strategies were developed in recent years. Stochastic approaches, derived from statistical physics and originally employed to describe reaction–diffusion systems, provide the basic methodology to simulate translation in silico [9, 10, 11]. In combination with their deterministic counterparts [12, 13], these stochastic models contributed to the understanding of protein translation as one of the crucial steps in synthesising proteins [14]. However, both approaches often neglect the conditions a cell encounters and their impact on essential components of the translation machinery. In vitro experiments demonstrate that especially when a cell experiences nutritional stress, the building blocks of proteins become rare [15]. The limitation of these cellular resources makes precise regulation of translation crucial to survival [14] and has not been accounted for so far. This work outlines the shortcomings of current modelling approaches to protein translation in the context of constrained cellular resources and nutritional stress conditions. By deriving extended mathematical models and comparing them to in vivo experiments, we demonstrate the importance of accounting for the supply of amino acids in such a mathematical description. Furthermore, numerical simulations highlight how the codon configuration of an mRNA determines its susceptibility to limitations imposed by the environment. Analysis of realistic sequences relate the findings to protein translation in S. cerevisiae as our model system. In this regard, biased codon usage emerges as one control mechanism to regulate the impact of nutrient starvation on protein expression. We reason that evolution might have shaped the genetic code and hence the usage of codons in mRNAs with respect to competition for limited cellular resources within the transcriptome of a cell.

Chapter 2. Biological and Theoretical Background

2

Chapter 2

Biological and Theoretical Background

The following pages summarise essential biological aspects of protein bio–synthesis in eukaryotes and outline protein translation with respect to its modelling. Subsequent to a brief discussion of sensitivity analysis, section 2.4 then introduces the concept of competition for rare resources in the context of protein translation. Eventually section 2.5 highlights the Michaelis–Menten equation as a description for enzymatic reactions.

2.1. Protein Bio–synthesis in Eukaryotes Proteins perform numerous tasks in living systems, ranging from catalysing chemical reactions to forming cellular structures, and thus facilitate life itself [16]. Their production is the multi–step process of protein bio–synthesis whose regulation is vital for organisms and their smallest entity, the cell [14]. Protein synthesis starts with nuclear DNA which provides the genetic information required. Sections of DNA, the so called genes, contain the blueprint of proteins in coding regions, whereas non–coding regions regulate the gene’s activity. Figure 2.1 illustrates transcription and translation as the two sub–processes involved in synthesising a linear chain of amino acids (or polypeptide chain), i.e., the protein. Initially RNA polymerases transcribe a coding region into a messenger molecule, the mRNA. Once completely processed, this messenger molecule travels into the cytoplasm where the translation of its nucleotide sequence into a corresponding amino acid chain takes place. Whilst biological systems are known to use various transcription level strategies to regulate protein synthesis, mechanisms interfering on a translational level also constitute an important factor [17, 18]. The exact extent to which an mRNA’s structure contributes to this regulation is still not fully understood. We address this issue by means of a mathematical modelling approach to protein translation.

2.2. Protein Translation Forming a chain of amino acids based on the mRNA template, a process referred to as protein translation, comprises of the three distinct phases: (i) Initiation, (ii) Elongation and (iii) Termination (figure 2.2). These phases correspond to the attachment, movement and dissociation of molecular machines, called ribosomes, to an mRNA strand


Figure 2.1.: Scheme of eukaryotic http://estrellamountain.edu).

protein

bio–synthesis

3

(source:

(see section 2.2.4). In addition to the mRNA that carries genetic information and ribosomes which act as translators, another key component is necessary to realise translation, namely transfer RNAs (tRNAs). In this context, the genetic code emerges as a fundamental principle of life.

2.2.1. The Genetic Code Defining the mapping between a nucleotide sequence and the corresponding amino acid chain, i.e., between mRNA and protein, the genetic code (figure 2.3a) is highly conserved throughout life [19, 20]. It represents a key for encoding all 20 standard amino acids by only four different nucleotides which form an mRNA. To achieve this, groups of three nucleotides, known as codons, encode one amino acid. The tRNAs (shown in figure 2.3b and discussed in detail in section 2.2.3) provide the biochemical basis of this mapping [21].

2.2.2. Ribosomes As some of the oldest and most conserved molecular machines [22], ribosomes consist of protein and RNA components and perform the production of proteins. They move along an mRNA while translating its nucleotide sequence [22]. To produce multiple copies of the same protein, multiple ribosomes (the polysome) elongate simultaneously. Experiments confirm that ribosomes protect a discrete footprint of 9 to 10 codons on an


4

(a)

(b)

(c)

Figure 2.2.: The three phases of protein translation. (a) Initiation, (b) Termination and (c) Elongation (source: http://kvhs.nbed.nb.ca). mRNA from nuclease digestion [23]. Hence, they are extended particles which not only block enzymes from interacting with a section of codons, but more importantly each other during elongation. This phenomenon is referred to as steric hindrance or blocking of ribosomes. With respect to the mechanism of elongation, ribosome movement on an mRNA resembles a traffic problem with extended particles that interact exclusively [24, 25]. Recent articles studying the distribution of ribosomes on mRNAs find a relatively high ribosome content within the first 50 to 100 codons (figure 2.4), but reveal a low overall density at the same time [26, 27]. On average one ribosome is found every 50 codons, which results in about 1/5 of the maximum packing density [26]. Density profiles like the one in figure 2.4 provide experimental evidence that translation is mainly controlled by initiation, preventing large amounts of ribosomes from starting translation. Steric hindrance significantly contributes to the dynamics of elongation within the first 100 codons of an mRNA [7, 12]. A detailed discussion of how initiation and elongation control ribosome densities follows in chapter 3.

2.2.3. Transfer RNAs As chemical mediators, tRNAs (figure 2.3b) are the key for the genetic code. They contain a specific sequence of three nucleotides, an anti–codon, which binds to codons on an mRNA. Transfer RNAs exist in two states: charged with an amino acid as aminoacyl– tRNAs (aa–tRNAs); or in their uncharged, amino acid free form. Specificity In most organisms the number of different tRNA species exceeds the number of available amino acids. The set of tRNAs is redundant, i.e., an amino acid is


(a)

5

(b)

Figure 2.3.: The genetic code. (a) Codons and their respective amino acids (source: http://faculty.irsc.edu). (b) Structure of a tRNA molecule meditating the translation of codons into amino acids (source: http://www.wiley.com). occasionally encoded by more then one tRNA species. Hence, mRNAs with distinct codon configurations can encode the same protein. Furthermore, some tRNAs bind to several codons, allowing the decoding of all 43 = 64 possible nucleotide combinations. Only through an inaccurate codon–anticodon or wobble base pairing, binding to more than one codon is facilitated. Particularly the 3’ base of a codon and the 5’ base of the respective anticodon form these wobble base pairs. Suboptimal codon–anticodon pairing results in a change of binding kinetics and reduces the rate with which binding occurs [28, 29]. Codon Frequency and tRNA Abundance Resembling the occurrence of amino acids in proteins, codon species are non–uniformly distributed within an mRNA [30, 31]. Qin et al. demonstrate their overall frequency to vary between different genomes [32]. Kurland suggests a role of this phenomenon, know as codon bias, in the regulation of individual gene expression [33]. Since nucleotide triplets and their corresponding tRNAs are tightly related, the varying abundance of different tRNA species reflects this biased codon usage [34, 35]. Two studies even report the intracellular concentration of tRNAs to correlate with the usage frequency of their codons in specific genes of E.coli and S. cerevisiae [35, 36]. In this regard, the redundancy of genes encoding individual tRNA species is important. Identification and classification of 274 yeast tRNA genes show that up to 16 copies of a single tRNA species’ gene exist and that gene copy number and abundance are related [37]. In absence of direct measurements, we utilise this relation for modelling purposes


6

Figure 2.4.: Ribosome reader density, i.e., density of the ribosome’s A site which receives a cognate tRNA to the current codon, as a function of position taken from literature [27]. Ribosome densities were determined by ribosome–profiling based on the deep sequencing of mRNA fragments protected from nuclease digestion. Well–expressed genes were each individually normalised and then averaged with equal weight. to approximate the abundance of yeast tRNAs presented in table 2.1. Note that the gene copy number and hence the tRNA abundances vary by a factor 16. The calculation of specific translation rates in table 2.1 takes aspects into consideration which are presented in section 2.2.4, and is explained in later sections. Table 2.1.: Abundances of tRNA species in one S. cerevisiae cell. Calculated abundances of all 42 yeast tRNAs based on gene copy number (GCN) measured by Percudani et al. [37] and a total of 3 · 106 tRNA molecules in a cell [38]. Specific translation rates are scaled to ensure an average translation rate of 10 codon/s. tRNA

aa

Anti-codon

Ala1 Ala2 Arg2 Arg3 Arg4 Arg5

A A R R R R

IGC UGC ICG CCG UCU CCU

Codon Recognized GCU, GCC GCA, GCG CGU, CGC, CGA CGG AGA AGG

GCN

tRNAs molecules/cell

tranlation rate codon/s

11 5 6 1 11 1

120438 54745 65693 10949 120438 10949

16.86 7.66 9.20 1.53 16.86 1.53

Table 2.1 – continues on next page


7

Table 2.1 – continued from previous page tRNA

aa

Anti-codon

Asn Asp Cys Gln1 Gln2 Glu1 Glu2 Gly1 Gly2 Gly3 His Ile1 Ile2 Leu1 Leu2 Leu3 Leu4 Lys1 Lys2 Met f1 Met f2 Phe Pro1 Pro2 Ser1 Ser2 Ser3 Ser4 Thr1 Thr2 Thr3 Trp Tyr1 Val1 Val2A Val2B

N D C Q Q E E G G G H I I L L L L K K M M F P P S S S S T T T W Y V V V

GUU GUC GCA UUG CUG UUC CUC GCC UCC CCC GUG IAU UAU UAA CAA GAG UAG UUU CUU CAU CAU GAA IGG UGG IGA UGA CGA GCU IGU UGU CGU CCA GUA IAC UAC CAC

Sum Average

Codon Recognized AAC, AAU GAC, GAU UGC, UGU CAA CAG GAA GAG GGC, GGU GGA GGG CAC, CAU AUC, AUU AUA UUA UUG CUC, CUU CUA, CUG AAA AAG AUG AUG UUC, UUU CCU, CCC CCA, CCG UCU, UCC UCA UCG AGC, AGU ACC, ACU ACA ACG UGG UAC, UAU GUC, GUU GUA GUG

GCN

tRNAs molecules/cell

tranlation rate codon/s

10 15 4 9 1 14 2 16 3 2 7 13 2 7 10 1 3 7 14 5 5 10 2 10 11 3 1 4 11 4 1 6 8 14 2 2

109489 164234 43796 98540 10949 153285 21898 175182 32847 21898 76642 142336 21898 76642 109489 10949 32847 76642 153285 54745 54745 109489 21898 109489 120438 32847 10949 43796 120438 43796 10949 65693 87591 153285 21898 21898

15.33 22.99 6.13 13.80 1.53 21.46 3.07 24.53 4.60 3.07 10.73 19.93 3.07 10.73 15.33 1.53 4.60 10.73 21.46 7.66 7.66 15.33 3.07 15.33 16.86 4.60 1.53 6.13 16.86 6.13 1.53 9.20 12.26 21.46 3.07 3.07

274 6.52

3 · 106 71428

10


8

Charging of tRNAs Outlining the role of tRNAs in translation demonstrates the importance of correctly charging these molecules with amino acids that match their anti– codon. A family of 20 enzymes (one for each amino acids), the aminoacyl tRNA synthetases (aaRS), mediates the formation of aminoacyl–tRNAs. Aminoacylation is a two step process and uses ATP, an uncharged tRNA and the respective amino acid (aa) as substrates [39, 40]. The two steps are aaRS + aa + ATP −→ (aa–AMP : aaRS ) + PP i (aa–AMP : aaRS ) + tRNAaa −→ aatRNAaa + AMP + aaRS

and

(2.1) (2.2)

with aa aaRS aatRNAaa AMP ATP PP i tRNAaa

amino acid aminoacyl tRNA synthetase charged tRNA adenosinmonophosphat adenosintriphosphat pyrophosphate uncharged tRNA

in in in in in in in

mol mol mol mol mol mol mol.

This mechanism not only influences the overall fidelity of protein synthesis, but determines the tRNA charging level, i.e., the ratio of aa–tRNAs to total tRNA molecules of a given species [41, 42].

2.2.4. Phases of Protein Translation With the essential components discussed, the following section focuses on the sub– processes of protein translation, namely initiation, elongation and termination (figure 2.2). Here, we introduce kinetic considerations, which are later applied in modelling. Initiation Translation initiation (figure 2.2a) subdivides in several steps itself, all of which are facilitated by translation initiation factors [43]. The exact mechanism that enables ribosomes to find the initiation signal in combination with these factors is still controversial [44]. Consequently modelling has to abstract this process. According to the “scanning” hypothesis, a 40S ribosome–Met–tRNA complex recognises and binds to the 5’ end of an mRNA and scans for the AUG codon. Once this initiator codon is located, the 60S subunit joins the complex and forms a 80S ribosome, ready to mediate the first peptide bond [44]. Theoretical and experimental studies suggest that initiation is one of the limiting steps of protein translation [45, 46, 47]. Elongation Shortly after the ribosome complex has been formed, elongation of the polypeptide chain takes place (figure 2.2c). In each elongation step the ribosome recruits an aa–tRNA, whose anti–codon pattern complements the codon, to its A site (later referred to as the reader). It then forms a peptide bond between the amino acid delivered and the nascent chain [48, 49]. The ribosome itself has to run through several internal


9

states, which involves the consumption of high energy bonds provided by ATP and GTP. Several mathematical models address this concept [25, 50]. Once a peptide bound has been formed, the ribosome moves one codon forward and releases an uncharged tRNA originating from the previous step. Various publications describe the waiting time for a cognate tRNA as the limiting step of elongation [45, 46, 47, 51]. The waiting time in turn depends on the concentration of the respective tRNA species. This is in agreement with the observation that different codons types show varying translation rates [28, 29, 51, 52]. In this context, nucleotide triplets corresponding to an abundant tRNA species are commonly referred to as fast codons. In contrast, the low translation rate of so called slow codons, i.e., codons decoded by a rare tRNA, causes ribosomes to slow down. Hence, elongation resembles a traffic problem including the possibility of queue formation. Combining observations from the previous paragraph with section 2.2.3, we reason that it is not the concentration of tRNAs, but rather the abundance of charged aa–tRNAs which affects codon translation rates. The calculation which is used to derive table 2.1 utilises this concept to find the specific translation rate of a codon. By considering that approximately 85 % of all ribosome are engaged in translation [26] and that the efficiency of protein translation per ribosome is 8.8 aa/s [53], one can calculate the average macroscopic translation rate as hrtr i = 8.8 aa/s ·

1 b 10 codon/s. ≈ 10 aa/ (s · Rib) = 0.85 Rib

Using the average number of tRNA molecules htRNAi then yields a new parameter, the specific translation rate per available tRNA molecule rtr,tRNA =

hrtr i . htRNAi

(2.3)

Multiplication by the abundance of charged tRNA molecules of species i results in the specific translation rate of its corresponding codon rtr,i = rtr,tRNA · aatRNAi

(2.4)

with aatRNAi hrtr i rtr,i rtr,tRNA htRNAi

abundance of the charged tRNA species i average translation rate specific translation rate of codons decoded by tRNAi translation rate per available tRNA molecule average number of tRNA molecules per species

in in in in in

molecules codon/s codon/s codon/ (s · molecules) molecules.

The rates in table 2.1 are derived using these equations and assuming complete charging of the entire cellular tRNA population. By introducing rtr,tRNA , we assume codon translation rates to solely depend on the waiting time for cognate aa–tRNA molecules and neglect other tRNA species characteristics. This implies that all intrinsic state changes a ribosome undergoes add up to


10

a total rate constant, which is independent of a tRNA’s identity. Fluitt et al. however demonstrate an influence of proof reading steps and near cognate tRNA concentrations on elongation time [50]. To correct for this simplification, one could calculate the time a ribosomes needs to pass through all internal states. Accounting for proof–reading requires the derivation of a tRNA species specific rtr,tRNA [25, 54]. A second concern emerges in context of deriving rtr,tRNA . By employing the average amount of tRNA molecules htRNAi, we neglect biased codon usage. In S. cerevisiae the macroscopically observed average translation rate hrtr i includes different codon frequencies. Thus, weighting the influence of tRNA abundance in htRNAi by the respective codon’s frequency would improve the estimation. However, for the sake of simplicity and in absence of accurate measurements for most of the required parameters we employ (2.3) in the following. Termination Two distinct scenarios can cause a termination of the translation process. As soon as the ribosome encounters one of three possible stop codons and its corresponding release factors, it releases a complete amino acid chain and dissociates from the mRNA (figure 2.2b). Also processivity errors can result in premature termination and in an incomplete polypeptide [55, 56]. False terminations are rare and affect the entire mRNA population homogeneously [57]. Hence, detailed modelling would not yield significantly different results. Additionally, termination occurs quickly compared to initiation and elongation, allowing its simplification in a translation model.

2.3. Sensitivity Analysis In this work we employ a sensitivity analysis to determine how variation in the output of our models can be appointed to different sources, i.e., different model parameters [58]. Studying changes in steady–state simulation results provides evidence for the importance of parameters in terms of modelling accuracy. In later sections, the sensitivity also yields a mathematical explanation for the observed system characteristics. We calculate local normalised sensitivities of state variables with respect to parameters according to w¯i,j =

∆yi pj dyi pj · ≈ · dpj yi ∆pj yi

(2.5)

with pj w¯i,j yi

parameter j normalised sensitivity of state variable i to parameter j state variable i.

By monitoring the system’s output, one can utilise this equation to assess the impact of parameter perturbation. In this context, the difference in numerical simulation results ∆yi can approximate the derivative dyi . Application of the presented normalisation allows a comparison of sensitivities with respect to state variables and parameters of different scales.


11

2.4. Competition for Rare Resources Since life hardly ever exists in optimal conditions it often encounters limitations imposed by its environment. Gaining the maximum benefit out of scarce resources is one of the driving forces of evolution and emerges throughout all scales of life. The constraint of resources frequently results in competition for them — a concept which even affects molecular processes. Charged tRNAs represent a limiting factor for synthesising proteins and form a finite pool of molecules within a cell. A high translational activity or an impaired tRNA charging process due to inadequate supply with nutrients, i.e., free amino acids, could yield a reduced availability of these molecules (figure 2.5). Current models neglect this fact by assuming a pool of constantly charged tRNAs, which is not affected by translation or nutritional stress. However, as up to 15 ribosomes simultaneously translate a single mRNA [26] and a yeast cell contains approximately 15,000 mRNA strands [59], a large number of ribosomes have to compete for this finite amount of molecules. Changes in the availability of aa–tRNAs would affect this population of ribosomes significantly. Moreover, depending on the codon configuration of the mRNA strand they translate, some elongating ribosomes might perform better in the competition caused by the constraint of cellular resources. We presume competition to become important in case of low tRNA charging levels that are most likely to occur when the charging process is impaired due to nutritional stress. Accordingly, we are investigating the performance of ribosomes on an mRNA, i.e., the protein production rate, in response to the limitation of available aa–tRNAs by amino acid starvation (figure 2.5). Studies show that extra–cellular amino acids, representing

Figure 2.5.: Simplified scheme of protein translation and the influence of amino acid starvation on tRNA charging.


12

a major nutrient a cell has to take up, influence the charging process [15, 42]. Under starvation conditions, uncharged tRNAs act as down–regulating factors of global gene expression [60]. However, observations show an up–regulated production of proteins involved in amino acid synthesis [61]. Neglecting signalling events or metabolic changes, we address the question as to whether a specific distribution of codons on mRNAs which encode these proteins yields a lower sensitivity to limited resources. In this regard evolution might have also shaped the codon usage of mRNAs with respect to ribosomal competition for rare aa–tRNAs.

2.5. Michaelis–Menten Kinetics A central concept of modelling limited cellular resources is the expressions used to describe their supply and demand. As we focus on protein translation under starvation conditions, the supply of aa–tRNAs is of particular interest [41, 42]. Assuming the production of tRNAs to be in balance with their degradation, the overall pool of tRNAs remains constant. Hence, only charging increases the abundance of aa–tRNAs. The Michaelis–Menten kinetics apply to many enzymatically catalysed reactions and in the here presented case to the charging process, which is facilitated by aminoacyl–tRNA synthetases. Simplifying (2.1) and (2.2) to a single reaction step with an uncharged tRNA and its respective amino acid as the two substrates (assuming that ATP is not limiting), one obtains v aa + tRNAaa → − aatRNAaa . (2.6) For the reaction velocity (v) a Michaelis–Menten like equation with double substrate limitation and a maximum reaction velocity (vmax ) applies. It is given by v = vmax

tRNAaa tRNAaa + Km,tRNAaa

!

aa aa + Km,aa

!

(2.7)

with Km,aa Km,tRNAaa v vmax

Michaelis–Menten constant for an amino acid Michaelis–Menten constant for an uncharged tRNA reaction velocity maximum reaction velocity

in in in in

mol mol mol/s mol/s.

Equation (2.7) shows the essential characteristics (figure 2.6) observed in biological experiments and mathematical models [41, 42]. For instance, if either of the two substrates is depleted (tRNAaa = 0 or aa = 0), the reaction does not occur, i.e., v = 0. In contrast, with tRNAaa Km,tRNAaa and aa Km,aa the reaction velocity reaches its maximum v = vmax . Both Michaelis–Menten constants (Km ) represent their respective , assuming the substrate’s concentration at which the reaction velocity reaches v = vmax 2 other substrate is not limiting. Parameters Km,tRNAaa , Km,aa and vmax are specific for an aminoacyl–tRNA synthetase and can be found in enzyme databases such as Brenda [62]. The Michaelis–Menten constants are explicitly stated. With the turnover number (kcat ), which refers to the


13

Figure 2.6.: Example of the Michaelis–Menten like equation (2.7) with double substrate limitation and vmax = 1, Km,tRNAaa = 1 and Km,aa = 1. number of substrate molecules processed by one enzyme per second, and the enzyme’s abundance in our model organism S. cerevisiae [63], we calculate the maximum reaction velocity using vmax = kcat [E] Vyeast (2.8) with [E] kcat Vyeast

enzyme concentration turn over number volume of a yeast cell

in mol/L in 1/s. in L

Chapter 3. Deterministic Modelling Approach

14

Chapter 3

A Deterministic Approach to Modelling Protein Translation

Modelling protein translation has been the focus of two opposing strategies over the past decades. Beside stochastic approaches (see chapter 4), it is also possible to use deterministic models. Based on such a deterministic description of protein translation in E.coli, we present an extended mathematical framework which explicitly accounts for the tRNA charging process. We examine the dependence of aa–tRNA availability and protein production on the supply and demand for these molecules. Later sections employ numerical simulations to study different artificial mRNA configurations and improve the model with respect to the results. This helps us to understand the role of codon usage in the dynamics of translation.

3.1. Deterministic Model Neglecting Steric Hindrance For now we neglect ribosomal blocking to derive a simple mathematical description. There are two situations where this approach is reasonable: an initiation controlled system, where there is a low overall ribosome density; or when there is a homogeneous translation rate for all codons.

3.1.1. Mathematical Model Based on a model for prokaryotic protein translation developed by Kremling [13] and with respect to the simplified scheme in figure 3.1, we first set up a network of reaction equations. These equations represent the three sub–processes of initiation, elongation and termination. In addition to the original model, they also account for tRNA charging and discharging. As shown in figure 3.1, ribosomes located in the cytosol initially bind to the 5’ leading region of an mRNA with rate kbi and form a ribosome–mRNA complex (RRNA). Scanning for the start codon, they hop onto the open reading frame (ORF) with rate kα , which yields a ribosome occupying the first codon1 (X1 ). By using cognate aa–tRNAs (aatRNACi ), the ribosome then elongates from codon i to the next codon i+1. Eventually, it returns to the cytosol while releasing a protein (PR), where translation of the stop codon and binding of release factors is neglected (see section 2.2.4). In this 1

excluding the start codon, since its translation is part of the initiation process


15

Figure 3.1.: Simplified scheme of protein translation employed to derive a deterministic translation model. context, the elongation rate ktr resembles rtr,tRNA (section 2.2.4). We choose parameters and initial conditions in a way to ensure that Xi can be interpreted as a mean ribosome density on the respective codon i. Please note that in the here presented case, initiation consists of two separate sub– processes. Ribosomes first form a ribosome–RNA complex and then hop onto the ORF. Thus, we are not only following the original model of Kremling, but allow its extension with different initiation regimes. For instance, permitting an occupation of the 5’ leading region with several ribosomes at once could account for batch–wise initiation. Tertiary structures in this region occasionally prevent a single ribosome to pass and cause this phenomenon. Hence, later studies could employ the equation system to derive a more realistic model of initiation. The reaction equations we use are kbi

− * R + RNA − ) − − RRNA k

α RRNA −→ X1 + RNA

k

tr X1 + aatRNAC1 −→ tRNAC1 + X2 .. .

k

tr Xn−1 + aatRNACn−1 −→ tRNACn−1 + Xn

k

tr Xn + aatRNACn −→ R + PR + tRNACn

vj

tRNAj + aa j − → aatRNAj vj = vmax,j with

tRNAj tRNAj + Km,tRNAj

!

aa j aa j + Km,aaj

!

(3.1)


aatRNACi kα kbi ktr PR R RRN A tRNACi Xi

16

abundance of the cognate charged tRNA of codon i initiation rate binding rate translation rate abundance of proteins abundance of ribosomes abundance of ribosome–mRNA complexes abundance of the cognate uncharged tRNA of codon i ribosome on codon i

in in in in in in in in in

molecules 1/s 1/(molecules · s) 1/(molecules · s) molecules molecules molecules molecules molecules.

Utilising this network of reaction equations, one can now obtain a set of ordinary differential equations (ODEs). Due to the conservation laws RNA + RRNA = RNAt = const and tRNAj + aatRNAj = tRNAj,t = const, equations (3.2) and (3.6) simplify, giving R˙ = ktr · aatRNACn · Xn − kbi · R · RNA ˙ = kα · RRNA − kbi · R · RNA RNA ˙ ˙ RRNA = −RNA

(3.2)

˙ = ktr · aatRNAC · Xn PR n ˙ X1 = kα · RRNA − ktr · aatRNAC1 · X1

(3.3)

X˙ 2 = ktr (aatRNAC1 · X1 − aatRNAC2 · X2 ) .. . X˙ n = ktr aatRNACn−1 · Xn−1 − aatRNACn · Xn ˙ j = vmax,j aatRNA


− ktr · aatRNAj ·

X

!

Xi

aa j aa j + Km,aaj

(3.4) !

(3.5)

CaatRNAj

˙ j = −aatRNA ˙ j tRNA

(3.6)

with CaatRNAj

codons requiring aatRNAj .

Most mathematical models for protein translation assume a constant pool of aa– tRNAs. We expect this to be a particularly poor assumption under starvation conditions and thus lead to the possibility of variations in the tRNA charging level. Derivation of an equation for the steady–state amount of charged tRNAs based on the presented ODE–system elucidates whether our working hypothesis is justified.


17

3.1.2. Analytic Solution for the steady–state tRNA charging level Assuming initiation, elongation and charging to be fast compared to the overall time scale of protein translation yields steady–state conditions: ˙ ˙ aatRNA j = −tRNAj = 0 ˙ = −RRNA ˙ RNA =0

(3.7) (3.8)

X˙ i = 0 .

(3.9)

In combination with (3.3) this results in X1 =

RRNA kα · . ktr aatRNAC1

(3.10)

Utilising (3.5) and (3.7), one then obtains vmax,j


!

aa j aa j + Km,aaj

!

= ktr · aatRNAj ·

X

Xi .

(3.11)

CaatRNAj

Solving this equation requires two simplifications. First, we linearise the Michaelis– Menten equation (3.1) in its substrate limitation term accounting for uncharged tRNAs, which yields ! vmax,j aa j vj = tRNA . Km,tRNAj aa j + Km,aaj This assumption holds for tRNA Km,tRNAj . We then assume that only one codon on the mRNA is decoded by the aatRNAj . Thus, the ribosome density on this codon is XCaatRNAj . These simplifications let us rewrite (3.11) to aa j vmax,j tRNA Km,tRNAj aa j + Km,aaj

!

= ktr · aatRNAj · XCaatRNAj .

Considering the conservation equation tRNAj,t = tRNAj + aatRNAj yields −1



Km,tRNAj ktr



aatRNAj = tRNAj,t  1 +

vmax,j

aa j aa j +Km,aaj

 XCaatRNA  j

.

(3.12)

To evaluate equation (3.12), we must obtain an equation for the ribosome density XCaatRNAj = f (aatRNAj ). Assuming the codon Ci to be decoded by the aatRNAj , let us use aatRNAj = aatRNACi and XCaatRNAj = Xi . We then employ (3.4) at n = i and (3.9) to obtain Xi =

aatRNACi−1 Xi−1 . aatRNACi


18

Back–calculating the ribosome density to the 5’–end of the mRNA–strand yields Xi =

aatRNACi−1 aatRNACi−2 aatRNACi−2 · Xi−2 = Xi−2 aatRNACi aatRNACi−1 aatRNACi

Xi =

kα RRNA. ktr · aatRNACi

and with (3.10)

Using the original notation results in XCaatRNAj =

kα RRNA. ktr aatRNAj

(3.13)

By considering (3.2), (3.8) and the conservation law RNAt = RNA + RRNA, as well as assuming that the number of ribosomes exceeds the number of free mRNA binding sites (R RNA), resulting in R ≈ R0 , we obtain the following equation RRNA =

kbi R0 (RNAt − RRNA) , kα

which resolves to RRNA = RNAt

kα 1+ kbi R0

!−1

.

With (3.13) this yields XCaatRNAj

kα RNAt kα = 1+ ktr aatRNAj kbi R0

!−1

.

(3.14)

Finally we combine (3.12) and (3.14) to derive 

Km,tRNAj ktr



aatRNAj = tRNAj,t  1 +

vmax,j

aa j aa j +Km,aaj

kα RNAt kα 1+ ktr aatRNAj kbi R0

!−1

−1   

,

which simplifies to an equation for the steady–state tRNA charging level kα Km,tRNAj RNAt

aatRNAj = tRNAj,t −

vmax,j

aa j aa j +Km,aaj

1+

kα kbi R0

.

(3.15)

Please note that Equation (3.15) is not defined for aa j = 0 or vmax,j = 0, assuming kbi , kα ≥ 0 and R0 > 0. In these cases, employing (3.5) yields the solution 

aatRNAj (t) = aatRNAj (t = 0) exp  −ktr · t

 X CaatRNAj

Xi  

(3.16)


19

and with ktr > 0 and t → ∞ this results in aatRNAj (t) =

 aatRNA 0

j

(t = 0) for Xi = 0 for Xi > 0.

Contrary to a biological system, the solution of (3.15) reaches negative charging levels. This results from linearising the term that accounts for uncharged tRNAs as a substrate in the Michaelis–Menten equation. In fact, when high concentrations of uncharged tRNAs occur (tRNA Km,tRNAj ), the conservation law tRNAj,t = tRNAj + aatRNAj is violated by (3.15). To compensate this violation, we artificially introduce the constraint aatRNAj ≥ 0 and set aatRNAj = 0 in case negative values occur. For the transition aatRN Aj = 0 in dependence of aa j one obtains the following relation (−1)

 

kα =  

Km,tRNAj RNAt

tRNAj vmax,j

aa j aa j +Km,aaj

−

1   kbi R0 

.

(3.17)

Assessment of the solution With the presented solution, we focus on analysing possible variations in the relative charging level of a tRNA species, i.e., the fraction of charged to total tRNAs. In this context, the free amino acid concentration, which is determined by the nutritional conditions a cell experiences, affects aa-tRNA supply by influencing the recharging capacity. Furthermore, the initiation rate emerges as a second important parameter. Dependent on the concentration of translation initiation factors, several regulatory mechanisms are known to influence kα [5, 6]. Hence the initiation rate, controlling the number of elongating ribosomes and therefore the demand for charged tRNAs, is variable and treated as such in our analysis. For these reasons we focus on kα and [aa] when evaluating (3.15) with the parameter set in table A.1 in order to understand the qualitative behaviour of aa–tRNA availability (figure 3.2). Having a direct influence on the aminoacylation reaction velocity, we expect amino acid starvation to be a major cause of impaired charging levels. Accordingly, the relative charging level in our model decreases dramatically with the reduction of available amino acids. The same effect is observed with an increasing initiation rate kα . The more ribosomes enter the system the higher the demand for transfer RNAs, causing the steady– state solution to tend toward a less charged state. With the constraint aatRNA ≥ 0 introduced in the previous section, the solution reaches a plateau of aatRNA = 0 for large kα values or low amino acid concentrations. Equation (3.17) gives the parameter pairs kα and [aa] for passing into this plateau. Hence, the derived solution not only predicts a non-constant pool of aa–tRNAs, but its complete depletion in specific parameter regimes. Evaluation of equations (3.5) and (3.16) for the limits [aa] = 0 and kα = 0 yielding 0 and tRNAt , respectively, is in agreement with the presented analytic solution (figure 3.2) and reflects data published by Elf et al. [41]. To validate our working hypothesis that variations in tRNA charging level significantly affect protein translation, we proceed with studying a more realistic mathematical description and the protein production of mRNAs.


20

Figure 3.2.: Analytic solution for the steady–state tRNA charging level in response to changes in initiation rate kα and amino acid concentration [aa] calculated with (3.15) and the parameter set in table A.1. The employed equation assumes a deterministic model neglecting steric hindrance of ribosomes. An mRNA strand of arbitrary length contains one codon decoded by the presented tRNA species which carries the amino acid affected by starvation.

3.1.3. Numerical Simulations of Artificial Sequences The presented analytic solution assumes a linearised Michaelis–Menten equation and only holds for tRNA Km,tRNAj . Thus, we next study numerical simulations that employ the exact equation and a more realistic parameter set for tRNA charging (table A.3). Such simulations in the current and following chapters use artificial mRNA configurations. These artificial strands consist of codons decoded by an abundant aa–tRNA (fast codons), whose charging level is constant and not affected by amino acid starvation. A second codon type decoded by a rare tRNA species (slow codons) interrupts these fast sites. Amino acid starvation influences the charging level of this rare tRNA. Therefore, it varies in time and parameter space. Since protein translation resembles a one dimensional traffic problem, the slowest codon on an mRNA, i.e., the codon corresponding to the rarest tRNA species, limits the overall ribosome flow. Hence, such configurations yield a first approximation of realistic sequences and highlight the principle dynamics in protein translation. Following the system used to derive (3.15), we here consider an mRNA of 100 codons length with a single slow codon. Besides the tRNA charging level, numerical simulations ˙ yield the mRNA’s ribosome current which refers to its protein production rate kpr = PR (figure 3.3). tRNA Charging Level Comparing the analytic prediction in figure 3.2 to its numerical counterpart (figure 3.3a), one observes the same qualitative behaviour, although employ-


(a)

21

(b)

Figure 3.3.: Steady–state tRNA charging level in (a) and protein production rate in (b) for the deterministic model neglecting steric hindrance of ribosomes. An mRNA strand of 100 codons length contains a single slow codon, which is decoded by the presented tRNA species that carries an amino acid affected by starvation. ing a different parameter set alters the quantitative result. Again, the limits [aa] = 0 and kα = 0 result in a relative charging level of 0 and 1, respectively. Together with a larger aatRNA = 0 plateau for high initiation rates and low amino acid concentrations, a sharp transition between the two extremes arises. Since no negative charging levels occur, the region where aatRNA = 0 has not to be introduced artificially. Thus, numerical simulations of the presented deterministic model not only confirm the analytic solution, but again predict a non-constant pool of tRNAs. The part of our model concerned with tRNA charging resembles previous modelling approaches [41]. More importantly, it is validated by biological experiments conducted by Zaborske et al. [15], who demonstrate a decrease of charging due to amino acid starvation (figure 3.4). However, in contrast to in vivo experiments, figure 3.3a shows a depletion of aa–tRNAs. We conclude that the availability of amino acids affects tRNA charging levels in vivo. In contrast to existing translation models, our description accounts for a non–constant pool of aa–tRNAs but under estimates charging levels. Based on this knowledge the question arises to which extent protein production is affected by these facts. Protein Production Rate Figure 3.3b depicts how the protein production varies with initiation and amino acid concentration. Interestingly kpr shows a first–order phase transition in response to an increase in kα (figure 3.5a), i.e., a non–continuous change in the kpr = f (kα ) profiles. This transition is confirmed by a plot of the derivative (data not shown) and further discussed in section 3.2. In contrast, kpr = f ([aa]) profiles follow a distinct pattern presented in figure 3.5b. For high initiation rates, protein production depends smoothly on changes in amino acid concentrations. Together with a very low charging level this suggest that the ribosome flow is limited by the availability of aa–tRNAs, i.e., by the translation rate of the slow


22

Figure 3.4.: Relative charging level of different tRNA species in response to amino acid starvation measured by Zaborske et al. [15]. The yeast strain WY795 was cultured in SC medium depleted for leucine for 15 min and tRNA charging was measured using a microarray method. Relative levels of tRNA charging are presented as the charging ratio of each tRNA prepared from the strain cultured in SC medium devoid of leucine for 15 minutes compared to cells grown in SC medium containing all amino acids codon. Hence, an unsaturated state is present in which a higher tRNA charging capacity, caused by an increase in [aa], leads to a direct increase in kpr . However, in case of low

0.2

−1

[aa] = 0.012 mM [aa] = 0.037 mM

0.2

kα = 0.12 s

−1

kpr in protein/s

kpr in protein/s

kα = 0.37 s 0.15

0.1

0.05

0 0

0.15

0.1

0.05

0.1

0.2

0.3 −1

kα in s

(a)

0.4

0.5

0 0

0.01

0.02

0.03

0.04

0.05

[aa] in mmol/L

(b)

Figure 3.5.: A first–order phase transition in the steady–state protein production of the deterministic model neglecting steric hindrance is caused by (a) elongation or (b) initiation becoming the limiting factor. Profiles of kpr in figure 3.3b against (a) initiation rate and (b) amino acid concentration are shown.


23

initiation rates the system reaches a saturated state. A high charging level and a constant kpr for increasing [aa], following the first–order phase transition, characterise this state. It is caused by initiation becoming the limiting factor for ribosome flow. Thus, tRNA charging exceeds the demand for aa–tRNAs and ribosomes elongate with almost their maximum rate ktr · tRNAt on the slow codon. The transition between saturated and unsaturated state is given by kα = ktr · aatRNA and only occurs for kα ≤ ktr · tRNAt . It marks the balance between demand and supply of aa–tRNAs at which neither initiation nor elongation exclusively limit protein production. Hence, we find the presented deterministic model to show two distinct regions of protein production dependence on amino acid concentration (figure 3.5b). Additionally, the already known phase transition in response to changing initiation rates (figure 3.5a and section 3.2) is present. Ribosome Density To gain a more comprehensive picture of the model, figure 3.6 shows the ribosome density on an example mRNA. Here an increased influx of ribosomes 4

x 10

i = 25 i = 50 i = 75

3

2

1

0 0

0.1

i = 25 i = 50 i = 75

4

ribosome density

ribosome density

4

0.2

0.3 −1

kα in s

(a)

0.4

0.5

3

2

1

0 0

0.01

0.02

0.03

0.04

0.05

[aa] in mmol/L

(b)

Figure 3.6.: Steady–state ribosome density for the deterministic model neglecting steric hindrance. An mRNA of 100 codons length contains a single slow codon at position i = 50, which is decoded by a tRNA species that carries an amino acid affected by starvation. Density on codon i is depicted at [aa] = 0.05 mmol/L in (a) and kα = 0.5 s−1 in (b). Plots for i = 25 and i = 75 coincide. (figure 3.6a) results in a higher density on all codons, most pronouncedly on the slow side. Contrary to biological reality, up to four ribosomes occupy this codon simultaneously in the steady–state, demonstrating the draw back of a model neglecting ribosomal blocking. The simulation also shows that the slow site’s ribosome density is inversely proportional to the availability of charged tRNAs (figure 3.6b). Hence, the mathematical description reflects the assumption that tRNA abundance influences translation rates and therefore a ribosome’s waiting time on a given codon. The direct effect of a reduced aa–tRNA amount (caused by an increase in kα or decrease in [aa]) in this case is an increased ribosome density especially on the slow codon.


24

3.2. Phase Transition in Protein Production Rates In figure 3.5a protein production rates show a discontinuity in their first derivative with respect to kα , i.e., a first–order phase transition. A recent article that employs the later discussed TASEP model describes a similar observation for the influence of the initiation probability α ([64] and figure 3.7). Romano et al. therein divide the mRNAs of cellular

ribosome current

Type I Type II

α

Figure 3.7.: Protein types according to the dependence of ribosome current on initiation probability α as presented in literature employing a TASEP model [64]. proteins into two classes dependent on the slowest codon’s position. With the slowest codon positioned at the start of an mRNA, the ribosomal current is said to display type II behaviour as α is varied, whereas if the slowest codon is at a position i > 1, the behaviour is described as type I (see figure 3.8b). These classes match the biological function, namely as ribosomal (type II) and non–ribosomal (type I) proteins.

(a)

(b)

Figure 3.8.: Different mRNA configurations correspond to distinct protein types. Fast codons are depicted as rectangles without labelling and the leftmost site of the slowest codon type is labelled with A. Type I mRNAs in (a) correspond to non–ribosomal proteins and type II mRNAs in (b) to ribosomal proteins [64]. Romano et al. also derive equations describing the ribosome flow J along the found mRNA types, which corresponds to their protein production rate. This flow depends on the initiation probability α which is equivalent to our parameter kα . Although we discuss stochastic TASEP models in chapter 4 in more detail, introduction of these equations at this point gives an impression of the difference between the two types. Assuming


25

M slow sites (codons) with hopping probabilities {q1 , . . . , qm }, i.e., translation rates, to be distributed over an mRNA, one can distinguish two different cases. These cases are solved in their mean field approximation in literature [64]. Case 1 The first slow site q1 is at position i > 1. Assuming that the slow sites are far apart and separated by fast codons the following solution is obtained: J=

 α qmin

if α < αc otherwise

(3.18)

with α αc = qmin J qmin = min {q1 , . . . , qm }

initiation probability critical initiation probability ribosome current minimum hopping probability.

Case 2 Considering the same configuration as before but now the first slow site q1 is at position i = 1, one can distinguish two sub–cases: (i) if q1 = qmin , then αq1 for 0 ≤ α ≤ 1 (3.19) J= α + q1 − αq1 and (ii) if, in contrast, q1 > qmin , then  

αq1 α+q1 −αq1

J = qmin where αc =

if α < αc and otherwise

(3.20)

q1 qmin . q1 − qmin + q1 qmin

Considering these equations, one finds two main results. Not only does the type an mRNA belongs to depend on the slowest codon’s position, but the respective translation rate qmin determines the ribosome flow J and therefore the protein production. In the context of aa–tRNA abundance controlling a codon’s translation rate, this fact becomes even more significant. Contrary to literature [64], the effective translation rate of a specific codon in our model, i.e., the hopping probability in a TASEP, is no longer constant, but influenced by the charging level of its respective tRNA species. Hence, the position of the slowest codon can change dynamically and the critical kα value causing a phase transition depends directly on the availability of amino acids (figure 3.5a). It is for these reasons that an extension of the existing model by considering a more realistic translation process and accounting for the charging of tRNAs can lead to a new insight into protein translation. In contrast to figure 3.7, simulations carried out with our deterministic model neglecting steric hindrance and positioning the slowest site on the first codon (data not shown) resemble results in figure 3.3. They only provide evidence for type I proteins. Romano


26

et al. describe queueing of ribosomes as the main reason for an occurrence of two distinct protein types. Additionally, our approach yields unrealistic high ribosome densities (figure 3.6). Hence, we next investigate an improved deterministic model accounting for steric hindrance.

3.3. Deterministic Model Accounting for Steric Hindrance A model which neglects exclusive interactions of ribosomes does not account for non– uniform codon translation rates or high ribosome densities on an mRNA. Particularly, under amino acid starvation, one expects the translation of specific codons to slow down. Hence, ribosomes are likely to queue in front of slow codons, densities increase (figure 3.6) and steric hindrance becomes important. Implementation of a more realistic translation process including the blocking of individual ribosomes by others occupying adjacent codons accounts for such a behaviour.

3.3.1. Mathematical Model We introduce steric hindrance by reducing the translation rate of a given codon i dependent on the occupation density of its consecutive codon i + 1. The employed term has to fulfil two criteria: translation of a codon i should not occur in case a ribosome occupies the consecutive codon (Xi+1 = 1); and it should reach ktr given the next codon is vacant (Xi+1 = 0). For the sake of simplicity, a linear dependence is used. However, other implementation such as quadratic or Michaelis–Menten like approaches are reasonable as well. A linear term of the form ktr (1 − Xi+1 ), which yields the required behaviour, results in kbi

−− * R + RNA ) − − RRNA kα (1−X1 )

RRNA −−−−−→ X1 + RNA ktr (1−X2 )

X1 + aatRNAC1 −−−−−−→ tRNAC1 + X2 .. . ktr (1−Xn )

Xn−1 + aatRNACn−1 −−−−−−→ tRNACn−1 + Xn k

tr Xn + aatRNACn −→ R + PR + tRNACn


27

and the ODE systems changes to R˙ = ktr · aatRNACn · Xn − kbi · R · RNA ˙ = kα · RRNA (1 − X1 ) − kbi · R · RNA RNA ˙ ˙ RRNA = −RNA ˙ = ktr · aatRNAC · Xn PR n ˙ X1 = kα · RRNA · (1 − X1 ) − ktr · aatRNAC1 · X1 · (1 − X2 ) X˙ 2 = ktr (aatRNAC1 · X1 · (1 − X2 ) − aatRNAC2 · X2 · (1 − X3 )) .. .

X˙ n = ktr aatRNACn−1 · Xn−1 · (1 − Xn ) − aatRNACn · Xn ˙ j = vmax,j aatRNA



X

!

aa j aa j + Km,aaj

Xi (1 − Xi+1 )

!

for CaatRNAj 6= Cn

(3.21)

CaatRNAj

˙ j = vmax,j aatRNA



X

!

Xi

aa j aa j + Km,aaj

!

for CaatRNAj = Cn

CaatRNAj

˙ j = −aatRNA ˙ j. tRNA Due to the inter–linking of ribosome densities Xi with densities on their preceding codons Xi−1 , deduction of an analytical solution that describes the tRNA charging level is complicated. Hence, we employ numerical simulations to analyse this ODE–system. Note that by introducing exclusively interacting ribosomes, the equation system above resembles the later presented TASEP process in its mean field approximation. Neglecting stochastic effects, both descriptions should yield similar results. However, a low abundance of aa–tRNAs increases the importance of stochasticity, presumably leading to different predictions.

3.3.2. Numerical Simulations of Artificial Sequences In order to understand the effect of codon position and to resemble the two distinct protein types presented in literature [64], we study two different mRNA configurations. They contain the slowest codon, i.e., the codon decoded by the rare tRNA species, which is affected by amino acid starvation, at position i = 1 and i > 1, respectively. A Rare tRNA Decodes a Codon at Position i > 1 For an mRNA of 100 codons length and the slowest codon at position i = 50, we expect a type I protein to emerge. The system should yield similar results to the simulations presented in section 3.1.3, given the parameter set in table A.3 is used. Figure 3.9b


28

(a)

(b)

Figure 3.9.: Steady–state tRNA charging level in (a) and protein production rate in (b) for the deterministic model accounting for steric hindrance of ribosomes. An mRNA strand of 100 codons length contains a single slow codon at position i = 50, which is decoded by the presented tRNA species that carries an amino acid affected by starvation. shows the predicted behaviour, although kpr is generally lower caused by a decrease in ribosome flow due to steric hindrance. Again the first–order phase transition occurs, but in comparison to figure 3.3b at a lower critical kα (figure 3.11a). The tRNA charging level in figure 3.9a shows a characteristic kink at parameter pairs corresponding to the phase transition. As described before, this kink marks the transition between an initiation and a translation controlled system. This time however, no aatRNA = 0 plateau emerges, but a smooth dependence of charging level on [aa] in the presented parameter regime. Hence, a model accounting for steric hindrance of ribosomes yields results in good agreement with the experimental data in figure 3.4. It does not predict total aa–tRNA depletion unless [aa] = 0. Simulation results on ribosome densities explain this observation. Ribosome Density Without accounting for steric hindrance, the slow codon’s ribosome density X50 increases limitless (figure 3.6). This causes the demand for aa–tRNAs (De aatRNAj ) given by (3.5) as De aatRNAj = ktr · aatRNAj

X

Xi

CaatRNAj

to increase as long as aatRNAj 6= 0. By introducing steric hindrance, the ribosome density is limited to Xi ≤ 1 and thus ribosomes gradually slow down and queue in front of the slow site (figure 3.10). This results in an upper bound for the demand. Restricting the number of ribosomes requiring a specific aa–tRNA prevents its complete discharging with the here employed parameter set unless [aa] = 0. In this context, a queue starts to form as soon as the translation rate of the slow site, determined by the tRNA charging level and therefore by [aa], falls below kα (figure 3.10a). Note that


(a)

29

(b)

Figure 3.10.: Steady-state ribosome density for the deterministic model accounting for steric hindrance. An mRNA of 100 codons length contains a single slow codon at position i = 50, which is decoded by the tRNA species that carries an amino acid affected by starvation. Ribosome density against amino acid concentration for kα = 0.12 s−1 in (a) and kα = 0.37 s−1 in (b). the start of queue formation corresponds to the phase transition in figure 3.9. More precisely, for high initiation rates or low amino acid concentrations the system passes into the unsaturated (translation controlled) state. Not only does a low charging level and a smooth dependence of protein production on [aa] characterise this state, but queue formation (figure 3.10b) occurs. In contrast, the initiation controlled system, i.e., the saturated state, yields a relatively high charging level and no queueing. Phase Transition To support the drawn conclusions and to allow a comparison to the model neglecting steric hindrance, figure 3.11 presents the phase transition in protein production rates. Just like in figure 3.5, one observes the same qualitative behaviour, but again a generally lower kpr due to ribosomal blocking. The two distinct regions of protein production rate dependence on [aa] persist and a more pronounced region of smooth dependence emerges. A Rare tRNA Decodes a Codon at Position i = 1 We next study an mRNA configuration with the slowest codon in first2 position. Numerical simulations yield the tRNA charging level and protein production rate presented in figure 3.12. In comparison to the system with a slow codon positioned at i > 1, one observes a reduced kpr most pronouncedly for low initiation rates. More importantly, this mRNA configuration results in a type II protein as described in section 3.2. It shows no first–order phase transition and yields a smooth dependence of protein production on initiation rate and amino acid concentration (see also figure 3.13). Our deterministic model accounting for steric hindrance thus resembles the two distinct mRNA types 2

excluding the start codon, since its translation is part of the initiation process


0.1

0.1 0.08 accounting for steric hindrance neglecting steric hindrance

0.06 0.04 0.02 0 0

kpr in protein/s

kpr in protein/s

0.12

30

0.08 k = 0.12 s−1 α

0.06

k = 0.37 s−1 α

0.04 0.02

0.1

0.2

0.3 −1

kα in s

(a)

0.4

0.5

0 0

0.01

0.02

0.03

0.04

0.05

[aa] in mmol/L

(b)

Figure 3.11.: Phase transition in protein production for the deterministic model accounting for steric hindrance and a slow codon at i = 50. (a) Profiles of protein production against initiation rate at [aa] = 0.012 mmol/L in figure 3.9b compared to the model neglecting steric hindrance in figure 3.3b. (b) Profiles of protein production rate in figure 3.9b against amino acid concentration. described in literature [64]. Phase Transition To support this observation, figure 3.13 shows the profiles of kpr against initiation rate and amino acid concentration. The absence of a first–order phase transition in figure 3.13a indicates that the system is not exclusively elongation controlled. In fact, due to two consecutive slow processes, namely initiation and elongation

(a)

(b)

Figure 3.12.: Steady–state tRNA charging level in (a) and protein production rate in (b) for the deterministic model accounting for steric hindrance of ribosomes. An mRNA strand of 100 codons length contains a single slow codon at position i = 1, which is decoded by the presented tRNA species that carries an amino acid affected by starvation.


0.1

[aa] = 0.012 mM [aa] = 0.037 mM

31

0.1 islow = 1

kpr in protein/s

kpr [protein/s]

0.08

0.06

0.04

0.08

islow = 50

0.06 0.04

0.02 0.02 0 0

0.1

0.2

0.3 −1

0.4

0.5

kα in s

0 0

0.01

0.02

0.03

0.04

0.05

[aa] in mmol/L

(a)

(b)

Figure 3.13.: Phase transition in protein production for the deterministic model accounting for steric hindrance and a slow codon at i = 1. Profiles of protein production rate, (a) in figure 3.12b against initiation rate and (b) in figure 3.9b and figure 3.12b against amino acid concentration for kα = 0.12 s−1 . over the slow site, neither of the two mechanisms solely limits translation. Thus, variations of the correspondent parameters kα and [aa] affect protein production smoothly (figure 3.13b). A slow first codon in combination with steric hindrance prevents large numbers of ribosomes from initiating. This also prohibits queue formation which literature [64] describes as the reason for a first–order phase transition. As a consequence, kpr maintains its smooth dependence even outside the presented parameter range (data not shown). Evaluating the occurrence of a first–order phase transition in figure 3.11 and 3.13, we can determine whether a system is controlled by initiation or elongation. A change of protein production rate in response to a varying amino acid concentration (figure 3.11b and 3.13b) characterises an elongation controlled system. Accordingly, if kpr against kα (figure 3.11a and 3.13a) does not remain constant, initiation control is present. Table 3.1 summarises the findings. Table 3.1.: System behaviour of type I and II mRNAs in response to changing initiation rate and amino acid concentration. Type I mRNAs contain the codon affected by amino acid starvation at position i > 1 and type II mRNAs at i = 1. The system is initiation controlled if changes in kα affect the protein production rate. Accordingly, elongation control leads to a dependence of kpr on [aa]. Type I [aa] low kα low kα high

elongation & initiation elongation

[aa] high initiation elongation

Type II [aa] low [aa] high elongation & initiation elongation & initiation


32

3.3.3. Relative Protein Production To assess the possible biological reason for the occurrence of two different mRNA types, we next study the relative protein production rate, i.e., the protein production rate of an mRNA normalised to its value at a high amino acid concentration (optimal growth conditions). Focusing on the data already presented in figure 3.13b, the two mRNA types result in two distinct responses to a reduction of available amino acids (figure 3.14). With the codon representing an amino acid affected by starvation in first position, the

1 islow = 1 islow = 50

relative k

pr

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

relative [aa]

Figure 3.14.: Relative protein production in response to amino acid starvation. An mRNA of 100 codons length contains a single slow codon, which is positioned as indicated and decoded by a tRNA species that carries an amino acid affected by starvation. Results are normalised to their values at a high amino acid concentration, i.e., at optimal growth conditions. Insert depicts magnification of phase transition. respective mRNA maintains a higher relative current for lower amino acid concentrations. Furthermore, kpr decreases smoothly over the entire presented parameter range. This type is thus more robust to changing nutritional conditions when considering lower values of [aa], but more sensitive at optimal growth conditions (figure 3.15). In contrast, kpr for the type I mRNA (islow = 50) is not sensitive to changes at high [aa], but less robust at low amino acid concentrations. The transition between the two states depends on the initiation rate and occurs at a critical charging level aatRNA = kα/ktr of the tRNA affected by starvation. Figure B.2 in the appendix depicts examples for the sensitivity of deterministic kpr predictions in response to other simulation parameters. In this regard vmax , Km,aa and ktr emerge as the parameters affecting protein production rates the most. In contrast, the initiation rate kα exerts no influence in parameter ranges corresponding to the unsaturated, translation controlled state. This fact changes for initiation controlled systems.


normalised sensitivity

1

33

islow = 1 islow = 50

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

relative [aa]

Figure 3.15.: Normalised sensitivity of protein production rate to changes in amino acid concentration. Normalised sensitivity of kpr to [aa] calculated with equation (2.5) for the deterministic model accounting for steric hindrance. An mRNA of 100 codons length contains a single slow codon, which is positioned as indicated and decoded by a tRNA species that carries an amino acid affected by starvation. Insert depicts magnification of phase transition as shown in figure 3.14.

3.3.4. Influence of the Charging Enzyme Activity Knowing that our model is sensitive to the charging enzyme activity, we study how its variation affects protein production. Numerical simulations employing a constant amino acid concentration and altering vmax result in the two distinct mRNA types already presented (figure B.1). Implementation of the charging process via a Michaelis–Menten like equation (3.21) explains this fact. Decreasing either [aa] or vmax results in a decreased charging capacity. Hence, our model predicts that mutant forms of a tRNA species or a aminoacyl–tRNA synthetase that impair the protein substrate fit result in the same translational dynamics than amino acid starvation. We expect a reduction of tRNA abundance in tRNA gene deletion mutants or due to a reduced stability of these molecules to yield a similar effect.


34

3.4. Effects of Two Codon Species and Competing Ribosomes To gain a comprehensive picture of translation and further approach realistic mRNA sequences, we highlight two aspects in this chapter, namely the competition for rare resources and the influence of multiple codon species. Relating to the latter, the previous sections focused on mRNAs consisting of fast codons interrupted by one slow side. The slow codon’s cognate tRNA carried an amino acid subject to starvation. This scenario corresponds to starvation for an amino acid delivered by one of the six rarest tRNA species in yeast (see table 2.1). In contrast, we now present the consequence of decreasing charging levels of an abundant tRNA by introducing two different codon species in our simulations. The second part of this section then elucidates the competition for limited cellular resources and its effect on translation by means of studying several simultaneously translated strands. The presented numerical simulations again employ mRNAs consisting of fast codons which are decoded by an abundant tRNA with a constant charging level. This time however, two distinct tRNA species with variable charging levels are used to translate two different codon types that interrupt the fast sites (figure 3.16). The more abun-

(a)

(b)

Figure 3.16.: Different mRNA configurations employed during numerical simulations. An mRNA consists of codons which are decoded by an abundant tRNA with constant charging level (rectangles without numbering). Two different codons with variable translation rates interrupt these fast sites. Codons of type A are decoded by an abundant tRNA and represent an amino acid subject to starvation, whereas type B codons are the slowest codons under optimal growth conditions. dant tRNA delivers an amino acid subject to starvation and decodes codons of type A. In contrast, type B codons are decoded by a rare tRNA carrying an amino acid with constant concentration. We consider such a configuration, since the ribosome flow along an mRNA is limited by the slowest codon [65]. Thus, under optimal growth conditions, codons decoded by the rarest tRNA species determine the protein production rate. Starvation decreases charging levels and translation rates, changing which codon species limits protein production.

3.4.1. Two Codon Species on a Single mRNA Employing the deterministic model which accounts for steric hindrance to study the configuration presented in figure 3.16b yields the relative abundance of both aa–tRNAs (figure 3.17). As seen before, the charging level of a tRNA species delivering the amino


(a)

35

(b)

Figure 3.17.: Steady–state tRNA charging levels for an mRNA strand containing two codon species (figure 3.16b). An mRNA of 100 codons length contains a slow codon (type B), which is decoded by a tRNA that does not carry an amino acid subject to starvation (depicted in (b)) at position i = 25. A second, faster codon (type A) in position i = 75 represents the amino acid affected by starvation. Its cognate aa–tRNA level is shown in (a). amino subject to starvation (figure 3.17a) decreases with increasing kα or a reduced availability of amino acids. The second tRNA species in contrast, shows an increased charging level at lower values of [aa] and kα (figure 3.17b). The abundance of the respective aa–tRNA solely depends on the demand for it given by translation, since this tRNA delivers an amino acid not affected by starvation. Hence, a diminished ribosome flow (see protein production rate in figure 3.18b), caused by starvation and decreasing the demand for aa–tRNAs, leads to an increased charing level. In fact, in vivo experiments in figure 3.4 show such an increase for methionine tRNA’s and several other species. Figure 3.18b depicts the protein production rate in response to these changes of relative charging level. It resembles results previously presented for type I mRNAs (figure 3.9b), i.e., mRNAs containing a slow codon at positions i > 1, with one exception. Since the slowest codon on an mRNA limits the overall ribosome flow, protein production rates are constant (for high kα ’s) until the decreasing translation rate of the type A codon becomes the limiting factor. That leads to a plot similar to figure 3.9b but restricted above by a section plain kpr = ktr · aatRNArare with aatRNArare representing the rarest charged tRNA at high amino acid concentrations. Therefore, kpr now shows two phase transitions. One in response to changes of kα which represents the transition between an initiation and a translation controlled system and a second that marks the amino acid concentration at which both codon species are translated with the same rate. Below this concentration, kpr is affected by starvation since the aa–tRNA species that carries the respective amino acid is flow limiting. In contrast, figure 3.18a presents a system controlled by initiation at high amino acid concentrations and by elongation at low [aa] as well as high kα values. The transition between both states is marked by a sharp kink, which represents the critical initiation rate that changes with the amino acid concentration. However, this system does not show a clear phase transition for a change


(a)

36

(b)

Figure 3.18.: Steady–state protein production for an mRNA strand containing two slow codon species. An mRNA contains two different codon types, whose cognate tRNA’s charging levels are presented in figure 3.17. (a) configuration shown in figure 3.16a with a codon of type B at i = 1, (b) configuration shown in figure 3.16b with a codon of type B at i > 1. of the limiting codon species (see next section, equation (3.19) and (3.20) for details).

3.4.2. Simultaneously Translated mRNAs and Ribosomal Competition for Rare aa–tRNAs Section 2.4 briefly outlines the concept of competition for rare cellular resources in the context of protein translation. In summary, different ribosomes encountering the same codon species while translating mRNAs compete for the limited amount of available aa– tRNAs. To demonstrate the qualitative effect of ribosomal competition, we consider the mRNA configurations already presented in the previous section (figure 3.16). However, ribosomes now translate both mRNAs simultaneously and thus have to compete for limited resources. Since we again employ the parameter set in table A.3, aa–tRNA supply (influenced by vmax , Km,aa and Km,tRNA ) remains unchanged. However, two codons of each species now demand the mediator molecules, which causes a lower overall protein production when both strands are translated simultaneously (figure 3.19) in contrast to separate translation (figure 3.18). Hence, a higher demand for aa–tRNAs causes an impaired protein production. Increased Protein Production in Response to Changes in Initiation Rate Comparing the performance of an mRNA containing the slowest codon (type B) at position i > 1 with competition (figure 3.19b) and without (figure 3.18b) results in figure 3.20. Therein, we normalise kpr to a high initiation rate, which corresponds to translational activity. An increased relative kpr emerges for a high amino acid concentration near the phase transition in response to kα . In this parameter range, codons of type B (not representing


37

(a)

(b)

Figure 3.19.: Steady–state protein production for two mRNA strands containing two codon species. Two mRNAs are translated simultaneously and contain a slow codon (type B), which is decoded by a rare tRNA that carries an amino acid not subject to starvation and a second faster codon (type A) that represents the amino acid affected by starvation. (a) configuration shown in figure 3.16a with a type B codon at i = 1, (b) configuration shown in figure 3.16b with a type B codon at i > 1. the amino acid subject to starvation) are limiting. Their respective tRNA’s charging level, defining the limiting translation rate, depends on the demand for aa–tRNAs and is therefore determined by the ribosome flow along both mRNAs (figure 3.19a and 3.19b). As presented in section 3.2, ribosome currents of type I mRNAs (figure 3.19b) are not sensitive to changes in initiation rate above the critical value that causes the phase transition. Thus, type I mRNAs gain a comparative advantage over type II mRNAs for high [aa] values and kα ’s above the critical rate. More precisely, a decreasing kα affects

without competition with competition

1.2

relative k

pr

1 0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6 −1

0.8

1

kα in s

Figure 3.20.: Relative protein production in response to kα with and without competition. Protein production at [aa] = 0.022 mmol/L from figure 3.18b (without competition) and figure 3.19b (with competition) was normalised to a high initiation rate. An mRNA of 100 codons length that contains codons as presented in figure 3.16b is simulated.


38

the ribosome flow along type II mRNAs (figure 3.19a), yielding a decrease in aa–tRNA demand and thus an increased charging level of the limiting tRNA (see figure 3.17b for comparison). Since type I mRNAs are robust to these changes of kα in a specific parameter range, their ribosomes are able to use a higher availability of the limiting aa–tRNA species to enhance protein production. Increase Protein Production in Response to Changes in Amino Acid Concentration A second interesting phenomenon emerges in response to amino acid starvation. Figure 3.19a (depicting a type II mRNA) shows an increased protein production for decreasing amino acid concentrations in a specific kα regimes (figure 3.21). 0.07

[aa] = 0.015

[aa] = 0.016

[aa] = 0.018

kpr in proteins/s

0.06 0.05 0.04 0.03 0.02 0.01 0 0

0.2

0.4

0.6 −1

0.8

1

kα in s

Figure 3.21.: Protein production of a type II mRNA in response to competition and changes in amino acid concentration. Protein production against initiation rate in figure 3.19a for the indicated amino acid concentrations in mmol/L and an mRNA as presented in figure 3.16a that competes with a type I mRNA (figure 3.16a). Insert depicts magnification. Again this is caused by an increase in the tRNA charging level of the species not carrying an amino acid subject to starvation (see figure 3.17b for comparison). The mRNA configuration in figure 3.16a is more sensitive to changing translation rates of the respective type B codon in intermediate parameter ranges of [aa] (figure 3.15). Hence, if more aa–tRNA become available, the protein production increases for specific kα ’s. Therefore, type II mRNAs can gain an advantage over type I strands when resources become scarce. Furthermore, we find that a collectively used pool of aa–tRNAs leads to coupled ribosome flows along simultaneously translated mRNAs (figure 3.22b). Mathematical Considerations Given that the deterministic model qualitatively resembles a TASEP (see section 4.2.1), equations (3.18) to (3.20) provide a mathematical


1 type I type II

1

39

type I type II

pr

relative k

relative k

pr

0.8

0.6

0.4

0.2

0 0

0.8 0.6 0.4 0.2

0.2

0.4

0.6

0.8

1

0 0

0.2

0.4

0.6

relative [aa]

relative [aa]

(a)

(b)

0.8

1

Figure 3.22.: Relative protein production of simultaneously translated mRNAs is coupled through the pool of available aa–tRNAs. Protein production against amino acid concentration at kα = 0.12 s−1 from figure 3.18 (simultaneous translation) in (a) and figure 3.19 (separate translation) in (b) for two mRNAs of 100 codons length each and configurations presented in figure 3.16 (3.16a type I and 3.16b type II). framework to explain both increases in protein production (figure 3.19). In this context, protein production corresponds to the ribosome flow J, kα represents α and the least abundant of both aa–tRNAs determines qmin , which changes dynamically with the availability of amino acids. With respect to competition, we observe two opposing effects, (i) an increase in charging level of the tRNA species that is limiting under optimal growth conditions, which is caused by (ii) a decrease in ribosome flow. A lower ribosome flow in turn is the consequence of either a decreased initiation (α) or a slower translation of the limiting codon (qmin ) caused by amino acid starvation. For a type I strand (figure 3.19b), equation (3.18) applies and demonstrates that given an α above the critical value, the ribosome flow solely depends on qmin and is independent of α. At high amino acid concentrations, i.e., the range an increase protein production is observed, the aa–tRNA that corresponds to type B codons shows an increased charging level and determines qmin . Hence, decreasing initiation only affects the type II mRNA (equation (3.19)) and procures resources (aa–tRNAs) for the type I strand. For the mRNA in figure 3.19a equation (3.19) applies for high amino acid concentrations and (3.20) for low [aa]. As long as the first codon is the slowest or the critical α value has not been reached, a higher qmin can lead to an increased ribosome flow. But, since J is also affected by initiation in the respective parameter range, this only applies for specific values of α. In this context, we can also explain why figure 3.18a does not show a transition when the limiting codon species changes. Equation (3.19) and (3.20) yield the same ribosome flow independent of the position of the limiting codon, as long as the first codon is slow and the critical α has not been reached. Thus, now visible transition occurs. On the other hand, to cause an increased flow in figure 3.19a, the charging rate of the tRNA


40

not delivering the amino acid subject to starvation has to increase, which requires type A codons to be limiting. Hence, an increased ribosome flow along the mRNA presented in figure 3.19a occurs between the change of the slowest codon species q1 = q2 = qmin , corresponding to the curvature at the hill’s foot and α = αc determining the top of the hill. To support this hypotheses, figure B.4 in the appendix shows a system with two simultaneously translated mRNAs, whereby one of them contains a codon representing the amino acid subject to starvation in first position. The higher susceptibility of the respective strand results in more aa–tRNAs becoming available under starvation condition. Thus, a greater protein production increase of the second, competing mRNA occurs.

3.5. Discussion The here presented deterministic model of eukaryotic protein translation extends current approaches. It explicitly accounts for the tRNA charging process and for codon translation rates affected by the abundance of aa–tRNAs. In this context, a steady–state solution of the derived ODE–system demonstrates a significantly decreased charging level due to amino acid starvation (figure 3.2). Although, theoretical studies [41] and in vivo experiments ([15] and figure 3.4) confirm these results, current models largely neglect this fact. The Response to Amino Acid Starvation In response to nutritional stress our model predicts an increase in a ribosome’s waiting time on codons representing the missing amino acid (figure 3.6). Since regulatory mechanisms are known to support the dissociation of ribosomes from mRNAs in case of significantly prolonged codon occupation times [66, 67], starvation could lead to more premature terminations. Furthermore, with respect to an mRNA’s protein production rate, the system shows two states (figure 3.5). The transition kα = ktr · aatRNA marks a balance between demand (translation) and supply (charging) of aa–tRNAs in which neither initiation nor elongation exclusively limit protein production. If demand exceeds supply, translation of codons representing the amino acid subject to starvation is the limiting factor and an unsaturated, elongation controlled state is present. This causes low charging levels and kpr can increase with a higher availability of amino acids. In contrast, an initiation controlled system results in a saturated state. A first–order phase transition with increasing amino acid concentration (figure 3.5b) characterises this system and protein production remains unchanged even if more aa–tRNAs become available. Initiation and Elongation Control In table 3.1, we present how different mRNA configurations respond to changes of the initiation rate kα and the amino acid concentration [aa]. These system responses may relate to biological mechanisms. Two publications describe changes in initiation and thus kα as the result of translational control [6, 7]. Our model predicts an effect of this type of regulation only for an initiation controlled


41

system. However, if the slowest codon is limiting and hence elongation control present, protein production is independent of kα and affected by the amount of available amino acids. Ribosome Density and Steric Hindrance With the knowledge that decreasing charging levels reduce codon translation rates, the nature of ribosomes as extended objects becomes important. Section 3.3 introduces the concept of steric hindrance and extents our model to account for it. As a consequence, ribosomes form queues in front of slow codons (figure 3.10), which represents another characteristic of elongation controlled systems. Models that neglect variable translation rates do not consider this queue formation and may therefore overestimate protein production. In general, ribosomal blocking causes a decrease in protein production and prevents a complete depletion of aa–tRNAs in our simulations. This in turn is in agreement with the experimental data shown in figure 3.4. Distinct mRNA Types Considering the slowest codon’s position on an mRNA, two distinct types of strands emerge (figure 3.13b and table 3.1). Romano et al. describe the effect of an increased initiation with respect to these mRNA types [64] (section 3.2). In addition, we find them to result in distinct sensitivities to amino acid starvation (figure 3.15). Thus, the type an mRNA belongs to determines whether it is susceptible to translational control or nutritional stress conditions. An Increased Demand for tRNAs In section 3.4.2, we introduce multiple codons of one type. This increases the demand for a specific tRNA species, affects charging levels and generally impairs protein production (figure 3.18 and figure 3.19). With respect to protein over–expression, where multiple gene copies or changes in promoter regions result in a higher abundance of a specific mRNA, this could become important. Depending on the codon bias of an over–expressed mRNA, the overall frequencies of codons in a cell and thus the demand for aa–tRNAs could be altered. Our model outlines the importance of the slowest codon’s position, i.e., the codon decoded by the rarest aa– tRNA, in determining whether translational control is possible. Thus, abnormal charging levels, which are caused by a bias toward a specific codon type, could change the reaction of many mRNAs to regulatory mechanism and may also cause toxic effects in the context of over–expression. A sensible countermeasure would be to shift charging to normal levels by adding more tRNA synthetases or more amino acids of a specific type. In section 3.3.4, the maximum charging reaction velocity shows the same influence on protein production than amino acid concentration. Hence, we conclude that modification of the structure of tRNAs or synthetases, which impairs the enzyme substrate fitting, may result in amino acid starvation phenotypes. Multiple Codon Species and Competition Numerical simulations show no effect of starvation on the ribosome flow until the translation rate of the codon species representing a missing amino acid becomes limiting. Its respective aa–tRNA’s abundance has to fall below that of the rarest species under optimal growth conditions. Hence, starvation for amino acids delivered by rare tRNAs exerts the strongest influence on kpr . When


42

considering two codon species, amino acid starvation causes in an increased charging level of some tRNAs, which also emerges during in vivo experiments (figure 3.17b and figure 3.4). This phenomenon could not be explained in literature so far. However, the mathematical model indicates that it results from the balance of charging and translation being shifted in response to nutritional stress. Furthermore, in a group of several simultaneously translated strands, specific codon configurations can yield an increased protein production. The sensitivity of an mRNA to initiation rate and amino acid concentration affects its performances, which causes the outlined behaviour. We expect mRNAs coding for proteins that counteract amino acid starvation to be configured in such a way. Exemplary analyses are presented in section 4.3.4.

Chapter 4. Stochastic Modelling Approach

43

Chapter 4

A Stochastic Approach to Modelling Protein Translation

Having discussed the principle effects of amino acid starvation, we now derive a more realistic, stochastic description for protein translation. The previous chapter demonstrates that under starvation conditions cellular resources can easily become rare and thus limiting. Stochastic approaches are particularly advantageous when describing such systems, which involve small numbers of molecules. Furthermore, ribosomes elongate in discrete steps requiring discrete numbers of charged tRNAs. In contrast to deterministic models, a stochastic simulation accounts for this discreetness.

4.1. The TASEP Model Originating in the description of diffusion–driven systems, the asymmetric simple exclusion process (ASEP) plays a paradigmatic role in non–equilibrium statistical physics. It models particle systems characterised by stochastic dynamics and exclusive interactions [68]. These properties makes the ASEP capable of describing protein translation, assuming ribosomes to be particles moving along a one dimensional lattice of discrete sites, which corresponds to an mRNA and its codons [69]. Several studies extend the exact ASEP solution derived in the early 1990’s [65] to include particles of arbitrary size [70, 10, 71]. Such descriptions are reasonable since ribosomes cover several codons, i.e., several boxes of the lattice, while elongating. Incorporation of site dependent movement probabilities due to a varying tRNA abundance refines these approaches in later studies. Recent works also show how the distribution of slow codons [24] or the restriction by ribosome abundance [72] influences translation. As discussed in section 3.2, Romano and et al. [64] even demonstrate the effect of specific codon position and its biological importance in grouping proteins into classes of distinct translational dynamics related to their function. Considerations like this improve the ASEP’s application to realistic translation phenomena. All these ASEPs share the common aspect of using a lattice on which particles move in a stochastic manner and follow specific rules. In case of ribosome movement, a one dimensional lattice which consists of discrete boxes representing one codon, respectively resembles an mRNA strand (figure 4.1). As translation only proceeds from the 5’ to the 3’ end of an mRNA, particles exclusively hop in a single direction and thus conduct a totally asymmetric movement (TASEP). The system shown in figure 4.1 employs open


44

Figure 4.1.: Scheme of a TASEP resembling protein translation with ribosomes R, codons Ci , elongation probabilities pi , initiation probability α and termination probability β. boundary conditions, assuming ribosomes to form a pool in the cytosol from which they inject in the lattice with probability α. Hopping down the chain one side at a time with an elongation probability pi , which depends on the abundance of a codon’s cognate tRNA species, they eventually exit with probability β and release a complete amino acid chain. These three processes refer to initiation, elongation and termination, respectively. The excluded volume constraint, i.e., steric hindrance of ribosomes, is implemented by ensuring that each site can accommodate at most one particle. Note that the above stated description simplifies translation by employing three probabilities that represents its distinct sub–processes. Such an assumption is reasonable for termination, as it is a fast process and not limiting to translation (section 2.2.4). In case of initiation, experimental and theoretical studies outline its importance [26, 12, 46] and, considering the complex initiation mechanism, suggest that a more detailed model is required. However, we focus on extending the implementation of elongation with respect to tRNA charging and propose an improved TASEP algorithm in the following.

4.1.1. Improved TASEP Algorithm Earlier TASEP translation models often employ constant elongation probabilities pi , which are in some cases related to the abundance of tRNAs. However, as outlined in section 2.2.4, elongation depends primarily on the waiting time for cognate aa–tRNAs. Hence, constant elongation probabilities which are calculated with the total abundance of a tRNA species imply two things: the fraction of uncharged to charged tRNAs is negligible, and charging levels remain constant. Our results and in vivo studies during starvation conditions [15] show this to be violated. Therefore, we propose an improved TASEP algorithm consisting of three steps: (i) proportional to their abundance, charged tRNAs bind to ribosomes occupying a codon, (ii) ribosomes that have bound a tRNA move one codon further and (iii) tRNAs are recharged. Binding of aa–tRNAs Assuming the cell to be an ideally mixed system, aa–tRNA binding is unbiased and independent of a ribosome’s position or the mRNA it translates. To resemble this, the algorithm first generates a position list of all ribosomes occupying a codon. By applying a Fisher-Yates-Shuffle, it then computes a random permutation of the list, which serves as a basis for simulating tRNA binding. Considering each ribosome


45

separately as listed in the shuffled position list, the algorithm calculates whether an aa– tRNA binds in this step. Considerations in section 2.2.4 yield that the probability to find and bind a charged tRNA j is proportional to its abundance, pbi,aatRNAj ∝ aatRNAj . To couple the process to real time, we employ the translation rate per available tRNA (2.3) and a real time equivalent of one Monte–Carlo–Step (MCS) as proportionality factors. With the assumption that ribosomes only translate a single codon at ha time, i one can 1 0 define the codon translation rate of one ribosome as rtr,tRNA = rtr,tRNA · codon and thus 0 · cMCS,time · aatRNAj pbi,aatRNAj = rtr,tRNA

(4.1)

with aatRNAj cMCS,time pbi,aatRNAj 0 rtr,tRNA

abundance of charged tRNA j time equivalent of a MCS probability to bind the aatRNAj translation rate per ribosome and available tRNA

in in in in

molecules s % 1/ (s · molecules)

For codons subject to wobble base pairing this probability is reduced by 36% (see section 2.2.3 and [28, 29]). Note that the only constraint for choosing cMCS,time is 0 ≤ pbi,tRNAj ≤ 1 ∀i and that binding reduces the abundance of the respective aa–tRNA by one. Ribosome Movement We compute ribosome movement according to a particle– ordered–sequential update rule. All ribosomes bound to a tRNA are able to move one codon further with a constant probability pmove given the next codon is vacant. The update itself starts with the right most codon and sequentially considers each particle until it reaches the start codon. If a ribosome occupies the stop codon, it leaves the lattice with probability β. Accordingly, a particle can enter the system with probability α given the start codon is vacant. Only in case a ribosome moves its tRNA is released, presuming elongation to be a fast process compared to the dissociation of a tRNA–ribosome complex. An update rule as described above follows experimental observations which show that the movement of ribosomes is fast in comparison to the waiting time for aa–tRNAs. We thus assume that ribosomes move with an infinite velocity, i.e., in a time interval δt = 0, from one codon to the next, resulting in pmove = 1. Hence, all ribosomes bound to an aa–tRNA hop within the same Monte–Carlo–Step, given the next codon is not occupied by a ribosome without a tRNA. Commonly used random update rules do not account for such a behaviour. In algorithms using them, situations might occur in which a randomly chosen ribosome is blocked by a ribosome that occupies the adjacent codon and is itself bound to an aa–tRNA (figure 4.2). Beside random and particle–ordered– sequential updates, a number of other rules is present in literature [73]. tRNA Recharging Lastly, uncharged tRNAs have to be recharged. Employing (2.7) this process depends on the level of uncharged tRNAs and the availability of amino acids.


46

Figure 4.2.: A Random update of two ribosomes that have bound a tRNA can cause the leftmost of them to be blocked in case it is chosen first. The amount of charged tRNAs in the next MCS n + 1 is then calculated by aatRNAj (n + 1) ≈ aatRNAj (n)+vmax,j · cMCS,time

tRNAj (n) tRNAj (n) + Km,tRNAj

!

aa j aa j + Km,aaj (4.2)

!

with aa j concentration of the amino acid carried by tRNAj aatRNAj (n) abundance of charged tRNA j in MCS n cMCS,time time equivalent of a MCS Km,aaj Michaelis–Menten constant of the amino acid carried by tRNAj Km,tRNAj Michaelis–Menten constant of the uncharged tRNAj vmax,j maximum reaction velocity of aaRS charging tRNAj tRNAj (n) abundance of uncharged tRNA j in MCS n and aatRNAj , tRNAj ∈ N.

in in in in in in in

mol molecules s mol molecules molecules/s molecules

4.1.2. Analytic Solution for the Ribosome Flow We next discuss a general solution of the above stated algorithm. Employing the law of total probability yields a description of the event of ribosome movement (Move) dependent on the event of binding a tRNA (Bind)

P (Move) = P (Move|Bind) P (Bind) + P Move|Bind P Bind . Assuming ribosomes to move with P (Move|Bind) = pmove if they have bound a tRNA and with P Move|Bind = 0 if not, this simplifies to P (Move) = pmove · P (Bind) = pmove · pbi,aatRNA . Thus, the effective hopping probability of a ribosome on codon i is pi,eff = pmove · pbi,aatRNACi


47

and since ribosome movement is fast compared to tRNA binding (pmove = 1) 0 pi,eff = pbi,tRNACi = rtr,tRNA · cMCS,time · aatRNACi

(4.3)

with aatRNACi cMCS,time pbi,tRNACi pi,eff pmove 0 rtr,tRNA

abundance of the cognate charged tRNA of codon i time equivalent of a MCS probability to bind the cognate charged tRNA of codon i effective hopping probability of a ribosome on codon i probability that a ribosome with tRNA moves translation rate per ribosome and available tRNA

in in in in in in

molecules s % % % 1/ (s · mol).

The result is that we have a TASEP with variant hopping probabilities. In its steady state it shows the same characteristics as discussed in literature [64] and equations (3.18) to (3.20) apply. However, the effective hopping probability pi,eff in (4.3) is influenced by the aa–tRNA availability and thus by codon abundance and tRNA recharging capability. More precisely, if a codon is abundant and charging of its respective tRNA species impaired, lower charging levels cause the binding probability and therefore pi,eff to decrease. In contrast, rare codons will be translated with a translation rate that is determined by the overall abundance of their cognate tRNA species if the recharging capacity is sufficient. Thus, under optimal conditions hopping probabilities (4.3) reach values previously used by Romano et al. [64]. Starvation for amino acids reduces these rates.

4.2. Numerical Simulations of Artificial Sequences Following the deterministic system, this section discusses numerical simulations considering a single mRNA with several fast codons interrupted by a single slow site. Again, fast codons are decoded by an abundant tRNA with constant charging level and slow codons by a rare tRNA species with a varying charging level affected by amino acid starvation. By employing the improved stochastic TASEP approach, we are not only able to assess the effect of stochastic aa–tRNA binding to ribosomes, but also to extend the model by implementing more realistic ribosomes covering multiple codons in later sections. Please note that results presented in this chapter employ sequences with a slow codon at position i = 25 or i = 250, respectively, which represent type I mRNAs. Figure B.5 in the appendix confirms that this position change has no influence compared to a stochastic simulation with a slow codon at i = 50, i.e., the codon position used during deterministic simulations.

4.2.1. Ribosomes Covering One Codon To compare the stochastic and the deterministic system, we first study ribosomes covering only one codon (figure 4.3). Again, a slow codon in first position (figure 4.3a and


48

(a)

(b)

(c)

(d)

Figure 4.3.: Steady–state charging level in (a) and (c) and protein production in (b) and (d) for the stochastic TASEP model with ribosomes covering one codon. A single mRNA strand of 100 codons length contains a single slow codon at position i = 1 in (a) and (b) and i = 25 in (c) and (d). This codon is decoded by the presented tRNA, which carries the amino acid affected by starvation. 4.3b) results in a smooth dependence of relative charging level and protein production rate on both, the initiation rate and the amino acid concentration. The absence of a first– order phase transition is in good agreement with the deterministic results accounting for steric hindrance. With the codon representing an amino acid affected by starvation positioned at i > 1 (figure 4.3c and 4.3d), a first–order phase transition as observed in figure 3.9 occurs. Hence, our improved stochastic TASEP model qualitatively resembles the deterministic description accounting for steric hindrance. Quantitative differences To evaluate the quantitative differences between the stochastic and deterministic model, we scale initiation probabilities and protein production rates to real time by multiplication with the time equivalent of a Monte–Carlo–Step cMCS,time . However, due to the different nature of both approaches, this linear scaling provides


49

only a rough estimation. Different update rules which we do not study here are likely to influence the results. Nevertheless, figure 4.4 presents the deviation of both models when predicting kpr and the relative charging level.

(a)

(b)

(c)

(d)

Figure 4.4.: Quantitative differences in steady–state protein production rate ((b) and (d)) and tRNA charging level ((a) and (c)) employing the deterministic model accounting for steric hindrance and the stochastic TASEP description. Protein production rate multiplied with cMCS,time , and charging level predicted by the TASEP model were subdued by the values predicted by the deterministic description. An mRNA of 100 codons length contains a single slow codon at position i = 1 in (a) and (b) and i > 1 in (c) and (d). This codon is decoded by the tRNA species that carries an amino acid affected by starvation. Protein production rates and tRNA charging levels predicted by both models show a 10% deviation and are therefore in reasonable agreement. The prominent peaks occuring for low initiation rates and amino acid concentrations in figure 4.4a and 4.4c are most likely caused by numerical inaccuracies or simulations not reaching steady–state. With exception of these peaks, the TASEP model generally yields lower charging levels. Deviations especially occur at low amino acid concentrations, where aa–tRNAs are rare.


50

A stochastic models should lead to more accurate results at such conditions. The effect of these lower charging levels on protein production is non–linear considering two facts: the protein production is increased in some parameter regimes although fewer aa–tRNAs are available; and the factor in which kpr and charging level predictions differ changes (data not shown). In comparison to the deterministic approach, the TASEP model shows a reduced kpr for low [aa] and an increased production rate for high amino acid concentrations. The latter occurs most pronouncedly for a high kα , which corresponds to high ribosome densities. This behaviour may result from the implementation of steric hindrance. In the TASEP model all ribosomes that have bound a tRNA can move in the same Monte–Carlo–Step. Especially for high tRNA binding probabilities, i.e., high charging levels, the influence of densely packed ribosomes and blocking decreases, as most ribosomes find an aa–tRNA. For instance, if the entire mRNA is occupied by ribosomes, all elongate in a wave–like movement within the same MCS as soon as the last ribosome has encountered its cognate aa–tRNA. In the deterministic model however, this situation results in an impaired protein production, since high ribosome densities dramatically decrease the effective elongation rates.

4.2.2. Ribosomes Covering Multiple Codons As described in section 2.2.2, ribosomes cover more than one codon while elongating. Accounting for this, we improve the stochastic simulation by implementing a ribosome size of nine lattices sites, i.e., nine codons. The ribosome’s A site, which binds a cognate aa–tRNA to the currently translated codon and is referred to as the reader, resides in the middle of our simulated ribosomes. Consideration of mRNAs consisting of 1000 codons maintains the relation between ribosome size and codon number used in section 4.2.1, but overestimates the average mRNA length in eukaryotes [74]. Following previous simulations, we study one mRNA with a single codon in two different positions. This codon is decoded by a rare tRNA species, which is affected by amino acid starvation. Figure B.6 in the appendix shows that no significant changes occur with the configurations already presented compared to ribosomes cover only one side. However, if the slowest codon is in close proximity to the start of an mRNA, there is different behaviour with larger ribosomes (figure 4.5). We see that for a slow codon at position i = 15 and ribosomes covering only one codon, we have a type I mRNA and a first–order phase transition. Larger ribosomes in contrast, maintain the smooth dependence of kpr on α (figure 4.5) and [aa] (data not shown) even in case the first sites are fast. Thus, ribosomes covering multiple codons extend the range of positions in which a slow site can prevent a first–order phase transition. Furthermore, extended ribosomes alter the impact of clustered slow sites on ribosome flow and protein production [10, 71].

4.3. Numerical Simulations of Realistic Sequences After having discussed the principle dynamics in translation, we here apply the stochastic TASEP approach with extended ribosomes to realistic mRNA sequences. As our model


51

0.012

pr

k in protein/MCS

0.01

one codon nine codons

0.008 0.006 0.004 0.002 0 0

0.01

0.02

α

0.03

0.04

0.05

Figure 4.5.: Steady–state protein production for the stochastic TASEP model considering different ribosome sizes. A single mRNA strand of 1000 codons length contains a single slow codon at position i = 15. Results are presented for [aa] = 0.025 mmol/L and ribosomes that cover codons as indicated. organism, S. cerevisiae provides an ideal biological system to theses studies. Not only are its genomic [75] and proteomic information [63, 76] freely available, but Percudani et al. demonstrate a correlation between tRNA gene copy number and abundance [37]. Furthermore, in vivo experiments confirm that charging levels are affected by amino acid starvation [42, 15] and Kurland suggests an influence of biased codon usage in translational regulation of protein expression [33]. We focus on starvation for a single amino acid and assume the charging level of all other tRNAs to be constant. To match biological conditions, the employed parameter set is further improved in the beginning of this chapter. We choose several mRNAs coding for yeast proteins which have different biological functions to serve as examples for our analysis. Discussion of mRNA configurations yields a first impression of the expected translational dynamics. Subsequently, the calculation of ribosome flows along the strands predicts protein production rates. Eventually, this leads to an assessment of how codon distribution influences protein expression under nutritional stress conditions.

4.3.1. Derivation of a Realistic Parameter Set With some experimental and theoretical data already available, our numerical simulations focus on the amino acid leucine and its respective tRNA species. S. cerevisiae possesses four different types of these isoacceptors, i.e., tRNAs carrying the same amino acid (see table 2.1), which differ in their abundance by approximately 11, 000 to 110, 000 molecules. Leu3 is the rarest, whereas Leu2 is the most abundant species. A difference in abundance of a factor of 10 makes Leu–tRNAs an ideal subject for theoretical studies, since it causes distinct patterns in tRNA isoacceptor charging [41].


52

Amino acid specific tRNA synthetases facilitate the charging process (section 2.2.3) and hence all four leucine isoacceptors bind to the same charging enzyme, the leucine– tRNA ligase (EC 6.1.1.4). This enzyme has the kinetic characteristics presented in table A.2, (given in Brenda [62]). In combination with the amount of leucine–tRNA ligase molecules in a yeast cell [63], equation (2.8) yields its maximum charging reaction velocity vmax = 1.06 · 104 molecules/s. However, this value applies to the charging of tRNAs required to translate 15, 000 mRNAs [59] with an average of 379 coding codons [74]. We rescale vmax to account for a reduced number of codons in our analysis. Hence, one obtains the maximum charging rate dependent on the number of simulated codons as vmax,codon

1.06 ∗ 104 molecules/s = 0.019 molecules/(s · codon) = 15, 000 mRNAs · 379 codons/mRNA

(4.4)

With respect to the Michaelis–Menten constants, we can use the Km,aa value in table A.2. However, since our model describes discrete tRNA molecules, Km,tRNA requires recalculation. Brenda provides different estimations of this parameter. With the volume of a yeast cell [77] and the Avogadro constant, one obtains its lower bound as Km,tRNA = 1 ∗ 104 mmol/L · 29 ∗ 10−15 L · 6.022 ∗ 1023 mol−1 = 1750 molecules. Considering the highest estimation, the Michaelis–Menten constant is thus in a range of 1750 − 10500 molecules and we chose Km,tRNA = 5000 molecules for further considerations. Please note that all four isoacceptors contribute to the concentration of uncharged tRNA molecules used in combination with Km,tRNA in (2.7). Taking the cellular abundance of tRNAs into account (table 2.1), it is thus sufficient to allow charging of leu–tRNAs under optimal conditions, even if only the least abundant of the four isoacceptors is discharged. Table 4.1 shows the parameter set used in this section, and table 2.1 the abundance of each tRNA species in S. cerevisiae. Table 4.1.: Parameters employed in stochastic simulations of realistic mRNA sequences with the improved TASEP algorithm. Parameter

Value

β cMCS,time Km,aa Km,tRNA ktr M CSdiscarded M CSsampled vmax,codon

1 0.1 0.02 5000 14 · 10−5 500, 000 1, 000, 000 0.019

Unit s mmol/L molecule 1/(molecules · s)

molecules/(s · codon)

4.3.2. Example Proteins and mRNA Configurations We analyse three example proteins with distinct functions regarding the cellular reaction to starvation in S. cerevisiae. First the leucine–tRNA ligase which charges leu–tRNAs


53

with an amino acid. As an enzyme degrading leucine, we then present the branched– chain–amino–acid transaminase. Eventually, translation elongation factor 1 beta (Efb1) which is part of a complex facilitating aa–tRNA binding to ribosomes is introduced. Leucine–tRNA Ligase As one of the enzymes charging tRNAs, the leucine–tRNA ligase (LeuRS) is known to show an enhanced expression under amino acid starvation conditions [61]. We expect nutritional stress to influence its production on a translational level. With 894 coding codons it is longer than the average mRNA. This aspect makes translational control even more likely. In order to quickly react to changing environmental conditions, stockpiling of some mRNAs in the cytoplasm, whose translation could be switched on if necessary, is reasonable. Such a mechanism saves the time that is needed for transcription and nuclear export, which is of particular significance for large mRNAs [61, 6]. In fact many of these longer mRNAs were found in the cytoplasm — associated with only a few ribosomes [26] — which suggests that translational control plays a role. From the theoretical considerations of previous chapters, we find that the distribution of slow codons, which correspond to rare tRNA species, and codons representing an amino acid affected by starvation, is of particular interest. Hence, an assessment of the mRNA sequence with respect to these two facts gives an impression of the translational dynamics (figure 4.6). In this regard the CUU codon emerges as one of the most im-

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Figure 4.6.: Position of codons decoded by rare tRNAs and of codons representing leucine in an mRNA coding for the leucine–tRNA ligase. Position of codons (a) decoded with less than 1/3 of the average translation rate and (b) representing leucine, is depicted with red bars. Arrows indicate the position of flow limiting (under optimal growth conditions) CUU codons. portant. Not only does it represent leucine and its respective tRNA species is very rare, but wobble base pairing additionally impairs its translation rate. Hence, figure 4.6 also


54

depicts the position of CUU codons as the slowest and therefore flow limiting codons under optimal growth conditions. With respect to the distribution of slow codons in figure 4.6a, a bias toward early positions emerges. Clusters of multiple slow sites are located in direct proximity to the initiation region. One such cluster which resides at the 60th nucleotide triplet is of particular size and contains the flow limiting codon. Hence, we expect an initiation controlled system with no significant queue formation under optimal growth conditions and a relatively high ribosome density on slow codons most pronouncedly when clustered. However, since the first codon is fast and considering the conclusion of section 4.2.2, the simulated ribosome size might influence the behaviour. The distribution of leucine representing nucleotide triplets shows no significant bias in position (figure 4.6b). In contrast, figure 4.7a presenting the biased usage of leucine codons demonstrates a low CUA

CUA

CUC

UUG

UUA

CUG

UUG CUU UUA

(a)

(b)

Figure 4.7.: Biased usage of codons representing leucine in an mRNA coding for leucine– tRNA ligase in (a) and Efb1 in (b). Fraction of each codon species to the total amount of all codons representing leucine. abundance of the flow limiting CUU s. Moreover, the two most common leucine codons in this specific example, namely UUA and UUG, are decoded by the abundant Leu1 and Leu2 tRNAs, respectively. This is in agreement with the confirmed relation of codon usage and tRNA abundance [34, 35], Branched–chain–amino–acid transaminase Increasing the expression of leucine–tRNA ligase in response to starvation would enhance the aminoacylation process and counteract low charging levels. In contrast, the branched-chain-amino-acid transaminase (EC 2.6.1.42) degrades L–leucine to 4–methyl– 2–oxopentanoate and thus causes a further reduction of available amino acids and aa– tRNAs. Hence, we expect this enzyme’s mRNA to show a fundamentally different response to starvation conditions. As a second difference, the transaminase mRNA is with 393 coding codons significantly shorter than a LeuRS coding sequence.


55

Neither slow codons (figure 4.8a) nor leucine coding nucleotide triplets (figure 4.8b)

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Figure 4.8.: Position of codons as depicted in figure 4.6 for the branched–chain–amino– acid transaminase. (a) codons decoded with less than 1/3 of the average translation rate, (b) codons representing leucine. Arrows indicate CUU codons. show a bias in distribution. But compared with the mRNA of leucine–tRNA ligase, which contains approximately 8% leucine representing codons, the content within a transaminase mRNA is with 10% higher. Since no clusters of slow sites occur in close proximity to the start of the sequence and the first flow limiting CUU codon is further downstream, we expect a classical type I protein and a first–order phase transition to emerge. Efb1 The third protein we investigate is Efb1, which is part of the EF-1 complex that facilitates aa–tRNA binding to the ribosomal A site [78]. As a crucial part of the translation machinery, we expect its expression to be highly robust to amino acid starvation. Translation of the respective mRNA was already determined by Romano et al. to be of type II. In this context, figure 4.7b reveals some special characteristics. The Efb1 mRNA avoids flow limiting CUU codons and uses primarily fast nucleotide triplets to code for leucine. Its translation completely abdicates Leu3. Moreover, it shows none of the codons translated with less than 1/3 of the average rate (data not shown). This indicates that Efb1 translation might be optimised to high ribosome flows.

4.3.3. Charging Level of tRNAs At optimal growth conditions, we find charging levels of 90% and above for all four isoacceptors (figure 4.9). Dittmar et al. confirm these results experimentally by detecting a similar aa–tRNA abundance in vivo [42]. The plots resemble a step–like function in response to starvation. Thus, variations of the amino acid concentration at optimal


56

conditions exert only a minor influence. The biological system may be robust to these fluctuations. However, below a specific amino acid supply, charging levels decrease dramatically. The exact point is determined by Km,aa in the Michaelis Menten equation (4.2) and hence dependant on the tRNA synthetase. Elf et al. demonstrate that a difference in tRNA isoacceptor abundance causes a distinct charging pattern in response to amino acid starvation [41]. The here presented numerical simulation resembles these theoretical results for the translation of a realistic sequence (figure 4.9). In this context, charging levels depend on the relation of demand

relative charging level

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relative [aa] Figure 4.9.: Relative steady–state charging level of tRNALeu isoacceptors in response to amino acid starvation. Charging level of all four leu–tRNAs at α = 0.025 against the relative amino acid concentration, i.e., the amino acid concentration normalised to optimal growth conditions. The simulation employs a stochastic TASEP model with ribosomes covering nine codons to predict the translation of one leu–tRNA ligase mRNA. (codon abundance) and supply (tRNA abundance) of aa–tRNAs. In particular low abundant species are likely to become uncharged, as they provide only a limited number of tRNA molecules as a substrate to the charging process (see equation (2.7)). However, the abundance of their cognate codons plays a role as well. In figure 4.9, Leu4–tRNAs are the least charged, although Leu3–tRNAs are less abundant. With respect to figure 4.7a, we find that more codons require Leu4. Hence, a high demand of a low abundant isoacceptor provokes a reduced availability of aa–tRNAs in response to starvation. This aspect is even more striking, since low abundant tRNAs often correspond to flow limiting codons. Abundant tRNAs on the other hand maintain a higher charging level. In this regard, the characteristics of an Efb1 mRNA are significant. By avoiding the usage Leu3 and containing only very few codons requiring Leu4, it employs primarily high abundant


57

tRNAs. Such a strategy allows a high protein production under starvation conditions.

4.3.4. Ribosome Density on Realistic mRNAs With respect to protein translation and an mRNA’s occupation with ribosomes, two opposing effects emerge. The transcription process requires cellular resources (nucleotides, polymerases and energy), which represent the costs of mRNA synthesis. To gain the maximum benefit, i.e., many proteins, a high ribosome density and flow is worthwhile. On the other hand, nucleotide sequences have a limited half–life [79] resulting in many premature terminations in case a densely occupied mRNA decays. Furthermore, regulatory mechanisms remove ribosomes stalled upon an mRNA due to a high ribosome density [66, 67], which yields the same effect. Leucine–tRNA ligase For artificial mRNA sequences our mathematical framework already demonstrated an increased waiting time of ribosomes translating slow codons (figure 3.10). Considering a realistic strand, the presented ribosome density under optimal growth conditions resembles such a behaviour (figure 4.10a). According to the distribution of slowly translated nucleotide triplets, peaks in the occupation with ribosomes occur, which are higher on clustered slow sites. The profile indicates no queue formation and averaging reveals that 1/3 of the maximum packing density is reached. Hence, our model predicts a higher value then the 1/5 measured by Arava et al. [26]. Since ribosomes cover nine codons, whereas their A site only occupies one, reader densities are generally lower and the respective peaks narrower. Comparing the theoretical prediction in figure 4.10a with experimental data from Ingolia et al. (figure 2.4), we find less pronounced peaks and a higher density on early codons in the latter. Theses observations may result from averaging over numerous mRNAs performed by Ingolia to derive his results. Such an approach smoothen stochastic fluctuations. Furthermore, averaging over mRNAs subject to translational control, which are associate with very few ribosomes mainly within the region of initiation [26], may cause the observed density increase. When our system encounters nutritional stress conditions, causing a decrease of tRNALeu charging levels, translation rates of the respective codons decrease. Hence, simulations yield an increased ribosome density (figure 4.10b). As predicted, the position of leucine coding CUU codons plays an important role. In particular at these sites queues form, which manifest themselves by an increased ribosome density prior to the slow codon and periodic peaks in reader densities. These peaks occur with a period of nine codons, i.e., the size of a ribosome used in simulations and result from steric hindrance by ribosomes occupying a consecutive slow site. Dong et al. describe this phenomenon in combination with a TASEP model neglecting the impact of tRNA charging [71].


58

reader ribosome

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Figure 4.10.: Predicted steady–state ribosome density on an mRNA coding for leucine– tRNA ligase. The stochastic TASEP model with α = 0.025 and ribosomes covering nine codons is employed to predict the density of ribosomes and readers, i.e., a ribosome’s A site. Density in (a) under sufficient leucine supply ([Leu] = 0.05 mmol/L) with slow codons as shown in figure 4.6a depicted with red bars and in (b) in response to amino acid starvation ([Leu] = 0.01 mmol/L) with leucine representing codons as shown in figure 4.6b presented in red. Arrows indicate the position of flow limiting (under optimal growth conditions) and leucine representing CUU codons.


59

Branched–chain–amino–acid transaminase The predicted ribosome density on an mRNA coding for the branched–chain–amino– acid transaminase shows queueing even under optimal growth conditions (figure 4.11a). Derivation of the maximum charging reaction velocity employed in numerical simula1

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Figure 4.11.: Predicted steady–state ribosome density on an mRNA coding for the branched–chain–amino–acid transaminase. Data as presented in figure 4.10. Density in (a) for [Leu] = 0.05 mmol/L and slow codons as shown in figure 4.8a depicted with red bars and in (b) for [Leu] = 0.01 mmol/L with leucine representing codons as shown in figure 4.8b presented in red. Arrows indicate the position of CUU codons. tions gives an explanation for this observation. Averaging over all codons in a cell in


60

(4.4) implies that simulations use mRNAs with the average codon frequency. Hence, if simulated strands contain more leucine coding codons, an increased demand for aa– tRNAs causes lower charging levels (data not shown), which yields queueing. Apart form that, ribosome densities again outline the importance of flow limiting CUU codons. Just like in figure 4.10, amino acid starvation causes significant queues and a generally higher density on preceding sites. Our model even predicts an occupation with almost the maximum capacity in the first hundred codons of a transaminase mRNA. Ribosome densities further downstream of the 5’ region are relatively low, which is in good agreement with literature [27]. Efb1 In contrast to the other two proteins, an Efb1 mRNA shows no queue formation at [aa] = 0.01 mmol/L (figure B.8 in the appendix). In fact, the first signs of ribosomal queueing do not emerge until the amino acid concentration is significantly decreased. This provides another evidence for the Efb1 mRNA to be flow optimised and robust to starvation.

4.3.5. Protein Production Rate With respect to the protein production rate of a specific mRNA, we focus on two characteristics. Its reaction to changing initiation rates, which represents an mRNA’s susceptibility to translational control and the influence of amino acid starvation. Considering the former and given optimal growth conditions, our model is equivalent to a simple TASEP and should resemble the results published by Romano et al. [64]. Hence, the leucine–tRNA ligase and the transaminase as non–ribosomal proteins should show type I characteristics. In contrast, Efb1 is a ribosomal protein with type II translational dynamics in literature [64]. We find the expected behaviour for transaminase and Efb1 (figure 4.12a). However, the translation of LeuRS shows a smooth dependence of kpr on α. This discrepancy to the analysis of Romano et al. results from employing extended ribosomes in our studies (see section 4.2.2). A simulation with ribosomes covering only one codon yields a type I protein (figure B.7 in the appendix). Considering optimal growth conditions, the translation of Efb1 reaches the highest protein production rate. Thus, employing primarily fast codons results in the predicted flow optimisation. Even under starvation conditions (figure 4.12b), kpr of this mRNA remains almost unchanged. However, we find Efb1 to show a first–order phase transition with increasing α under nutritional stress. Furthermore, translation of the leu–tRNA ligase mRNA changes to type I characteristics as well. Thus, both are no longer sensitive to changing initiation probabilities over the entire parameter range. Such changes in the response to α demonstrate that amino acid starvation alters the translational dynamics of specific mRNAs in our model. Utilising the linear scaling between a TASEP simulation and real time (see section 4.2.1), the model predicts a production of one protein every 5.2 s (Efb1) to 7.9 s (transaminase) under optimal conditions. To highlight the influence of limited cellular resources, figure 4.13a presents the re-


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Figure 4.12.: Steady–state protein production rate against initiation probability for the separate translation of an mRNA coding for leucine–tRNA ligase, the branched–chain–amino–acid transaminase and Efb1, respectively. The improved TASEP algorithm with ribosomes covering nine codons predicts kpr at optimal growth conditions ([aa] = 0.05 mmol/L) in (a) and in response to starvation ([aa] = 0.01 mmol/L) in (b). sponse of all three proteins to starvation when translated separately. Again, Efb1 production shows a remarkable robustness. For its translation, a first–order phase transition with increasing amino acid concentration emerges. We observe a similar behaviour for

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Figure 4.13.: Steady–state protein production rate against amino acid concentration for the translation of mRNAs coding for the leucine–tRNA ligase, the branched–chain–amino–acid transaminase and Efb1, respectively. The improved TASEP algorithm with ribosomes covering nine codons predicts kpr at α = 0.025. Separate translation (neglecting competition) is presented in (a) and simultaneous translation (competition of all three strands for aa– tRNAs) in (b). The maximum charging reaction velocity vmax was adjusted according to the number of simulated codons.


62

the leu–tRNA ligase mRNA. Both maintain a constant protein production over a wide range of [aa]. In contrast, kpr of the transaminase is directly proportional to parameter variations. Hence, a decreasing availability of aa–tRNAs due to nutritional stress would impair the production of this amino acid degrading enzyme. This indicates that its expression may be regulated by codon configuration on a translational level. With respect to the relative kpr in figure 4.14 at high [aa], translation of LeuRS is again more robust

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Figure 4.14.: Normalised protein production rate of an mRNA coding for the leucine– tRNA ligase and the branched–chain–amino–acid transaminase. Protein production rate from figure 4.13a was normalised to optimal growth conditions ([aa] = 0.05 mmol/L). than that of transaminase (see section 3.3.3). Thus, nutritional stress affects these two mRNAs differently and the fraction of charging enzyme to amino acid degrading enzyme increases when resources become scarce. Even the net production of a single mRNA of LeuRS becomes higher compared with that of transaminase for a decreasing amino acid concentration (figure 4.13a).

4.3.6. Effects of Competing Ribosomes Since the presented realistic mRNAs show different translational dynamics, competition among elongating ribosomes might affect their performance. Figure 4.15 shows kpr for the simultaneous translation of leucine–tRNA ligase and transaminase. We again find coupled ribosome flows and an increased protein production of transaminase for an initiation rate just above the critical value (see section 3.4.2). Furthermore, the LeuRS mRNA reacts less to competition. Since it contains more codons, a reduced ribosome flow yields a substantial reduction of demand for aa–tRNAs. Hence, a moderate decrease in kpr of a large mRNA or several mRNAs can free more aa–tRNAs used to increase the flow along specific nucleotide sequences. Numeric simulations also demonstrate the importance of an mRNA’s competitors in determining its translational dynamics (figure 4.16). The leucine–tRNA ligase follows type II characteristics if translated separately. A similar behaviour emerges in case it


63

−3

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Figure 4.15.: Steady–state protein production for the simultaneous translation of two mRNAs coding for leucine–tRNA ligase and transaminase, respectively. The improved TASEP model with ribosomes covering nine codons predicts kpr at [aa] = 0.01 mmol/L. competes with an mRNA coding for the branched–chain–amino–acid transaminase. As both use codons with similar frequency, tRNA charging levels are not significantly altered by competition. However, although we adjust vmax to the number of simulated codons,

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Figure 4.16.: Steady–state protein production of an mRNA coding for leucine–tRNA ligase in response to competition with different realistic sequences. Protein production rate at [aa] = 0.025 mmol/L predicted by the improved TASEP with ribosome covering nine codons. An mRNA coding for the leucine–tRNA ligase is either translated separately or simultaneously with the mRNA of transaminase or Efb1, respectively. The maximum charging reaction velocity vmax was adjusted according to the number of simulated codons.


64

the protein production rate decreases in figure 4.16. This results from the usage of slow leucine codons in the mRNAs of leucine–tRNA ligase and transaminase. In contrast, simultaneous translation of LeuRS with Efb1 yields a higher kpr and a first–order phase transition with increasing α. Since the Efb1 mRNA employs primarily fast nucleotide triplets, tRNA charging levels differ to the system only simulating the single mRNA (data not shown). In the here presented case, this difference results in leucine–tRNA ligase showing type I behaviour in case it competes with Efb1. To conclude this section, figure 4.13b presents the competition of all three realistic mRNAs in response to amino acid starvation. In comparison with a system neglecting competition between the strands (figure 4.13a), a change of the translational dynamics and of the critical amino acid concentration that causes a phase transition occurs. Thus, the composition of the entire cellular mRNA set, i.e., the transcriptome, affects the response of mRNAs to starvation and translational control mechanisms (see also figure B.9 in the appendix).

4.4. Discussion Considering our working hypothesis of reduced aa–tRNA levels due to nutritional stress conditions, we utilise a stochastic approach to model translation in this chapter. Especially the low molecule numbers and the problem’s discreetness makes such an approach more realistic. Improving an ordinary TASEP by accounting for the tRNA binding, charging and discharging process (section 4.1.1) yields a system with variable hopping probabilities. In contrast to the constant pi ’s, employed in most models, we find them to depend on aa–tRNA supply and codon abundance (figure 4.3a). In this context, the improved stochastic description shows qualitatively similar results to the deterministic model in section 3.3 (figure 4.3). Quantitative differences occur in particular for low amino acid concentrations and are likely to be influenced by the used update rule and simulation parameters (figure 4.4). Especially at starvation conditions, we expect a TASEP model to be more accurate. Extended Ribosomes To approach the realistic biological system, we introduce extended ribosomes covering nine codons in section 4.2.2. Apart from yielding a lower ribosome flow, our model predicts a change in translational dynamics compared with the behaviour presented in literature [64]. For a slow codon in close proximity to an mRNA’s 5’ end, larger ribosomes provoke type II characteristics (figure 4.5). Thus, they extend the range of slow codon positions preventing a first–order phase transition. Re– evaluation of the results obtained in literature could detect non–ribosomal proteins that utilise type II behaviour in relation to their function. Varying tRNA Isoacceptor Abundance Focusing on an amino acid which is delivered by multiple tRNAs that bind to the same charging enzyme, we find selective isoacceptor charging (figure 4.9). Elf et al. predict this phenomenon theoretically without considering the consequence for protein production [41]. In this context, the supply and demand for


65

an aa–tRNA controls its response to starvation. Rare species that correspond to a commonly used codon are most likely to become uncharged. Abundant tRNAs maintain their charging level instead. In Efb1 we find a protein of S. cerevisiae that utilises this concept. By avoiding rare tRNA species and employing primarily fast codons, it shows a remarkable robustness to amino acid starvation (figure 4.13a). The analysis of Cyc3 (YAL039C) as a second ribosomal protein leads to similar results (data not shown). Hence, our model is capable of detecting mRNAs with a high robustness to starvation. Genome wide analysis of the entire yeast transcriptome could discover proteins with this characteristic, which may relate to their biological function. It should be mentioned that an increased demand for tRNAs can cause similar effects than starvation. Hence, we recommend to use codons decoded by tRNAs with a robust charging level in mRNAs of over–expressed recombinant proteins. This would result in a higher production rate. The here presented stochastic model could support these approaches by calculating the impact of a specific codon configuration on the aa–tRNA abundance. Ribosome Density Section 4.3.4 presents the predicted ribosome density under optimal and starvation conditions for three realistic mRNA sequences of S. cerevisiae. Two of them show significant ribosome queues at the flow limiting CUU codons when resources become scarce (figure 4.10b and figure 4.11b). An increased waiting time at these sites can cause frame shifts in vivo [29]. Furthermore, regulatory mechanisms remove ribosomes stalled upon an mRNA. Thus, amino acid starvation may cause an increase in premature terminations. In this context, the here presented mathematical description can predict the most likely places for queueing and frame shifts to occur. It is also able to identify the Efb1 mRNA as a flow optimised sequence which lags these sites (figure B.8a). Generally our stochastic model predicts a higher density under optimal conditions than measured by Arava et al. [26]. This indicates that in vivo initiation rates might be lower than the here employed. Protein Production The improved TASEP shows a steady–state production of one protein every 5.2 s to 7.9 s for the translation of realistic mRNAs under sufficient amino acid supply. To our knowledge no experimental data exist for this rate in vivo. However, considering the average translation rate of 10 aa/s (see section 2.2.2) and that literature states one ribosome every 50 codons [26] such a value is reasonable. In reality, the time needed to reach steady–state, i.e., fill the mRNA with ribosomes, might decrease the overall protein production rate. In combination with a model for protein and mRNA decay, our description could be utilised to predict protein expression based on the cellular mRNA set. We find the distinct protein types theoretically predicted in chapter 3 to emerge in the context of realistic nucleotide sequences. In case of an mRNA coding for leucine– tRNA ligase, a difference in robustness in response to starvation causes a relative and a net increase in the production compared with that of transaminase (figure 4.13). Since both proteins possess contrary functions regarding the tRNA charging level, this could indicate regulation on a translational level. Moreover, changing the ration of two proteins


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in response to starvation via codon usage, e.g. of cell cycle regulators or phosphatase– kinase systems, could influence other cellular functions. We also find that amino acid starvation alters the translation dynamics of specific mRNAs (figure 4.12). This provides another theoretical evidence for a role of codon configuration in translational control. Competition for Rare Resources In the last section, we account for the competition of ribosomes for aa–tRNAs. As demonstrated in section 3.4.2, simultaneous translation of multiple proteins alters the charging level. Hence, dependent on the composition of a cell’s mRNA set, changes of the translational dynamics or of critical parameter values, which cause a first–order phase transition, may occur (figure 4.13). We also find the competitor strands of an mRNA to affect its response to changes in initiation (figure 4.16). A simulation of the entire yeast transcriptome is thus necessary to gain a comprehensive understanding. However, significant changes in the transcriptome, e.g. during different cell cycle phases, could alter the translational dynamics of whole protein families. For instance, making proteins robust to changes in initiation enables a cell to produce initiation factors prior to cell devision without changing protein expression.

Chapter 5. Outlook

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Chapter 5

Outlook

In this work, we developed two mathematical models of protein translation. Both extend current approaches and provide new insight in the dynamics of this fundamental process. The feasibility of analysing realistic mRNAs offers a range of perspectives for future applications. Deterministic Model In some areas, the deterministic model is less realistic than the presented stochastic description. The model could be improved by the inclusion of extended ribosomes. Also, an analytic solution which avoids the simplification of having only one codon species would be able to predict the impact of codon bias in realistic sequences on the tRNA charging level. Applying a sensitivity analysis to such an analytic solution could also elucidate whether specific processes limit translation in certain parameter regimes. The presented results suggest that charging levels are less sensitive to the initiation rate than to the availability of amino acids (figure 3.2). Extension of the deterministic model to study realistic sequences would also allow comparison with the stochastic approach. Evaluating the parameter range in which significant differences occur and elucidating the impact of distinct TASEP update rules would increase our knowledge of the system. This would enable us to determine the parameter domain that allows an application of the deterministic model, which has advantages in terms of calculation time. Stochastic Model With the improved TASEP algorithm, we developed a tool to analyse how mRNAs respond to amino acid starvation and to predict the steady–state protein production rate. Scanning the entire transcriptome of yeast for sequences which are sensitive to nutritional stress can detect regulatory proteins and the mechanisms they participate in. Particularly robust or flow optimised proteins may be involved in fundamental cellular processes. Section 4.3.6 demonstrates the role of competition between multiple realistic mRNAs for aa–tRNAs. We expect this to be one form of translational regulation under nutritional stress conditions. In this context, a genome wide simulation of the entire yeast transcriptome could also divide mRNAs into two classes: mRNAs which encode proteins whose translation is down–regulated due to limited resources, and mRNAs which show an increased protein production under these conditions (figure 3.21). The latter may correspond to enzymes which the cell uses to counteract starvation.

Chapter 5. Outlook

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Scanning the set of mRNAs with respect to the reduction of kpr can detect proteins that suffer most from missing aa–tRNAs. This helps identifying nucleotide sequences affected in knockout mutants or by structural changes in tRNAs and synthetases which impair tRNA charging levels. Combining these predictions with in vivo experiments may lead to a better understanding of translation in S. cerevisiae. The improved stochastic description for translation could also be combined with models for gene regulation, transcription and protein decay. Such an approach may provide a comprehensive mathematical model which predicts the proteome from genetic information.

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75

List of Symbols α β aa aatRNAj aatRNAXi Ci cMCS,time De aatRNAj [E] J kα kbi kcat Km,aa Km,tRNAaa kpr ktr pbi,aatRNAj pj pi,eff pmove PR q R RNAt RRN A hrtr i rtr,i rtr,tRNA 0 rtr,tRNA t htRNAi tRNAj tRNAj,t tRNAXi v vmax vmax,codon

initiation probability termination probability abundance of an amino acid abundance of the charged tRNA j abundance of the cognate charged tRNA of codon i codon i time equivalent of a MCS demand for charged tRNA species j enzyme concentration ribosome current initiation rate binding rate turn over number Michaelis–Menten constant of an amino acid Michaelis–Menten constant of an uncharged tRNA protein production rate translation rate probability to bind the aatRNAj parameter j effective hopping probability of a ribosome on codon i probability that a ribosome with tRNA moves abundance of proteins elongation probability of a slow codon abundance of ribosomes total mRNA 5’ leading regions abundance of ribosome–mRNA complexes average translation rate specific translation rate of codons decoded by tRNAi translation rate per available tRNA molecule translation rate per ribosome and available tRNA time average abundance of tRNAs abundance of an uncharged tRNA j total abundance of charged and uncharged tRNA j abundance of cognate uncharged tRNA of codon i reaction velocity maximum reaction velocity vmax used in numeric simulations

in in in in in

% % mol molecules molecules

in in in in in in in in in in in in

sec molecules mol/L ribosomes/MCS 1/s 1/(molecules · s) 1/s mol molecules protein/s 1/(molecules · s) %

in in in in in in in in in in in in in in in in in in in

% % molecules % molecules molecules molecules codon/s codon/s codon/ (s · molecules) 1/ (s · molecules) s molecules molecules molecules molecules mol/s mol/s molecules/(s · codon)

76

Vyeast w¯i,j yi Xi

volume of a yeast cell normalised sensitivity of state variable i to parameter j state variable i ribosome density on codon i

in L

in molecules

Appendix A. Simulation Parameters

77

Appendix A

Simulation Parameters

A.1. Analytic Solution for the Steady–State tRNA Charging Level Table A.1.: Parameter set employed to calculate the steady–state tRNA charging level in section 3.1.2. Index j in (3.15) has been omitted. Parameter

Value

Unit

kbi Km,aa Km,tRNA R0 RNAt tRNAt vmax

1 0.5 0.5 1 1 1 1

1/(molecules · s) molecules molecules molecules molecules molecules molecules/s

A.2. Numerical Simulation of the Deterministic Model To derive a parameter set for tRNA charging, we consider tRNAs which deliver leucine (leu–tRNA), and their respective charging enzyme, the leucine–tRNA ligase (EC 6.1.1.4). The required kinetic parameters can be found in enzyme databases such as Brenda [62]. Table A.2.: Kinetic parameters of leucine–tRNA ligase according to Brenda [62]. Parameter

Value

Unit

kcat Km,Leu Km,tRNALeu

5 0.02 1-6 ∗ 10−4

1/s mmol/L mmol/L

With these values and a total of 2.12 ∗ 103 leucine–tRNA ligase molecules in one cell


78

of S. cerevisiae [63], equation (2.8) yields the maximum charging reaction velocity vmax = 5

molecules · 2.12 ∗ 103 = 1.06 ∗ 104 molecules/s. s

Employing the presented parameters is only reasonable when one simulates the entire yeast transcriptome, which includes realistic codon frequencies. However, our numerical evaluations of the deterministic system focus on artificial mRNA sequences with artificial codon distributions. In these cases, we use a different parameter set to account for the reduced number of mRNAs. For reasons of comparability, parameters should be scaled according to the relation between absolute codon number in a cell and the number of simulated codons. We use this method in section 4.3.1 to assess the translation of realistic sequences with a stochastic TASEP model. However, most of the parameters required for deterministic calculations are still subject to experimental investigation. Hence, we employ the following empirically estimated parameters to demonstrate the qualitative behaviour of the deterministic approach in prove of principle simulations. Table A.3.: Parameters used in numerical simulations of artificial sequences with the deterministic model. Parameter

Value

Unit

kbi Km,aa Km,tRNA ktr R RNAt tRNAt vmax

1 · 10−5 0.02 100 14 · 10−5 100, 000 1 1000 0.3

1/(molecules · s) mmol/L molecule 1/(molecules · s) molecules molecules molecules molecules/s

The number of free ribosomes R follows the assumption R RNA used in section 3.1.2 with RNA = 1 representing a single mRNA strand. In this context, the binding probability kbi ensures that one ribosome per second attaches to the 5’ leading region. This is based on experimental observations showing ribosomes to elongate with 8.8 aa/s [53] and to be 50 codons apart [26]. Thus, one ribosome has to attach at least every 5 s if steric hindrance is neglected. In the biological system, this value may be influenced by the recycling of ribosomes following termination [12]. Using equation (2.3) and table 2.1, we calculate the average translation rate per tRNA molecule as ktr = rtr,tRNA =

10 codon/s = 14 · 10−5 . 71428 molecules

Assuming that more tRNA molecules than cognate codons must be present, which allows a continuous translation and charging, yields an overall tRNA abundance of tRNAt = 1000. We estimate that 10% of these molecules are required to be uncharged in order to


79

reach half the maximum charging reaction velocity. Furthermore, vmax was estimated as of the number of codons that require the respective aa–tRNA, which determines the demand. 1/3

A.3. Numerical Simulation of the Stochastic Model To study artificial sequences in prove of principle simulations with the stochastic TASEP model, we employ the parameters presented in table A.4. Numeric simulations of realTable A.4.: Parameter set employed to simulate artificial sequences with the stochastic TASEP model. Parameter

Value

β cMCS,time Km,aa Km,tRNA ktr M CSdiscarded M CSsampled tRNAt,1 (rare and affected by starvation) tRNAt,2 (not affected by starvation) tRNAt,3 (constant charging level) vmax

1 0.1 0.02 100 14 · 10−5 500, 000 500, 000 1000 1100 10000 0.3

Unit s mmol/L molecule 1/(molecules · s)

molecules molecules molecules molecules/s

istic sequences use a more realistic parameter set which is derived and presented in section 4.3.1.

Appendix B. Additional Results

80

Appendix B

Additional Results

B.1. Deterministic Model

(a)

(b)

(c)

(d)

Figure B.1.: Steady–state tRNA charging level in (a) and (c) and protein production rate in (b) and (d) against kα and vmax for the deterministic model accounting for steric hindrance of ribosomes. An mRNA strand of 100 codons length contains a single slow codon at position i = 1 in (a) and (b) and i = 50 in (c) and (d). This codon is decoded by the presented tRNA species, which delivers the amino acid subject to starvation.


81

0.35 0.08

kpr in proteins/s

0.06 −5% −1 kα = 0.25 in s

0.05 0.04

+5% 0.03 0.02


0.3

0.07

0.2 0.15 0.1 0.05

0.01 0 0

0.25

0.01

0.02

0.03

0.04

0 0

0.05

0.01

[aa] in mmol/L

0.02

0.03

0.04

0.05

0.04

0.05

0.04

0.05

[aa] in mmol/L

(a)

(b) 0.1

0.08 −5% kα = 0.25 in s−1

0.06

+5% 0.04 0.02 0 0

0.01

0.02

0.03

0.04


kpr in proteins/s

0.1 0.05

0

−0.05

−0.1 0

0.05

0.01

[aa] in mmol/L

0.02

(c)

(d) 0



1

0.8

0.6

0.4

0.2

0 0

0.03

[aa] in mmol/L

0.01

0.02

0.03

[aa] in mmol/L

(e)

0.04

0.05

−0.2

−0.4

−0.6

−0.8

−1 0

0.01

0.02

0.03

[aa] in mmol/L

(f)

Figure B.2.: Local sensitivity of the deterministic model accounting for steric hindrance to selected parameters. Normalised local sensitivity was determined with equation (2.5) for parameter perturbations of 5% and an mRNA consiting of 100 codons. It contains a single slow codon at position i = 1 in (a) and (b) and i = 50 in (c), (d), (e) and (f). (a) and (c) depict kpr in response to changes of kα with the corresponding sensitivity in (b) and (d), respectively. (e) shows the sensitivity to vmax and (f) to Km,aa .


(a)

82

(b)

Figure B.3.: Different mRNA configurations employed in figure B.4. An mRNA consists of codons decoded by an abundant tRNA with constant charging level (rectangles without labelling). Two different codons with variable translation rates interrupt these fast sites. Codons of type A are decoded by an abundant tRNA and represent an amino acid subject to starvation, whereas type B codons are the slowest codons under optimal growth conditions.


83

(a)

(b)

(c)

(d)

Figure B.4.: Steady–state charging level and protein production for two mRNA strands containing two codon species. Two mRNAs contain a slow codon which is decoded by a rare tRNA that carries the amino acid not subject to starvation (depicted in (b)) and a second faster codon that represents the amino acid affected by starvation (cognate aa–tRNA depicted in (a)). Both strands are translated simultaneously. (c) configuration shown in figure B.3a with the codon which represents an amino acid with constant concentration in first position and (d) configuration shown in figure B.3b with the codon which represents the missing amino acid in first position.


84

B.2. Stochastic Model

Figure B.5.: Difference in steady–state protein production comparing two mRNAs which contain the slow codon at position i = 25 and i = 50, respectively. The slow codon is decoded by a rare tRNA species that is subject to amino acid starvation. Simulations use a stochastic TASEP algorithm with ribosomes covering one codon and a single mRNA strand of 100 codons.


85

(a)

(b)

(c)

(d)

Figure B.6.: Steady–state charging level in (a) and (c) and protein production in (b) and (d) for the stochastic TASEP model with ribosomes covering nine codons. A single mRNA strand of 1000 codons length contains a single slow codon at position i = 1 in (a) and (b) and i = 250 in (c) and (d), which is decoded by the presented tRNA that carries an amino acid subject to starvation.


one codon nine codons

0.015

0.01

pr

k in protein/MCS

0.02

86

0.005

0 0

0.01

0.02

α

0.03

0.04

0.05

Figure B.7.: Steady–state protein production against initiation probability for the translation of an mRNA coding for the leucine–tRNA ligase. The improved TASEP algorithm with ribosomes covering codons as indicated predicts kpr at optimal growth conditions ([aa] = 0.05 mmol/L).


87

0.7

reader ribosome

0.6

density

0.5 0.4 0.3 0.2 0.1 0

20

40

60

80

100

120

140

160

180

200

codon (a)

1

reader ribosome

density

0.8

0.6

0.4

0.2

0

20

40

60

80

100

120

140

160

180

200

codon (b)

Figure B.8.: Predicted steady–state ribosome density on an mRNA coding for Efb1. Density in (a) for [Leu] = 0.01 mmol/L and in (b) for [Leu] = 0.001 mmol/L.



0.018 0.016

k [protein/MCS]

0.01

pr

pr

k [protein/MCS]

0.015

88

0.005


0.014 0.012 0.01 0.008 0.006 0.004 0.002

0 0

0.01

0.02

α

(a)

0.03

0.04

0.05

0 0

0.01

0.02

α

0.03

0.04

0.05

(b)

Figure B.9.: Steady–state protein production of all three realistic mRNAs in response to competition and changing initiation probability. The mRNAs coding for leucine–tRNA ligase, transaminase and Efb1 are translated simultaneously and have to compete for aa–tRNAs. The improved TASEP with ribosomes covering nine codons predicts protein production against initiation rate at [aa] = 0.01 mmol/L in (a) and [aa] = 0.05 mmol/L in (b). The maximum charging reaction velocity vmax was adjusted according to the number of simulated codons.

A Mathematical Model of Protein Translation and

A Mathematical Model of Protein Translation and

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