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8th Vienna International Conference on Mathematical Modelling 8th Conference on Modelling February - 20, 2015. Vienna University of Technology, Vienna, 8th Vienna Vienna18International International Conference on Mathematical Mathematical Modelling 8th Vienna International Conference on Mathematical Modelling February -- 20, 2015. Vienna University of Technology, Vienna, Austria 18 Available online at www.sciencedirect.com February 18 20, 2015. Vienna University of Technology, February 18 - 20, 2015. Vienna University of Technology, Vienna, Vienna, Austria Austria Austria

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IFAC-PapersOnLine 48-1 (2015) 447–452

Development and analysis of a Development and analysis of a Development and analysis of Development and analysis of a a mathematical model for a synthetic mathematical model for a synthetic mathematical model for a synthetic mathematical model cell for a synthetic biological biological cell biological biological cell cell

Eugenia Schneider ∗∗ Michael Mangold ∗∗ Eugenia Schneider ∗∗ Michael Mangold ∗∗ ∗∗ Eugenia Eugenia Schneider Schneider Michael Michael Mangold Mangold ∗∗ ∗ ∗ Max-Planck-Institute for Dynamics of Complex Technical Systems, for Dynamics of Complex Technical Systems, ∗ ∗ Max-Planck-Institute Max-Planck-Institute for of Technical 39106 Magdeburg, Germany (e-mail: Max-Planck-Institute for Dynamics Dynamics of Complex Complex Technical Systems, Systems, 39106 Magdeburg, Germany (e-mail: 39106 Magdeburg, Germany (e-mail: [email protected]) 39106 Magdeburg, Germany (e-mail: [email protected]) ∗∗ [email protected]) for Dynamics of Complex Technical Systems, ∗∗ Max-Planck-Institute [email protected]) Max-Planck-Institute for of Technical ∗∗ ∗∗ Max-Planck-Institute for Dynamics Dynamics of Complex Complex Technical Systems, Systems, 39106 Magdeburg, Germany (e-mail: Max-Planck-Institute for Dynamics of Complex Technical Systems, 39106 Magdeburg, Germany (e-mail: 39106 Magdeburg, Germany (e-mail: [email protected]) 39106 Magdeburg, Germany (e-mail: [email protected]) [email protected]) [email protected]) Abstract: Synthetic biology as a new discipline is driven by progresses made in the understandAbstract: Synthetic biology as aand new is driven by progresses made in the understandAbstract: Synthetic as new discipline is by made the ing of microbiological processes itsdiscipline mathematical description by system biological models. Abstract: Synthetic biology biology as a aand newits discipline is driven driven by progresses progresses made in in the understandunderstanding of microbiological processes mathematical description by system biological models. ing of microbiological processes and its mathematical description by system biological models. The topic of synthetic biology is a systematic construction of artificial biological systems with ing oftopic microbiological processes andsystematic its mathematical description by system biological models. The of biology is systems with The topicproperties. of synthetic synthetic biology is a aofsystematic systematic construction ofanartificial artificial biological systems with tailored The subject this work construction is to developof artificialbiological cell model consisting The topic of synthetic biology is a construction of artificial biological systems with tailored properties. The subject of this work is to develop an artificial cell model consisting tailored properties. The subject of this work is to develop an artificial cell model consisting of functional biological devices like genome, transcriptome, proteome, metabolome, with a tailored properties. The subject of this work is to develop an artificial cell model consisting of functional biological devices like genome, transcriptome, proteome, metabolome, with aa of functional biological devices like genome, transcriptome, proteome, metabolome, with structure that guarantees a synchronization of the devices in a robust way. The mathematical of functional biological devices like genome,of transcriptome, metabolome, with a structure that guarantees synchronization the devices in aaproteome, robust way. The mathematical structure that guarantees a synchronization of the devices in robust way. The mathematical model shown in this worka involves the metabolism, the polymerization subsystem and the structure that guarantees a synchronization of the devices in a robust way. The mathematical model shown in this involves the polymerization subsystem the model shown in model this work work involves the metabolism, metabolism, the polymerization subsystem and the membrane. The structure enables an oscillatorythe behavior of the system with and a stable model shown in this work involves the metabolism, the polymerization subsystem and the membrane. The model structure enables an oscillatory behavior of the system with aa stable membrane. The model structure enables an oscillatory behavior of the system with stable limit cycle. Also the system provides a robust self-replicating state and a biological robustness membrane. The model structure enables an oscillatory behavior of the system with a stable limit cycle. Also the systemuncertainties. provides a robust self-replicating state and aa biological robustness limit cycle. the provides with respect to parameter limit cycle. Also Also the system systemuncertainties. provides a a robust robust self-replicating self-replicating state state and and a biological biological robustness robustness with respect to parameter with respect to parameter uncertainties. with respect to parameter uncertainties. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: nonlinear systems, simulation, synthetic biology, protocell, biocybernetics Keywords: Keywords: nonlinear nonlinear systems, systems, simulation, simulation, synthetic synthetic biology, biology, protocell, protocell, biocybernetics biocybernetics Keywords: nonlinear systems, simulation, synthetic biology, protocell, biocybernetics 1. INTRODUCTION Rasmussen et al., 2003). As pointed out by Rasmussen 1. INTRODUCTION INTRODUCTION Rasmussen et these al., 2003). 2003). Asmay pointed out by by Rasmussen 1. Rasmussen et al., pointed out et al. most of modelsAs be structured into three 1. INTRODUCTION Rasmussen et these al., 2003). Asmay pointed out by Rasmussen Rasmussen et al. most of models be structured into three three et al. most of these models may be structured into functional devices: a container forming the boundary of Enormous progresses have been made in the fields of mi- et al. most devices: of these amodels mayforming be structured into three functional container the boundary of Enormous progresses have been made in the fields of mifunctional devices: a container forming the boundary of the cell, a metabolism generating the building blocks Enormous have been in microbiology progresses and systems biology. The mathematical moddevices: a container forming the boundary of Enormous progresses have been made made in the the fields fields of ofmodmi- functional the cell, metabolism generating the building blocks crobiology and systems systems biology. The mathematical mathematical cell, aaa and metabolism generating building blocks of the cell, a programming partthe containing genetic crobiology and biology. The eling of biological systems is becoming increasinglymodim- the the cell, metabolism generating the building genetic blocks crobiology and systems biology. The mathematical modof the cell, and a programming part containing eling of for biological systems is becoming becoming increasingly proim- of the programming part genetic information and aaregulating the processes inside the cell eling of biological systems is increasingly important a deepened understanding of intracellular of the cell, cell, and and programming part containing containing genetic eling of for biological systems is becoming increasingly proim- information information and regulating the processes processes inside the the cell portant a deepened understanding of intracellular and regulating the inside cell (Rasmussen et al., 2003). portant for understanding of intracellular intracellular processes. Synthetic biology as a new biological field aims at information and the processes inside the cell portant for aa deepened deepened understanding of pro(Rasmussen et et al.,regulating 2003). cesses. Synthetic biology as of new biological field systems aims at (Rasmussen al., 2003). cesses. Synthetic biology as aa new biological field aims at the systematic construction artificial biological (Rasmussen et al.,work 2003). cesses. Synthetic biology as a new biological field aims at The issue of this is the definition of a structure that the systematic construction ofmain artificial biological systems The issuethe of this this work is is the the definition definition of aa structure structure that the construction artificial biological systems withsystematic tailored properties. Twoof working directions can The issue of work of the systematic construction of artificial biological systems masters synchronization of those three devices. that The The issuethe of this work is the definition of a structure that with tailored properties. properties. Two main the working directions can masters synchronization of those three devices. The with tailored Two main working directions can be identified in synthetic biology: top-down and the masters the synchronization of those three devices. The with tailored properties. Two main working directions can synchronization problem includes the question of robust masters the synchronization of those devices. The be identified in synthetic synthetic biology: the top-down and the synchronization problem includes theinthree question ofthat robust be identified in top-down andfrom the synchronization bottom-up approach. The biology: top downthe approach starts problem includes the question robust be identified in synthetic biology: the top-down and the functional dependence of the devices the senseof the synchronization problem includes the question of robust bottom-up approach. The top down approach starts from functional dependence of the devices in the sense that the bottom-up approach. The top down approach starts from an existing approach. micro-organism whose genome is manipulated the devices sense that the bottom-up The top down approach starts from functional material independence each of theof devices has in atthe least doubled at functional dependence of the devices in the sense that the an existing micro-organism whose genome is manipulated material in each of the devices has at least doubled at an existing micro-organism whose genome is manipulated and reduced intentionally based on acquired knowledge material in each of the devices has at least doubled at an existing micro-organism whose genome is manipulated the end of the cell cycle. In such a case we can assume material in the eachcell of cycle. the devices at least doubled at and hoping reducedtointentionally intentionally basedproperty(Keasling, on acquired acquired knowledge the end of Inartificial suchhas case we can assume and reduced based on achieve a desired 2012; the of the cell cycle. such aaa cell case can assume and reducedtointentionally basedproperty(Keasling, on acquired knowledge knowledge thatend self-replication of theIn iswe achieved. The the end of the cell cycle. In such case we can assume and hoping achieve a desired 2012; that self-replication of the the robustness artificial cell cell is achieved. achieved. Thea and hoping to achieve desired 2012; Jaschke et al., 2012). Inaacontrast, the bottom up approach self-replication of artificial is The and hoping to 2012). achieveIn desired property(Keasling, property(Keasling, 2012; that demand for the biological of the model, i.e. that self-replication of the robustness artificial cell is achieved. Thea Jaschke etthe al., contrast, the bottombiological up approach approach demand for the the biological biological of the the model, i.e. Jaschke al., 2012). In contrast, bottom up starts onet molecular level from the functional de- demand for robustness of model, i.e. aa Jaschke et al., 2012). In contrast, the bottom up approach certain independence of the qualitative system behavior demand for the biological robustness of the model, i.e. starts on the molecular level from functional biological decertain independence of the the qualitative system behavior starts on the molecular level from functional devices like genome, transcriptome, proteome,biological metabolome certain independence of qualitative system behavior starts on the molecular level from functional biological deon the kinetic parameter values, is justified by the aim to certain independence of the qualitative system behavior vices like genome, genome, transcriptome, proteome, metabolome on theable kinetic parameter values, is justified justified by the the aim to to vices like transcriptome, proteome, metabolome and attempts to assemble these biological parts into an ar- on the kinetic parameter values, is by aim vices like genome, transcriptome, proteome, metabolome being to deal with perturbations. on the kinetic parameter values, is justified by the aim to and attempts to assemble these biological parts into an arbeing able to deal with perturbations. and attempts to assemble these biological biological parts 2002). into an anThe ar- being able to deal with perturbations. tificial cell (Stano, 2011; Pohorille and Deamer, and attempts to assemble these parts into arable to dealofwith tificial cell (Stano, (Stano, 2011; Pohorille Pohorille and Deamer, Deamer, 2002). The being The dependence the perturbations. functional devices is well exemtificial cell 2011; and 2002). The idea behind the bottom-up approach is to obtain better tificial cell (Stano, 2011; Pohorille and Deamer, 2002). The The dependence of the functional devices is isbywell well exemidea behind the bottom-up approach is to obtain better The dependence of the functional devices plified by the Chemoton model described T. exemG´ anti idea behind the bottom-up approach is to obtain better understanding and improved predictability by building up The dependence of the functional devices isby well exemidea behind theand bottom-up approach is toby obtain better plified by the Chemoton model described T. G´ nti understanding improved predictability building up plified by the Chemoton model described by T. G´ aa nti (G´ a nti, 2003). The Chemoton comprises three functional understanding and improved predictability by building up something from zero. In reality, the bottom-up approach plified by the Chemoton model described by T. G´ a nti understanding and improved predictability by building up (G´ a nti, 2003). The Chemoton comprises three functional something from zero. Inproblems reality, the the bottom-up approach aanti, 2003). The Chemoton comprises three functional self-reproducing devices. The autocatalytic chemical cycle something reality, bottom-up still has to from solve zero. manyIn before reachingapproach the final (G´ (G´ nti, 2003). The Chemoton comprises three functional something from zero. In reality, the bottom-up approach self-reproducing devices. Thecycle autocatalytic chemical cycle still has to solve many many problems devices. The autocatalytic chemical cycle can be seen as the metabolic and produces precursor still solve before reaching the the final final self-reproducing goal has of ato self-reproducing artificialbefore cell. reaching self-reproducing devices. Thecycle autocatalytic chemical cycle still has to solve many problems problems before reaching the final can be seen as the metabolic and produces precursor goal of a self-reproducing artificial cell. can be seen as the metabolic cycle and molecules needed for reproducing the produces other twoprecursor devices. goal of aa self-reproducing artificial cell. can be seen as the metabolic cycle and produces precursor goal of self-reproducing artificial cell. molecules needed for reproducing the other two devices. In spite of existing mathematical models, which have been molecules needed for the two devices. One precursor is consumed as a monomer by the template molecules needed for reproducing reproducing the other other twotemplate devices. In spite of of for existing mathematical models, which haveNov´ been One precursor issubsystem consumed as aa monomer monomer by the In spite existing mathematical models, have been developed describing artificial cells (G´ awhich nti, 2003; ak One precursor is consumed as by the template In spite of existing mathematical models, which have been polymerization which serves as an information One precursor is consumed as a monomer by the template developed for describing artificial cells (G´ a nti, 2003; Nov´ a k polymerization subsystem which serves as aan an information developed describing cells aanti, Nov´ aak and Tyson,for 2008), thereartificial is a need for(G´ further theoretical subsystem which serves as information developed for2008), describing cells nti, 2003; 2003; Nov´ k polymerization carrier. The second precursor reacts with by-product of polymerization subsystem which serves as aan information and Tyson, thereartificial is aamodeling need for(G´ further theoretical carrier. The second precursor reacts with by-product of and Tyson, 2008), there is need for further theoretical analysis and mathematical for a deepened incarrier. The second precursor reacts with a by-product of and Tyson, 2008), there is a need for further theoretical the polymerization cycle into a membrane molecule. The carrier. The second precursor reacts with a by-product of analysis andfunctional mathematical modeling for aa Various deepened in- the polymerization cycle into a membrane molecule. The analysis and mathematical for deepened insights into devicemodeling interactions. qualpolymerization cycle into The analysis andfunctional mathematical for a Various deepened in- the membrane represents a container and growsmolecule. proportional the polymerization cycle into aa membrane membrane molecule. The sights into devicemodeling interactions. qualmembrane represents athe container and grows grows proportional sights into functional device interactions. Various qualitative artificial cell models have been proposed during membrane represents a container and proportional sights into functional device interactions. Various qualto the production of polymerization by-product unmembrane represents a container and grows proportional itative artificial cell models have been proposed during to the the production production of of the the polymerization polymerization by-product by-product unitative have been proposed the lastartificial decades cell (G´ amodels nti, 2003; Nov´ ak and Tyson,during 2008; to unitative artificial cell models have been proposed during the last last decades (G´ (G´ nti, 2003; 2003; Nov´ k and and Tyson, 2008; 2008; to the production of the polymerization by-product unthe aa aa the last decades decades (G´ anti, nti, 2003; Nov´ Nov´ ak k and Tyson, Tyson, 2008; Copyright © 2015, 2015,IFAC IFAC (International Federation of Automatic Control) 447Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © Copyright © 2015, IFAC 447 Copyright © 2015, IFAC 447 Peer review under responsibility of International Federation of Automatic Copyright © 2015, IFAC 447Control. 10.1016/j.ifacol.2015.05.018

MATHMOD 2015 448 February 18 - 20, 2015. Vienna, Austria Eugenia Schneider et al. / IFAC-PapersOnLine 48-1 (2015) 447–452

til the volume of the Chemoton is duplicated. Then the division of the protocell occurs spontaneously. It was shown by the simulation studies of Csendes (1984) that the Chemoton model displays self-sustained oscillations. This behavior could be verified by our own simulations (data not shown). However, if we wanted to reduce the complexity of the model structure these oscillations were not necessarily synchronized with cell growth, i.e. the time needed to double the total amount of material in the cell is not always an integer multiple of the oscillation’s period. The task of this work is to find a model structure with less complexity in comparison to the Chemoton and with an inherent mechanism guaranteeing the synchronization of container, metabolism and program for a wide range of kinetic parameter values. The minimal cascade model for the mitotic oscillator described by Goldbeter (1991) (Fig.1) provides such a model structure, which we have used as a starting point for the design of a new artificial cell model.

Fig. 1. Minimal cascade model for mitotic oscillations. (Goldbeter, 1991) 2. ARTIFICIAL CELL MODEL

Fig. 2. Artificial cell model based on Goldbeter’s minimal cascade including the functional devices of G´ anti’s Chemoton. System S1 represents the mother cell including the metabolism A, the polymer template Pt , the new growing polymer P , catalytically inactive membrane building blocks M ∗ and catalytically active membrane building blocks M . System S2 represents the daughter cell arising from degraded mother cell components. The daughter cell contains initially the metabolism component Ad , the polymer template Pt,d and the membrane composed of inactive membrane building blocks Md∗ . Arrows symbolize catalysis (→) and induction (⇢). of the polymer. This interaction forces a synchronization between metabolism and polymerization. In addition, the membrane subsystem serves as a self-replicating unit (Fig.3). First an inactive membrane building block becomes active catalyzed by a polymer. Then a second active membrane building block binds the first one and induces its duplication. After the copying process the inducing M molecule is free and able to induce another replication cycle. The newly generated M molecules become inactive during the separation of the daughter cell.

By combining the functional devices as proposed in the Chemoton model with the less complex structure of Goldbeter’s minimal cascade we developed a model of a selfreproducible artificial cell. Fig.2 shows a scheme of this model representing the mother cell (S1 ) and the daughter cell (S2 ). Three main functional devices form the mother cell: the metabolism (A), the polymerization subsystem (Pt , P ) and the membrane (M ∗ , M ). The activity of metabolism A is induced by the polymer template Pt with a constant rate vA and hence slows down towards the end of the polymer replication when there is hardly Pt in the cell. A catalyzes the polymer replication until Pt is completely consumed. The replicated polymer splits into the polymer template Pt and the new polymer Pt,d . During the polymerization the new growing polymer P catalyzes the conversion of catalytically inactive membrane building blocks M ∗ into the active state M . M triggers the transformation of A, P and M to daughter cell components Ad , Pt,d , Md∗ with constant degradation rates vdA , vdP and vdM . This initiates the cell division in our model. The proposed structure establishes a close interaction between the activation of the metabolism and the replication 448

Fig. 3. Self-replication of membrane building blocks. In the first instance, we assume that newly generated membrane building blocks M remain inside the mother cell, so the volume changes of the mother cell are neglected. 2.1 Model equations The proposed model system possesses three different operation modes, which may be represented as a Petri net with three nodes, as illustrated in Fig.4. First, we assume that the total quantity of the polymer template Pt and the newly synthesized polymer P is Pt +

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system as being in the P + Pt deficit state (Fig.4). In this state, it is assumed that the cell does not produce Pt,d for the daughter cell, but instead replenishes its own stock of P and Pt . Then (7) is replaced by: P

M vdP

2 Pt

(17)

(11) and (15) are replaced by:

Fig. 4. Representation of the protocell’s three operation modes as a Petri net. P = Ptot and the total concentration of inactive and active membrane building blocks amounts to M ∗ + M = Mtot . We call this case normal condition (Fig.4). Under normal conditions, the following chemical reactions take place inside the protocell: Pt vA

Pt P M∗

M A P M

A k1 k2 P k3 k4 M vdA M vdP M vdM

A

(1)

P

(2)

Pt

(3)

M

(4) ∗

(5)

Ad

(6)

M

Pt + Pt,d

M∗ + M∗d

k2 P dPt k1 APt + =− + 2vdP M P dt Km1 + Pt Km2 + P dPt,d = 0.0 dt

= v A Pt −

vdA M A KmdA + A k2 P k1 APt − = − vdP M P Km1 + Pt Km2 + P k2 P k1 APt + =− + vdP M P Km1 + Pt Km2 + P k4 M k3 P M ∗ − = − vdM M 2 ∗ Km3 + M Km4 + M k4 M k3 P M ∗ + =− + vdM M 2 ∗ Km3 + M Km4 + M vdA M A = KmdA + A = vdP M P = vdM M 2

(19)

For the case of membrane building blocks deficit (Fig.4 M + M ∗ deficit) we use the following equation instead of (8): M

M

vdM

2 M∗

(20)

This modifies (13) and (16) in the following way: k4 M k3 P M ∗ dM ∗ + =− + 2vdM M 2 dt Km3 + M ∗ Km4 + M dMd∗ = 0.0 dt

(21) (22)

The artificial cell model is solved with the parameter set listed in Table1. Table 1. Initial conditions and parameter set. Initial condition/parameter A0 Pt0 , M0∗ P0 , M0 , Ad0 , Pt,d0 , Md∗ 0 Ptot , Mtot vA KmdA Km 1 , K m 2 , K m 3 , K m 4 k1 k2 k3 k4 vdA , vdP , vdM

(7) (8)

Based on (1)-(8) for the normal condition case mass balances lead to the following set of differential equations: dA dt dP dt dPt dt dM dt dM ∗ dt dAd dt dPt,d dt dMd∗ dt

(18)

(9) (10)

Value 0.5 2.0 0.0 2.0 0.025 0.002 0.001 3.0 1.5 1.0 0.5 0.02

(11)

The system is implemented in ProMoT/DIANA (Ginkel et al. (2003); Mangold et al. (2014)).

(12)

2.2 Results

(13) (14) (15) (16)

From (10) and (11), or (12) and (13), respectively, it is obvious, that the total amount of P + Pt and the total amount of M + M ∗ are constant in the normal condition case. For the total amount of Pt + P < Ptot we consider the

449

By using normal conditions (Table 1), i. e. Pt + P = Ptot and M ∗ + M = Mtot , the system shows an autonomous oscillatory behavior with a stable limit cycle (Fig.5).

The growing polymer P increases when the activity of metabolism A increases, and only then the catalytically active membrane building blocks M increase as well. At high values of M the levels of A and P begin to decrease caused by conversion into daughter cell components catalyzed by M . The catalytic activity of M decreases during the degradation of the components Ad , Pt,d and Md∗ , i. e. during the separation of the daughter cell. As soon as the replicated polymer is split catalyzed by M the metabolism becomes active again and a new cycle starts.

MATHMOD 2015 450 February 18 - 20, 2015. Vienna, Austria Eugenia Schneider et al. / IFAC-PapersOnLine 48-1 (2015) 447–452

The self-reproduction study shows that the quantities of conversed materials are sufficient to initiate a new cell cycle of daughter cell after a certain phase of adaptation to new conditions (Fig.6). Based on the determined initial conditions for the daughter cell we have the case of Pd + Pt,d deficit and Md + Md∗ deficit, as a result of which the polymerization subsystem proliferates firstly until the condition Pd + Pt,d = Ptot is reached. Then the system merges to the proliferation of the membrane subsystem until it achieves the condition Md + Md∗ = Mtot . Only then the case of normal condition persists about after 200s and we can see the same dynamic behavior of the daughter cell as of the mother cell. Consequently, the model offers a robust self-replicating state.

Fig. 5. Simulation results of the mother cell components. Dynamic behavior of mother cell components A, P , M (a); corresponding parts of the polymerization subsystem P , Pt (b); inactive M ∗ and active membrane building blocks M (c); stable limit cycles (d,e). 2.3 Self-reproduction To test the system with regard to the self-reproduction we look at the degraded daughter cell components. We consider the mother cell as being able of self-reproduction if the degraded amounts of Ad , Pt,d , Md∗ at the time point of daughter cell separation are sufficient to start a new cell cycle. It is assumed that the cell division occurs, when the amount of M falls below a certain threshold due to its inactivation during the separation of the daughter cell at a time point tdiv . We determine the amounts of Ad , Pt,d , Md∗ at the time point of the first separation tdiv = 16.6s and define them as the initial conditions of the daughter cell cycle (Table 2). Table 2. Initial conditions for the daughter cell cycle determined from degraded materials Ad , Pt,d , Md∗ at time point tdiv = 16.6s. Initial condition A d0 P d0 Pt,d0 Md 0 Md∗ 0

Value 0.4629 0.0 0.6275 0.0 0.8671

450

Fig. 6. Dynamic behavior of daughter cell’s metabolism Ad , growing polymer Pd and catalytically active membrane building blocks Md in the deficit case. 2.4 Robustness with respect to parameter changes In addition to self-reproduction we can show that the model is robust with respect to parameter uncertainties. By changing the parameters by about ±20% compared to the case of normal condition the system indicates the same oscillating behavior (Fig.7). It is obvious that for parameter changes of −20% the period of the cycle is extended while for a +20% increase of the parameter values, the period becomes shorter, i.e. the cell divides faster. 3. CONSIDERATION OF VOLUME CHANGES So far, we have assumed that the cell volume is constant throughout the complete cell cycle. However, in reality the protocell’s volume will increase with its mass. To account A dV P dV M dV , , are for this effect, dilution terms V dt V dt V dt added to the model equations in the following. We assume that the volume changes are proportional to the change of the membrane building blocks M caused by insertion of the newly generated membrane building blocks M into the membrane. We consider the case of normal condition, so we examine the system of three differential equations (23)-(25) by assuming Pt = Ptot − P , M ∗ = Mtot − M and by using (26), (27).

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k3 P (Mtot − M ) k4 M dM = − − vdM M 2 dt Km3 + (Mtot − M ) Km4 + M M dV − (25) V dt

The volume V is determined from the affine equation (28). Thereby the factor σ defines the space captured by one membrane molecule M . 1 dV V0 σrM = (26) V dt V + V0 σM k3 P (Mtot − M ) k4 M rM = − − vdM M 2 (27) Km3 + (Mtot − M ) Km4 + M (28) V = V0 + V0 σM

The initial conditions and parameter set for the implementation of the volume expansion model are listed in Table 3. Table 3. Initial conditions and parameter set for the model including volume expansion. Initial condition/parameter A0 P0 M0 V0 vA KmdA Km 1 , K m 2 Km 3 , K m 4 k1 k2 k3 , k4 vdA vdP , vdM σ

Fig. 7. Parameter uncertainties study by changing the parameters by about ±20%. Parameter changes with respect to metabolist A (a), growing polymer P (b), active membrane building blocks M (c). vdA M A A dV dA = vA (Ptot − P ) − − dt KmdA + A V dt k1 A(Ptot − P ) k2 P dP = − − vdP M P dt Km1 + (Ptot − P ) Km2 + P P dV − V dt

(23)

(24)

451

Value 0.1402 0.1181 0.1609 100.0 0.025 0.00002 0.001 0.00011 4.5 0.395 0.6 0.02 0.002 0.5

The extended system shows stable periodic oscillations in the intracellular components of the mother cell in spite of the dilution due to the volume increase (Fig.8). At the maximal concentration of membrane building blocks M the volume reaches the maximal value as well, and it decreases during the separation of the daughter cell, when M decreases. An adjustment of the parameters compared to section 2 is needed, because the region of existence of periodic solutions is shifted in the extended model. The parameter adjustment causes a shorter cell cycle time compared to the model without volume expansion. However, periodic solutions still occur for wide parameter ranges (Fig.9). This means that the property of robustness is preserved in the extended model. 4. CONCLUSION The model shown in this work describes an artificial selfreproducible cell cycle, which is robust with respect to parameter uncertainties. Furthermore we considered the synchronization problem of intracellular changes with the volume expansion. It is found that the synchronization inside the cell is strongly dependent on the parameter set and the space proportionality factor σ. A further question, that we want to study in the future, is how the Chemoton model of T. G´anti (G´anti, 2003) can be

MATHMOD 2015 February 18 - 20, 2015. Vienna, Austria Eugenia Schneider et al. / IFAC-PapersOnLine 48-1 (2015) 447–452 452

polymerization subsystem similar to the Chemoton. In this way we expect to gradually approach these two models in order to better understand the synchronization mechanism of the Chemoton model. REFERENCES

Fig. 8. Simulation results of the model taking into account of volume changes. Dynamic behavior of mother cell devices: metabolism A, growing polymer P , newly generated catalytically active membrane building blocks M (a). Volume changes (b).

Fig. 9. Volume changes of the extended model with respect to parameter changes by about ±20%.

modified and reduced to accomplish similar synchronization of the concentration changes with the volume change as the model described in this work. In addition we want to expand the model described here with respect to the 452

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