Introducing Systems Biology to Bioscience Students through ...

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Introducing Systems Biology to Bioscience Students through Mathematical Modelling. A Practical Module Néstor V. Torres Grupo de Tecnología Bioquímica, Departamento de Bioquímica y Biología Molecular, Facultad de Biología, Universidad de La Laguna, San Cristóbal de La Laguna, Tenerife, Canary Islands, Spain

Corresponding author: Néstor V. Torres, Departamento de Bioquímica y Biología Molecular, Facultad de Biología, Universidad de La Laguna, San Cristóbal de La Laguna, Tenerife, Canary Islands, Spain. Email: [email protected], Phone: +34-922318334 Date recieved: 15 February 2013

Date accepted: 15 May 2013

Abstract Systems Biology, one of the current approaches to the understanding of living things, aims to understand the behaviour of living systems through the creation of mathematical models that integrate the available knowledge of the system’s component parts and the relations among them. Accordingly, model building should play a central part in any biology degree programme. One difficulty that we face when confronted with this task, however, is that the mathematical background of undergraduate students is very often deficient in essential concepts required for dynamic mathematical modelling. In this practical module, students are introduced to the basic techniques of mathematical modelling and computer simulation from a Systems Biology perspective. Keywords: Systems Biology, modelling, dynamics, simulation

Introduction The systemic approach to the study of living things is not new in the realm of biology, as it dates from the early twentieth century (Von Bertalanffy and Woodger 1933). However, it has been largely absent from most degree programmes in biology, which have been dominated by the reductionist approach. There is no room here to analyse the reasons behind this state of affairs, which can also be observed in other branches of science, but it should be noted that, in the case of biology, the need for a paradigm shift from the reductionist to the systemic approach is particularly pressing. In this context, Systems Biology (SB) must be understood as an approach to the study of living things that is different from the reductionist approach, but firmly based on the achievements and results of the latter. SB is an interdisciplinary approach to the study of living things, which seeks to explain and understand their functions and behaviours over time by drawing on both the knowledge of their elements and the (usually non-linear) relations that exist between them. In this view, functions ‘emerge’ from the web of system

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Bioscience Education, Vol 21, Issue 1 (December 2013) doi:10.11120/beej.2013.00012

N. V. Torres

components and interactions that occur in open systems very far from thermodynamic equilibrium. Model building, is a central part of any scientific endeavour, and one way to incorporate the systemic approach into biology degree programmes is by teaching this skill. In fact, it can be argued that the development of the ability to build models lies at the core of many sciences education programmes. What SB adds to the issue of modelling is the stress it places on mathematical formulation. This trend is already seen not only in the most mathematical branches of biology (e.g. physiology or ecology), but also in other disciplines such as zoology or botany, which have evolved from purely descriptive approaches to quantitative disciplines that rely on a rich range of mathematical models (see Otto and Day 2007). Building a model is a creative and rigorous task that involves the integration of knowledge and assumptions and requires intuition, imagination and independence of thought. Modelling, either in its classical version (the conceptual modelling) or in its more formalised form (the mathematical one) involves asking the questions, choosing the appropriate conceptual framework for formulating and testing hypotheses and making accurate assumptions and simplifications. Accordingly, teachers of biology need tools that are suitable for transferring the principles and techniques of model building to their students. However, faced with this need, we encounter the not insubstantial problem that the mathematical background of undergraduate students is very often deficient in essential concepts required for mathematical modelling dynamics. The challenge posed by this situation is: how do we introduce students of biology to the use of mathematical modelling? Here, I describe a practical exercise designed to introduce mathematical modelling and computer simulation to students of biology. The aim is to introduce mathematical modelling, in the context of an SB perspective, as a tool for integrating information and exploring the dynamics of the modelled system.

The Practical Module: Modelling of a Dynamic Biological System The course described here has been informed by iterative changes to the protocol used over more than five consecutive years (2006–2011), to teach a medium-sized group of around 20 students within a degree programme where students are normally taught a range of pure and applied biology subjects.

Educational goals The overall objective of the training proposal is to help undergraduate students of biology acquire an understanding and appreciation for the value and limitations of mathematical models and numerical integration methods. In particular, we seek to make students proficient in: (i) understanding and integrating the fundamental issues of modelling in biology; (ii) practicing converting diagrams into computable mathematical models; (iii) computing models in scenarios representing different environmental conditions or states; (iv) presenting, discussing and interpreting the results.

Programme deployment The module is widely applicable and could be taught in almost any course of any degree in science or engineering, provided that students have completed the first year of a science © 2013 G. Scott, The Higher Education Academy

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degree and at least one semester of biochemistry. This module has been taught by the author in two different programme locations. Originally, it was designed for and given to undergraduate biology students of the second year. At this point students were conversant in the fundamentals of mathematics, physics, chemistry and biology, as well as biochemistry, microbiology and cellular biology. Based in this experience it was also taught for engineering and science students, from mathematics to physics (both graduated and undergraduate) with similar results.

Activity 1. A Systems Biology primer. Introducing models and the role of modelling in the scientific process The first session (60 minutes) is devoted to the introduction of the concept of the model and its role in the scientific process. It explains what a model is, highlighting the fact those verbal arguments, graphs and pictures are different types of models, although of limited capacity in comparison to mathematical models and the computer simulations derived from them. It emphasises that the process of model development constitutes an educated exercise of simplification of the actual system reality, making explicit the assumptions adopted, emphasising the importance of variable selection and the ongoing critical dialogue between the experimental observations and the model’s representation of the observed facts. Subsequently, students are prompted to think about what models can be used for, what the advantages and limitations of any model might be and what is needed for a model to be useful. Models are not an aim in themselves, but a tool towards achieving the understanding of a biological function or its underlying mechanisms. Therefore, attention should be focused not on the mathematics, but on the usefulness of the description provided by the model in order to reach a better understanding of the system’s behaviour. At this point, students are ready to understand the value of an integrating modelling into scientific studies and thus to be introduced to the main tenets of SB. This seminar does not need to be tied directly to the subsequent practical exercise; in fact, it can be taught some time before. As assigned reading for this session, students are given the introductory chapter of an SB textbook (Klipp et al. 2005). Future delivery of the session will also employ the text recently published by Voit (2012).

Activity 2. Basic tools for mathematical modelling The second session (60 minutes) is intended to present the basic concepts and tools used in the development and simulation of biological models. First, the intuitive notion of instantaneous velocity and how this translates mathematically into the concept of the derivative is presented. On this basis, it is possible to start the representation of kinetic processes by differential equations. Figure 1 shows the metabolic pathway that is used for this purpose. It consists of a metabolic pathway involving three coupled reactions, two variables and a positive feedback loop. This scheme is simple enough to be accessible to most of the students and at the same time rich enough in terms of features and dynamic behaviour to be of pedagogical value. In fact there are some well known metabolic systems that are well represented for this scheme of reactions. These are the cases of the observed activation of the glycolitic enzyme phosphofructokinase by one of its products that prompts a variety of different dynamic behaviour in the glycolitic pathway and the feedback activation of the intracellular cAMP synthesis by the extracellular cAMP that operates in the Dictiostelium discoideum intercellular communication. In both cases the real systems displays the dynamics behaviour showed by the proposed regulatory structure. At this point the class is prompted to discuss the selection of variables, the distinction between variables and parameters and how it is possible to translate the changes in the variables into mathematical terms through a set of ordinary differential equations (ODEs). © 2013 G. Scott, The Higher Education Academy

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dA = V 0 − V1 dt dB = V1 − V 2 dt

V 0 = cont

5 V 0 10

A(t = 0) = 0 B (t = 0) = 0

V 1 = A·(1 + B 3) V 2 = 5·B

0

t

15

Figure 1 Scheme of the metabolic pathway used for the model presentation and selection of processes, rate expressions, variables, parameters and regulatory features and its translation into a set of ordinary differential equations. A and B variables represent two distinct species (metabolites, species, etc.) They interact in such a way that any increase in B activates its own synthesis, according with the V1 rate equation. V0 is the rate of synthesis of A which is constant. This will be the parameter used to explore how the system dynamics is affected for the parameter change. V2 is the rate of transformation of B. The values of A and B at time 0 (initial conditions) are indicated as well as the time range of the explorations.

The teacher guides the reflections and discussions based on student contributions. Then it is shown how the solution of this set of ODEs corresponds to the time evolution of the variables. The session ends with the question: how can we know the actual solutions of the ODEs systems which describe a biological system? The answer necessarily leads us to the concept of solution by numerical integration. The assigned readings for this session are certain sections from chapter 3 of the book by Klipp et al. (2005) or from chapter 2 of Voit (2012).

Activity 3. Computing simulation of biochemical system This session (90 minutes) starts with a presentation of the principles of numerical integration methods (Euler’s method). It is at this point when the meaning and need of the initial conditions come naturally as the numerical integration demands the values of the variables at time zero. Once the students know how the ODE system is solved numerically, they are presented with the ODE solver package to be used in the following exercise. The choice of simulation platform through which models are to be implemented is critical. The optimum choice should be a compromise between competing interests. On the one hand, the package should be powerful and versatile enough to be able to solve the models quickly. On the other hand, it should be simple, intuitive and capable of being installed on laptops. Given the profile of the students, the instrumental nature of the simulation program and the limited time available, the time investment required for them to master the basic elements needed for running simulations should be minimal. Among the several packages available, we use the Berkeley MadonnaTM software package, an all-purpose differential equation solver that runs on PCs or Macintosh which can be downloaded for free in a demo version from http://www.berkeleymadonna.com/download.html. Students are instructed to implement and compute the model developed in the previous section. Before the execution of the computing part of the practical exercise, they are asked to predict what behaviour the system will show for different values of the parameter that measures the strength of the positive feedback interaction. In discussions of how the system’s variables will evolve, students must use their intuition to predict results which they will compare with the results computed by the model. During the computing part of this session, students can explore the influence of changes in the model parameters as well as other questions that might arise as they explore the system. © 2013 G. Scott, The Higher Education Academy

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The following file may be copied directly into the software (see Figure 1 for meaning of variables). ; Technical parameters in Berkeley Madonna setting METHOD RK4

; method of integration

STARTTIME = 0 STOPTIME = 15 DT = 0.02

; initial time ; final time ; integration step

; Model differential equations d/dt (A) = V0 - V1 d/dt (B) = V1 - V2 ; Rate equations V0 = 7 V1 = A*(1 + B^3) V2 = 5*B ; Initial conditions (variables values at t = 0) INIT A = 0 INIT B = 0 For the proposed exercise, the following lines may be added: ; Parameter values V0 = 4 V0 = 5 V0 = 10 These commands reset the input rate of A (V0) at the indicated values. The corresponding output is shown in Figure 2. Other simulations could address the changes in the strength of the V1 activation (e.g. changing the exponent 3 in the V1 rate equation). The take-home message here is that if a system contains regulatory signals, it is not sound to rely on our intuition for making reliable predictions. Instead, we need to utilise computational models to determine how even the most simple systems work and how distinctly different responses are seen depending on the conditions under which they operate. As a supplementary assignment for this session, students are asked to read the simulation package tutorial.

Activity 4: Let’s make a model The next session (90 minutes) is devoted to the construction of a model by all students working together. The ideal size of the class for this part is about 10 students; in case the number were higher the class can be divided in two or more groups. The proposed system to model has the features of a classical predator–prey Lotka–Volterra system. The choice is based on the fact that it offers students the opportunity to model a different type of biological system that can be used to represent a wide range of biological systems. In this way, students can perceive how the same methodological approach and techniques can address problems of a very different nature. We started with a general introduction of the Lotka–Volterra predator–prey interaction. Directly afterwards, we introduced the corresponding dynamic model (Figure 3), without offering details on the ODEs, except for their functionality. At this point it was stressed that the proposed model, was a simplified version of reality that however it captured some aspects of the predator–prey interactions.


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Figure 2 Dynamics of the variables A and B for different values of V0. A. V0 = 4; B. V0 = 5. C. V0 = 10. © 2013 G. Scott, The Higher Education Academy

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A

r

N+P

h

P

B

N 2 P

z

dN = r · N – z· N ·P dt

dP = z· N ·P dt r = 0.07 z = 0.01 h =1 Conditions 1

Conditions 2

N (t = 0) = 0

N (t = 0) = 100

P(t = 0) = 0

P (t = 0) = 10

Figure 3 Simple model of the predator prey interaction. A. Diagram accounting for the interactions between predators (P) and prey (N). B. Model equations, parameter values and two set of initial conditions. Simulations starting at the first initial (Conditions 1) will show an (unstable) steady state while when the initial conditions are other (e.g. Conditions 2) the system will show periodic oscillations.

The aim at this stage is for students to go through the processes associated with model simplification and the choice of variables and parameters and acknowledge the difficulties involved in the experimental determination of parameters. The final outcome is the construction of a model that will be transferred by the students themselves to the simulation package. They are then able to observe the evolution of the variables over time and will analyse how they change depending on the initial conditions and parameter values. The following file may be copied directly into the Berkeley MadonnaTM software (see Figure 3 for meaning of variables). Figure 4 shows the corresponding output. ; Technical parameters in Berkeley Madonna setting METHOD RK4

; method of integration

STARTTIME = 0 STOPTIME = 100 DT = 0.02

; initial time ; final time ; integration step

; Model differential equations d/dt (N) = r*N - z*N*P d/dt (P) = z*N*P - h*P

; the prey population ; the predator population

; Parameter values r = 0.07 z = 0.01 h=1 ; Initial conditions (variables values at t = 0) that yields an unstable steady state INIT N = 0 INIT P = 0 © 2013 G. Scott, The Higher Education Academy

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Figure 4 Berkeley Madonna output for two different settings of initial conditions. A. N(0) = 0 and P(0) = 0; (unstable) steady state. B. N(0) = 100 and P(0) = 50; oscillating solution.

In another run the following initial values can be considered: ; Initial conditions (variables values at t = 0) that yields an oscillating solution INIT N = 100 INIT P = 50

Activity 5: Evaluation The grading of this activity is based on two components. Although the activities were carried out in groups, each student should, at the end of the module, develop a brief (five pages maximum) Report of Work representing 70% of the grade. This Report of Work should follow this general structure: general presentation of the system studied; model description; results obtained from the different initial conditions assayed; discussion and references. Since the most common observed pitfalls are related with the analysis of the simulation results, in the final assessment it will be the work and effort that is valued over the correction. The remaining 30% will be assigned based on the degree of participation throughout the sessions and the concerns and interest shown by them. Since the group of students were small (10–15) the participation assessment was done based (quantitatively) in the record of the interventions (registered in a worksheet on a student basis) and qualitatively, on the basis of the nature and extension of the participation. © 2013 G. Scott, The Higher Education Academy

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Comments from end-of-course evaluations suggested that the students were overwhelmingly positive about the module. They find it useful; the approach easy to follow and helpful in enhancing their understanding. Some student’s comments were illustrative of the valuable features of the module: I liked the methodology, especially the part in which we implemented the models and run some simulations. It helped me to see the meaning and usefulness of some of the concepts that initially did not fully understood. I found the practice very interesting because it allowed me to understand the importance of the models of biological systems and how through simplification of the actual systems they facilitate its understanding. The most interesting part of this practice has been that it has allowed me to understand that biological problems are global and therefore should not be addressed from a single discipline perspective. Concepts have been presented to us we had not had occasion to meet earlier. I would like the section dedicated to the exploration of model of biological systems already known by us to be extended. From the students comments we conclude that in general they found the module useful; the approach easy to follow and helpful in enhancing their understanding.

Discussion In many environments (biology, chemistry, engineering, physics or computing), there is a growing interest in dynamic mathematical modelling being seen among students. It is therefore necessary to respond to these demands. This practical module is designed to offer students a conceptual and practical introduction to this field. The ultimate aim is to demonstrate to students that modelling is a tool that they should consider in any scientific project and to instil in them the perception that modelling is an essential means of grasping the complexity of a system, the unpredictability of the dynamic response and how new properties emerge from the combined actions of a system’s elements. The module enables students to understand the process of model building and simulation-based research, without having to first learn programming. Also, it serves to increase student familiarity with computational tools and to enhance the biology student’s confidence in the increasingly quantitative realm of contemporary biology. It also serves to show, in practical terms, how a modelling task implies the mobilisation not only of many competences and skills (retrieving information, deciding among competing suitable hypothesis, research in a particular topic) but also prompts the students to integrate concepts and information from different disciplines, to explore the model dynamic outcome obtained from the implemented model and its subsequent analysis. This analysis would suggest new experimental designs and even some model modifications that would serve to test new explanatory hypothesis. As a result the student is involved in an activity that resembles an interdisciplinary and model driven research project. The degree to which scientific modelling concepts can be reinforced in students is restricted by the limited time available for this module, but the practical exercise presented allows them to acquire the necessary skills, even given the trade-offs in both time and attention. The experience gained over several years of teaching the module has shown that it allows biology students to master a minimum but sufficient set of concepts and techniques in


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dynamic modelling. All metrics have shown students to be overwhelmingly positive about the course. As stated before, this practical module has been run in the last five years by around 100 students of Biology, most of them without additional formal training in mathematics apart from the introductory courses at the first years of the degree. Eighty per cent of the students have approved the module; the average score given being six out of ten. Yearly surveys of the student’s satisfaction with the module showed that most of the students felt the course to be informative and useful about what they learnt and the way they learnt it. It is my view that we must gradually imbue our teaching of all biological disciplines with the modelling and systemic view, as well as with the perception that both the formulation and the implementation of the objectives of any research project would greatly benefit from the development of a mathematical model of the dynamics of the process.

Acknowledgements The author gratefully acknowledges the comments and feedback of the many students who participated in the development of this course and Guido Santos Rosales for his help in the design of the teaching materials. Also, the author would like to thank two anonymous reviewers for valuable comments on the manuscript. This work was funded by the research grants from Spanish MICINN, Ref. No. BIO2011-29233-C02-02 and Agencia Canaria de Investigación, Innovación y Sociedad de la Información, Ref. No. PIL2071001.

References Klipp, E., Herwig, R., Kowald, A., Wierling, C. and Lehrach, H. (2005) Systems Biology in Practice. Concepts, Implementation and Application. Weinheim, Germany: Wiley-VCH Verlag. Otto, S. and Day, T. (2007) A Biologist’s Guide to Mathematical Modeling in Ecology and Evolution. Princenton, NJ: Princenton University Press. Voit, E.O. (2000) Computational Analysis of Biochemical Systems. A Practical Guide for Biochemists and Molecular Biologists. Cambridge, UK: Cambridge University Press. Voit, E.O. (2012) A First Course in Systems Biology. New York, NY: Garland Science. Von Bertalanffy, L. and Woodger, J.H. (1933) Modern Theories of Development; an Introduction to Theoretical Biology. Oxford, UK: Oxford University Press. Berkeley MadonnaTM. Copyright ©1993–2001 Macey, Robert I. & Oster, George F., http://www.berkeleymadonna.com/download.html.


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