Cooperation and cheating in exoenzyme production by microorganisms: Theoretical analysis in view of biotechnological applications Stefan Schuster, Jan-Ulrich Kreft, Naama Brenner, Frank Wessely, Günter Theißen, Eytan Ruppin, Anja Schroeter

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Stefan Schuster, Jan-Ulrich Kreft, Naama Brenner, Frank Wessely, Günter Theißen, et al.. Co- operation and cheating in exoenzyme production by microorganisms: Theoretical analysis in view of biotechnological applications. Biotechnology Journal, Wiley-VCH Verlag, 2010, 5 (7), pp.751. ￿10.1002/biot.200900303￿. ￿hal-00552342￿

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Cooperation and cheating in exoenzyme production by microorganisms: Theoretical analysis in view of biotechnological applications For Peer Review

Journal: Biotechnology Journal

Manuscript ID: biot.200900303.R1

Wiley - Manuscript type: Research Article

Date Submitted by the 25-Apr-2010 Author:

Complete List of Authors: Schuster, Stefan; Friedrich-Schiller-Universität Jena Kreft, Jan-Ulrich; University of Birmingham, Centre for Systems Biology, School of Biosciences Brenner, Naama; Technion – Israel Institute of Technology, Dept. of Chemical Engineering Wessely, Frank; Friedrich Schiller University of Jena, Dept. of Bioinformatics Theißen, Günter; Friedrich Schiller University of Jena, Dept. of Genetics Ruppin, Eytan; Tel Aviv University, School of Computer Sciences & School of Medicine Schroeter, Anja; Friedrich Schiller University of Jena, Dept. of Bioinformatics

Primary Keywords: Biochemical Engineering

Secondary Keywords: Systems Biology

extracellular enzymes, evolutionary game theory, optimization of Keywords: enzyme synthesis

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1 1 2 3 4 5 biot.200900303 6 7 Research Article 8 9 10 Cooperation and cheating in exoenzyme production by 11 12 microorganisms – Theoretical analysis in view of 13 14 biotechnological applications 15 16 17 1 2 3 1 4 18 Stefan Schuster , Jan-Ulrich Kreft , Naama Brenner , Frank Wessely , Günter Theißen , 19 Eytan Ruppin 5, and Anja Schroeter 1,* 20 For Peer Review 21 22 1Dept. of Bioinformatics, School of Biology and Pharmaceutics, Ernst-Abbe-Platz 2, 23 24 Friedrich Schiller University of Jena, D-07743 Jena, Germany 25 26 e-mail: [email protected] , [email protected] , [email protected] 27

28 2 29 Centre for Systems Biology, School of Biosciences, University of Birmingham, 30 31 Edgbaston, Birmingham B15 2TT, UK 32 e-mail: [email protected] 33 34 35 3Dept. of Chemical Engineering, 36 37 Technion – Israel Institute of Technology, Technion City, Haifa 32000, Israel 38 39 e-mail: [email protected] 40

41 4 42 Dept. of Genetics, Philosophenweg 12, 43 44 Friedrich Schiller University of Jena, D-07743 Jena, Germany 45 e-mail: [email protected] 46 47 48 5School of Computer Sciences & School of Medicine, 49 50 Tel Aviv University, Tel Aviv 69978, Israel 51 52 e-mail: [email protected] 53 54 55 Key words: evolution of cooperation, evolutionary game theory, extracellular enzymes, 56 57 Prisoner’s Dilemma, snowdrift game 58 59 * 60 Correspondence

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2 1 2 3 4 5 Dept. of Bioinformatics, School of Biology and Pharmaceutics, Ernst-Abbe-Platz 2, 6 Friedrich Schiller University of Jena, D-07743 Jena, Germany 7 8 e-mail: [email protected] , Phone: +49 3641 949583, Fax: +49 3641 946452 9 10 11 12 Abstract 13 The engineering of microorganisms to produce a variety of extracellular enzymes, for 14 15 example for producing renewable fuels and in biodegradation of xenobiotics, has 16 17 recently attracted increasing interest. Productivity is often reduced by "cheater" mutants, 18 19 which are deficient in exoenzyme production and benefitting from the product provided 20 For Peer Review 21 by the "cooperating" cells. We present a game-theoretical model to analyze population 22 structure and exoenzyme productivity in terms of biotechnologically relevant 23 24 parameters. For any given population density, three distinct regimes are predicted: when 25 26 the metabolic effort for exoenzyme production and secretion is low, all cells cooperate; 27 28 at intermediate metabolic costs, cooperators and cheaters coexist, while at high costs, all 29 cells use the cheating strategy. These regimes correspond to the harmony game, 30 31 snowdrift game, and Prisoner’s Dilemma, respectively. Thus, our results indicate that 32 33 microbial strains engineered for exoenzyme production will not, under appropriate 34 35 conditions, be outcompeted by cheater mutants. We also analyze the dependence of the 36 population structure on cell density. At low costs, the fraction of cooperating cells 37 38 increases with decreasing cell density and reaches unity at a critical threshold. Our 39 40 model provides an estimate of the cell density maximizing exoenzyme production. 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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3 1 2 3 4 5 1 Introduction 6 7 Many biotechnologically relevant processes involve the action of extracellular enzymes. 8 An example of high commercial interest nowadays is biofuel production. It often 9 10 involves the exoenzyme, cellulase, for saccharification [1, 2]. Extracellular bacterial 11 12 cellulases, xylanases and amylases are important in digestive processes in many habitats 13 14 such as the rumen [3]. Saccharomyces cerevisiae secretes, among others, acid 15 phosphatase [4] and invertase [5]. Further examples are fungal lignolytic enzymes such 16 17 as lignin peroxidase, laccase, and manganese peroxidase [6]. Extracellular enzymes play 18 19 an important role in biodegradation of xenobiotics and, thus, in bioremediation of 20 For Peer Review 21 polluted areas. For example, lignolytic enzymes are also active in the breakdown of 22 23 toxic polycyclic aromatic hydrocarbons [6]. 24 The total production of extracellular enzymes by a population of microorganisms is 25 26 often diminished by the existence of a subpopulation of non-producing cells [3, 7]. 27 28 These cells do not invest the metabolic costs of producing and secreting these enzymes 29 30 while nevertheless benefitting from the substrates released by the action of the 31 exoenzymes generated by other cells (Fig. 1). An illustrative example is provided by 32 33 yeast invertase, encoded by the SUC genes. Some strains of S. cerevisiae carry a non- 34 35 functional SUC2 as the only SUC gene [8]. Moreover, the species S. italicus does not 36 37 harbour any SUC gene [9] while it can benefit from the etxracellular glucose generated 38 39 by the invertase secreted by other Saccharomyces species. Greig and Travisano [10] 40 studied wild-type cells of S. cerevisiae and cells in which the gene SUC2 had been 41 42 deleted. They observed coexistence between both cell types (although the 43 44 subpopulations were generated artificially in that case). Another example is provided by 45 46 Candida albicans. The activity of extracellular enzymes such as proteinase and 47 phospholipase differs significantly among three different genotypic strains [11, 12] and 48 49 among karyotypes of that fungus [13]. 50 51 Secretion of exoenzymes and partial or complete failure to do so can be regarded as 52 53 different strategies of cells in an evolutionary “game” on fitness (growth rates). The 54 55 strategies correspond to genetic or epigenetic states, and switches between them can 56 occur, for example, by mutations or gene silencing, respectively. Evolutionary game 57 58 theory [14 −18] can be used to determine equilibrium states of the population (i.e. 59 60 mixtures of strategies) when the fitness of an organism depends not only on its own

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4 1 2 3 4 5 strategy but also on the strategies of others. That theory has been used for analyzing 6 many properties of living organisms such as sex ratios [14]), the evolution of 7 8 cooperation [16, 18-20], biofilms [21], and the selection of biochemical pathways [22, 9 10 23]. Moreover, it can be used for classification in data analysis [24]. Game theory is 11 12 also a well-suited tool for analyzing and optimizing biotechnological setups in which 13 the balance between competition and cooperation determines productivity. Here we 14 15 present a mathematical framework for modelling exoenzyme production based on 16 17 evolutionary game theory. We analyze the difference in fitness of cooperating and 18 19 cheating cells in dependence on total cell density and on the costs of enzyme production 20 For Peer Review 21 and secretion. 22 In game theory, a number of typical games can be distinguished. Probably the most 23 24 famous is the Prisoner’s Dilemma [25]. In the traditional Prisoner’s Dilemma, defection 25 26 (cheating) is the only stable strategy. Thus, much work has been done on how 27 28 cooperation can evolve in a Prisoner’s Dilemma setting, for example, by stochastic or 29 spatial effects or iteration of the game [15, 16, 26, 27]. Another game is the snowdrift 30 31 game, also known as game of chicken or hawk-dove game [14, 15, 18, 20]. The name 32 33 snowdrift originates from a cover story in which two car drivers got stuck in a blizzard, 34 35 and shoveling snow by one of them is sufficient for both to move on. There, cooperation 36 occurs naturally because that game leads to two Nash equilibria, in which one player 37 38 cooperates and the other one defects. Nash equilibria are situations where none of the 39 40 players would benefit from switching strategy unilaterally [15, 28]. In the snowdrift 41 42 game, the payoff for the cooperating player would be reduced when switching to 43 44 defection because this player would then – in the cover story – remain stuck in the 45 snowdrift. An example from everyday life is the situation where two people want to 46 47 pass a narrow door. In the two Nash equilibria, one person is going first while the other 48 49 is waiting a moment. In a population, the two equilibria in the snowdrift game imply 50 51 that defectors and cooperators can coexist. Another relevant game is the harmony game 52 [23, 26], also called mutually beneficial game, in which the only stable solution is 53 54 where all players cooperate. 55 56 An advantage of using game theory rather than traditional dynamic simulation or 57 58 other techniques is that the complicated process of finding the stable solution is not 59 60 considered. For example, it usually takes some communication and, thus, some time,

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5 1 2 3 4 5 until the players agree upon the Nash equilibrium, as we know from the situation of 6 passing a door. Only the final situation is determined. For the present calculations, we 7 8 use the concept of “evolutionarily stable strategy” (ESS) [14, 15]. It is a generalization 9 10 of the Nash equilibrium to n-player situations with large n. An ESS is a strategy which, 11 12 if chosen by a population of agents (e.g. microbial cells), cannot be outcompeted by any 13 alternative strategy that is initially rare. The ESSs can be attained by different processes, 14 15 notably epigenetic processes or mutations, implying that cells switch between strategies 16 17 and, thus, between subpopulations, or by frequency-dependent selection among 18 19 subpopulations, implying varying growth rates. For our calculations, the exact nature of 20 For Peer Review 21 these processes is not relevant. In the Prisoner’s Dilemma, the only ESS is that all 22 players defect (which is a pure ESS), while in the snowdrift game, the only ESS is that a 23 24 certain subpopulation of cells cooperate and the remaining fraction of cells defect (a 25 26 mixed ESS) [15]. In the harmony game, the only ESS is that all cells use the cooperative 27 28 strategy. 29 In the experiment by Greig and Travisano [10], cheaters showed a higher or lower 30 31 fitness (growth rate) than cooperators when total cell density was high or low, 32 33 respectively (starting with equal fractions). The conclusion drawn by Greig and 34 35 Travisano [10] that this would correspond to a Prisoner’s Dilemma has been questioned 36 [17, 21], mentioning that it could rather be a snowdrift game (see also [27, 29]). As 37 38 mentioned above, stable cooperation arises naturally in the snowdrift game. Gore et al. 39 40 [30] showed by experiments (again with baker’s yeast) that a stable coexistence 41 42 between invertase-secreting and non-secreting yeast cells can be established even in 43 44 well-mixed cultures. They concluded that generally a snowdrift game applies and 45 supported this by a phenomenological mathematical model, which implies Prisoner’s 46 47 Dilemma, snowdrift, or harmony games depending on parameter values. 48 49 Identifying the correct game appropriate for describing a particular biological 50 51 situation is important because, as explained before, different games result in different 52 stable population structures. For biotechnological applications, it is of interest to 53 54 manipulate the system such that this population structure induces a maximal level of 55 56 exoenzyme production. To this end, we here present a mathematical model. It is based 57 58 on the Monod kinetics, which is widely used for quantifying growth in microbiology. 59 60 Moreover, our model takes into account the dependence on total cell density, a

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6 1 2 3 4 5 parameter that is crucial for optimizing the efficiency of exoenzyme production. This 6 enables us to describe mathematically the experimentally observed dependence of the 7 8 relative fitness of cheaters on total cell density in Greig and Travisano [10]. 9 10 The presented approach has a wide range of potential applications. This includes the 11 12 economically important production of renewable biofuels such as ethanol from non- 13 food sources. For this purpose, various microorganisms can be engineered to secrete 14 15 exoenzymes for degrading celluloses and hemicelluloses [1]. 16 17 18 19 2 Methods 20 For Peer Review 21 In accordance with the experiment by Greig and Travisano [10] and earlier modelling 22 23 studies [3, 7], we here consider two types of cells only: cooperating cells secreting a 24 given quantity of exoenzyme and complete defectors not secreting any exoenzyme at 25 26 all. Throughout, the term “enzyme” refers to the extracellular enzyme under study. The 27 28 product of that enzyme is often a growth substrate for the cell, for example, glucose in 29 30 the case of invertase and cellobiase (a special cellulase cleaving cellobiose). Therefore, 31 we make a distinction between “initial substrate” (of the enzyme) and “growth 32 33 substrate”. The latter may then be taken up into the cell and act as a substrate for further 34 35 biotransformations. We consider exponentially growing populations. The basic growth 36 37 rate is denoted by 0. This is the growth rate when the desirable growth substrate under 38 39 study is absent, while alternative substrates may be present, and it includes the death 40 rate of cells if this is non-negligible. The concentration of the desirable growth substrate 41 42 at the surface of a given cell is denoted by S. To quantify the dependence of the growth 43 44 rate on S, we use the Monod equation [31]: 45 46 47 ' ⋅S 48 '()S = max (1) 49 K + S 50 51 52 53 where the prime refers to the surplus in growth rate above 0 due to the desirable 54 55 external growth substrate. Note that 0 can be described by Monod kinetics in terms of 56 alternative substrates, but this dependence is irrelevant for the model because we 57 58 assume that alternative substrates (if any) have constant concentrations. K and 'max 59 60 denote the half-saturation constant and maximum increase in growth rate at saturating

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7 1 2 3 4 5 growth substrate levels, respectively. Note that this function is monotonically increasing 6 and concave. 7 8 Since most of the enzyme acts in the vicinity of the cell (e.g. in the periplasmic 9 10 space if any, Fig. 1), part of the growth substrate generated by it diffuses straight into 11 12 the enzyme-secreting cell. This can be regarded as a “privileged share” of the growth 13 substrate available to cooperating cells, even in the case of well mixed systems. 14 15 Moreover, in the case of spatially heterogeneous systems, cooperating cells have access 16 17 to an additional bonus due to spatial gradients, which imply higher growth substrate 18 19 levels around them (Fig. 1). This is another important “privileged share” factor in 20 For Peer Review 21 compensating for the costs of cooperation. The overall effect of the privileged shares for 22 a cooperating cell is an increase in concentration of growth substrate, here denoted by δ. 23 24 This quantity depends, in a monotonically increasing way, on the level of the initial 25 26 substrate and is zero if none is available. For simplicity’s sake, in our model, we assume 27 28 that the initial substrate and, thus, δ are constant. The variable x stands for the frequency 29 30 (fraction) of cooperating cells and ρ for total cell density. Both S and ρ relate to volume 31 32 in three-dimensional media (e.g., batch cultures) and to area in two-dimensional media 33 (e.g., agar plates). The total cost of enzyme production and secretion is counted by a 34 35 reduction, c, in growth rate. The growth rates of cooperating cells, C, and of defector 36 37 cells, D, can then be expressed as 38 39 40 41 C = 0 − c + '(qρ x + δ ) (2) 42 43 44 45 D = 0 + '(qρ x) (3) 46 47 48 49 where q is a proportionality constant relating the cell density of cooperators to the 50 growth substrate level. Like δ, q is considered constant throughout. 51 52 53 54 3 Results 55 56 3.1 Initial relative fitness 57 58 Using Eqs. (2) and (3), the relative fitness of defectors can be calculated as 59 60

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8 1 2 3 4 + '(qρ x) 5 D = 0 . (4) 6 C 0 − c + '()qρ x +δ 7 8 9 10 Together with Eq. (1), this equation gives the dependence of the relative fitness on cell 11 12 density. In Fig. 2 (note the logarithmic scale), that nonlinear dependence is plotted for 13 x = 0.5, in accordance with the initial conditions in the experiment of Greig and 14 15 Travisano [10]. They empirically fitted the data to a straight line. The curve plotted in 16 17 Fig. 2 is in agreement with the asymptotic behavior for very low and very high cell 18 19 densities. In the extreme cases, the curve tends to asymptotic values, which can be 20 For Peer Review 21 explained as follows. Defectors in isolation grow at a constant rate. At extreme 22 crowding, the enzyme concentration is in excess and the uptake of the growth substrate 23 24 is saturated, so that the growth rate is constant again. The nonlinear curve generated by 25 26 the model shows that the advantage of defectors and cooperators at high and low total 27 28 densities, respectively, can be readily computed and, thus, explained on theoretical 29 grounds. A quantitative comparison with the data of Greig and Travisano [10] will be 30 31 published elsewhere. 32 33 34 35 3.2 Equilibrium frequencies 36 Next , we calculate the frequency x* that is finally attained in the ESS. At equilibrium 37 38 (i.e., in an ESS), the frequency-dependent growth rates of cooperators and defectors are 39 40 equal. Thus, Eqs. (2) and (3) yield 41 42 43 44 0 – c + '(S *+δ ) = 0 + '(S *) (5) 45 46 with 47 48 S* = q ρ x* (6) 49 50 51 '(S *+δ ) – c = '(S *) (7) 52 53 54 55 where the asterisk refers to the situation in the ESS. Provided that δ is small, (S *+δ ) 56 57 can be expanded into a Taylor series up to the first-order term. Using eq. (7), this gives 58 59 60

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9 1 2 3 4 d'(S) 5 '(S *+δ ) = '(S *) + δ (8) 6 dS S* 7 8 9 10 d'(S) K = ' , (9) 11 dS max 2 12 S* ()K + qρ x * 13 14 15 Together with Eq. (7), we derive that the r.h.s. of Eq. (9) must equal c/δ. This leads to 16 17 the equilibrium frequency of cooperators: 18 19 20 For Peer Review 21 'max K δ c− K 22 x* = (10) 23 qρ 24 25 26 Interestingly, this equilibrium frequency does not depend on the basic growth rate, , 27 0 28 because it is the same for cooperators and cheaters. However, if 0 is very large in 29 30 comparison to ’, the computation of the equilibrium frequency of cooperators becomes 31 32 less robust because the difference in growth rates between cooperators and cheaters 33 34 does not exceed “noise”. 35 36 In case Eq. (10) gives values of x* lower than zero or larger than one, Eq. (5) cannot 37 be fulfilled. If the growth rate of cooperators (left-hand side of Eq. (5)) is larger than 38 39 that of the defectors (right-hand side), x tends to one, while in the opposite case it tends 40 41 to zero. Thus, the equilibrium frequencies x* in these two cases equal one and zero, 42 43 respectively. Accordingly, we can distinguish the following three cases (see Fig. 3). 44 45 46 Low-benefit case (compared to costs) : 47 48 The condition relevant in that case follows from Eq. (10) by assuming x* ≤ 0. This 49 50 leads to 51 52 53 c ≥ ' δ K = c . (11) 54 max 2 55 56 57 Pure defection is the only ESS in this case: x* = 0; the cells are trapped in a Prisoner’s 58 59 Dilemma (black lower plane in Fig. 3). The population can then survive only if 60 additional factors intervene, such as the presence of alternative growth substrates or

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10 1 2 3 4 5 intracellular enzyme forming growth substrate. Otherwise, the basic growth rate 0 6 would be negative, implying that the population, which then consists of cheaters only, 7 8 would become extinct. This fatal situation may no longer be observed in nature just 9 10 because the populations became extinct [32]. However, in biotechnological setups, it 11 12 can indeed occur, but should, of course, be avoided. 13 14 15 High-benefit case : 16 17 From Eq. (10), it follows that the frequency x* equals one if population density is lower 18 19 than the critical density threshold 20 For Peer Review 21 22 ' Kδ / c − K 23 ρ '' = max (12) 24 q 25 26 27 28 At these low densities there is effectively no interaction between cells and therefore the 29 30 cooperator cells take over the population by virtue of their higher growth rate. In terms 31 of the costs, the condition relevant in that case reads 32 33 34 35 'max Kδ 36 c ≤ 2 = c1 . (13) 37 ()qρ + K 38 39 40 Below the critical density ρ” given by Eq. (12), the cheater frequency tends to zero. In 41 42 the high-benefit case, the only ESS is pure cooperation: x* = 1 (white plateau in Fig. 3). 43 44 Thus, a harmony game applies. The curve of relation (13) in the equality case can be 45 46 seen by the curved black line on the upper end of the “cliff” in Fig. 3. 47 48 49 Intermediate-benefit case : 50 51 The relevant constraint reads 52 53 54 ' Kδ ' δ 55 c = max < c < max = c . (14) 56 1 ()qρ + K 2 K 2 57 58 59 60

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11 1 2 3 4 5 In this case, x* lies between 0 and 1. The dependence of x* on the costs of enzyme 6 secretion and cell density as given by Eq. (10) is depicted in Fig. 3. From Eq. (10), it 7 8 follows that for large population densities, the frequency of cooperators tends to zero, so 9 10 that practically pure defection occurs. Moreover, note that for low total densities, ρ, the 11 12 lower bound in relation (14) tends to the upper bound. Then, the intermediate-benefit 13 14 case practically disappears, so that the Prisoner’s Dilemma turns into a harmony game, 15 as can be seen in Fig. 3. 16 17 18 19 The critical density, ρ’, at which the cooperating and cheater cells have equal fitness 20 For Peer Review 21 and equal frequency (Fig. 2) can be determined by setting x* equal to ½ in Eq. (10): 22 23 24 ' Kδ / c − K 25 ρ'= 2 max = 2ρ '' (15) 26 q 27 28 29 30 This means that this critical density is twice as high as the critical threshold at which the 31 32 harmony game turns into a snowdrift game. It is worth noting that both Eqs. (12) and 33 34 (15) have been derived on the basis of a Taylor expansion. In Fig. 2, the more precise 35 36 value of ρ’ without using that approximation is shown. 37 38 39 3.3 Maximum productivity is attained at “the edge of harmony” 40 41 In view of technological applications, it is of great interest to maximize the effectivity 42 43 of exoenzyme production. The optimization problem is non-trivial because an increase 44 45 in total cell density also facilitates the occurrence of cheater cells. The question arises 46 47 whether the goal is to maximize specific productivity or volumetric productivity. Note 48 that the equilibrium density of cooperating cells is the (mathematical) product of total 49 50 equilibrium density and cooperator frequency (Fig. 4). (Equilibrium is here defined with 51 52 respect to frequencies rather than total growth). The volumetric enzyme productivity 53 54 can be assumed to be (nearly) proportional to the equilibrium density of cooperating 55 cells. The curve in Fig. 4 consists of two parts: at low densities, proportionality holds 56 57 because x* = 1, so that ρ x * = ρ. At high densities, the function is a constant, as can be 58 59 seen by multiplying Eq. (10) by ρ. It can be seen in Fig. 4 that the mathematical solution 60 to the optimization problem of maximizing exoenzyme production is not unique. It is

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12 1 2 3 4 5 sufficient to have any equilibrium density above the threshold value ρ”. This ambiguity 6 (degeneracy) of the solution can be resolved by maximizing specific productivity in 7 8 addition. This is achieved by decreasing cell density. Therefore, the best value is 9 10 reached at that critical value, ρ”, that is, at the upper end of the “cliff” in Fig. 3. The 11 12 same solution would be obtained by first maximizing the specific productivity and then 13 14 resolving the resulting ambiguity by maximizing volumetric productivity in addition. 15 16 17 4 Discussion 18 19 We have here presented a game-theoretical model of cooperation and competition 20 For Peer Review 21 among microbial cells upon secretion of exoenzymes and have discussed this from a 22 23 biotechnological perspective. As shown by our theoretical analysis, secretion of 24 25 exoenzymes is not always a Prisoner’s Dilemma. Under a wide range of physiological 26 conditions, it rather is a snowdrift game and can even turn into a harmony game. For the 27 28 special case of yeast invertase, this has been shown both experimentally and 29 30 theoretically by Gore et al. [30]. Notably, a snowdrift game occurs when cell density is 31 32 neither very low nor very high. From our model, it can be deduced that all cells 33 eventually cooperate (harmony game) when the process is run at constant low densities, 34 35 as has indeed been observed [33]. Another testable prediction is that, when cell density 36 37 increases upon growth, a transition from pure cooperation (harmony game) to 38 39 coexistence (snowdrift game) should occur. In summary, our results indicate that 40 41 microbial strains engineered for exoenzyme production will not, under appropriate 42 conditions, be outcompeted by non-producing mutants. Thus, our presentation 43 44 demonstrates that game theory is useful in designing biotechnological setups that are 45 46 robust against “takeover” by cheater mutants. 47 48 In both the snowdrift and harmony games, cooperation is a direct consequence of 49 the payoff structure of the game. Thus, more complex means to escape the Prisoner’s 50 51 Dilemma such as spatial effects are not required, in contrast to, for example, the 52 53 competition between microorganisms that use respiration for ATP production and those 54 55 using fermentation or respirofermentation [22, 34]. 56 57 For cooperation to occur it is crucial that the benefit (payoff) from the public goods 58 produced by a given cooperator exceeds the costs of its production. This is fulfilled if 59 60 each cell obtains a sufficient privileged share of the growth substrate. A privileged

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13 1 2 3 4 5 share, which has been taken into account here by a higher growth substrate 6 concentration, may arise when part of the substrate is retained by cooperating cells, for 7 8 example, when the exoenzyme is bound to the cell wall or to a periplasmic space and/or 9 10 if the medium is not well mixed. The local growth substrate around cooperating cells 11 12 will then be higher than around defecting cells [35]. In the important case of well mixed 13 systems, the privileged share only consists of the part directly retained by cooperating 14 15 cells. Thus, the same model can be applied, with a smaller privileged share, δ. This 16 17 leads to a shift of the boundaries between the three cases (games). In particular, the area 18 19 of the harmony game is smaller than in spatially heterogeneous setups because δ is then 20 For Peer Review 21 smaller, while the area of the Prisoner’s Dilemma is larger. Gore et al. [30] studied this 22 23 case both experimentally and theoretically. They modelled this by an efficiency of 24 capture, ε, and by raising the payoff for defectors and cheaters to an exponent α being a 25 26 phenomenological (not necessarily integer) parameter. Here, in contrast, we use the 27 28 Monod equation (Eq. (1)), an established kinetics in microbiology. 29 30 From the above reasoning about privileged shares (both in well mixed and 31 32 heterogeneous systems), it follows that “cooperators” operate in their selfish interest 33 (rather than for the well-being of the population, let alone by altruism). It is worth 34 35 noting that there are alternative possibilities of advantages for cooperators [35]. For 36 37 example, some secreting bacteria physically attach to a water-insoluble substrate and 38 39 can, thus, escape from predation by protozoa [3, 36]. 40 Upon variation of density, at most two games can occur for a given cost value: at low 41 42 costs, a harmony game or a snowdrift game can be observed, while at high costs, it is 43 44 always a Prisoner’s Dilemma independent of cell density (Fig. 3). Accordingly, in 45 46 biotechnological applications, first the costs of exoenzyme production and secretion 47 should be determined or at least estimated. Only if they are low enough to allow for 48 49 cooperation (snowdrift or harmony games), a further investment is worth being made. 50 51 Thus, enzyme production should not be upregulated to the extent that costs will be too 52 53 high and defecting mutants are likely to outcompete the engineered strain. Moreover, it 54 55 is useful to engineer the enzyme so as to remain attached to the cell, so that cooperators 56 get an advantage. 57 58 Our model is based on a number of simplifying assumptions. The first simplification 59 60 is the restriction to exponentially growing populations. The case of logistic growth

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14 1 2 3 4 5 eventually leading to stationary phases is left to future extensions of the model. The 6 simplifications allow a largely analytical treatment – the nonlinear dependencies of the 7 8 initial relative fitness on total cell density and of the equilibrium frequencies on both 9 10 cell density and the costs can be calculated in closed form. Our results are 11 12 complementary to, and consistent with, the numerical simulations based on a grid model 13 by Allison [7]. Those simulations predicted that cooperators would outcompete 14 15 defectors in the case of high benefit, while intermediate benefit leads to coexistence. 16 17 The assumptions of our model are, in spite of the simplification, justified on biological 18 19 grounds. 20 For Peer Review 21 The physiological advantage of extracellular hydrolysis is obvious for 22 macromolecular substrates such as cellulose because they cannot enter the cell. It is less 23 24 obvious for smaller substrates such as sucrose. Osmotic effects [37], foraging by 25 26 chemotaxis and the lower costs of uptake of glucose and fructose, which is performed 27 28 by facilitated diffusion in contrast to the active sucrose uptake [33] were suggested in 29 the literature. Moreover, intracellular space for proteins is severely limited due to 30 31 macromolecular crowding [38, 39]. Therefore, we hypothesize that secretion of 32 33 extracellular enzymes provides an advantage by relieving intracellular packing. 34 35 Of course, maximizing the volumetric productivity of exoenzyme secretion is of 36 interest for biotechnological applications. Our approach allows one to delineate specific 37 38 production conditions under which the synthesis of a desired exoenzyme would be 39 40 maximized by adjusting total cell density on the basis of estimates of parameters such as 41 42 costs of exoenzyme production/secretion. Our model shows, for the case of sufficiently 43 44 low costs, a degenerate maximum above the density threshold between snowdrift and 45 harmony games. Thus, somewhat counter-intuitively, the case of coexistence between 46 47 cooperators and cheaters allows a higher exoenzyme production than most situations of 48 49 pure cooperation. Since, for economic reasons, it is worth using a low density and, thus, 50 51 maximizing specific productivity in addition, the best value is reached at the “edge of 52 harmony”. This shows again that it is very helpful to analyze the types of “games 53 54 played” by microorganisms upon extracellular conversion of diverse substrates. 55 56 57 58 Support by the German-Israeli Foundation and the DFG (Germany, within the Jena 59 60 School for Microbial Communication) is gratefully acknowledged. The authors thank

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15 1 2 3 4 5 Sabine Hummert, Christoph Kaleta, Steffen Klamt and Thomas Pfeiffer for stimulating 6 discussions and two anonymous reviewers for helpful comments on a previous version 7 8 of the manuscript. 9 10 11 12 5 References 13 14 [1] Lee, S. K., Chou, H., Ham, T. S., Lee, T. S., Keasling, J. D., Metabolic 15 engineering of microorganisms for biofuels production: from bugs to synthetic 16 17 biology to fuels. Curr. Opin. Biotechnol . 2008, 19, 556-563. 18 19 [2] Wilson, D. B., Cellulases and biofuels. Curr. Opin. Biotechnol . 2009, 20, 295- 20 For Peer Review 21 299. 22 23 [3] Modak, T., Pradhan, S., Watve, M., Sociobiology of biodegradation and the role 24 of predatory protozoa in biodegrading communities. J. Biosci. 2007, 32, 775- 25 26 780. 27 28 [4] Arnold, W. N., Lacy, J. S., Permeability of the cell envelope and osmotic 29 30 behavior in Saccharomyces cerevisiae . J. Bacteriol. 1977, 131, 564-571. 31 [5] Carlson, M., Botstein, D., Two differentially regulated mRNAs with different 5' 32 33 ends encode secreted and intracellular forms of yeast invertase. Cell 1982, 28, 34 35 145-154. 36 37 [6] Haritash, A. K., Kaushik, C. P., Biodegradation aspects of polycyclic aromatic 38 39 hydrocarbons (PAHs): a review. J. Hazard. Mater . 2009, 169, 1-15. 40 [7] Allison, S. D., Cheaters, diffusion and nutrients constrain decomposition by 41 42 microbial enzymes in spatially structured environments. Ecol. Lett . 2005, 8, 626- 43 44 635. 45 46 [8] Naumov, G. I., Naumova, E. S., Sancho, E. D., Korhola, M. P., Polymeric SUC 47 genes in natural populations of Saccharomyces cerevisiae. FEMS Microbiol. 48 49 Lett . 1996, 135, 31-35. 50 51 [9] Schaefer, E. J., Cooney, C. L., Production of maltase by wild-type and a 52 53 constitutive mutant of Saccharomyces italicus . Appl. Environ. Microbiol . 1982, 54 55 43, 75-80. 56 [10] Greig, D., Travisano, M., The Prisoner's Dilemma and polymorphism in yeast 57 58 SUC genes. Proc. Biol. Sci. 2004, 271, S25-S26. 59 60

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16 1 2 3 4 5 [11] Sugita, T., Kurosaka, S., Yajitate, M., Sato, H., Nishikawa, A., Extracellular 6 proteinase and phospholipase activity of three genotypic strains of a human 7 8 pathogenic yeast, Candida albicans . Microbiol. Immunol . 2002, 46, 881-883. 9 10 [12] Nawrot, U., Skala, J., Wlodarczyk, K., Fonteyne, P. A., Nolard, N., Nowicka, J., 11 12 Proteolytic activity of clinical Candida albicans isolates in relation to genotype 13 and strain source. Pol. J. Microbiol . 2008, 57, 27-33. 14 15 [13] Tavanti, A., Campa, D., Bertozzi, A., Pardini, G., Naglik, J. R., Barale, R., 16 17 Senesi, S., Candida albicans isolates with different genomic backgrounds 18 19 display a differential response to macrophage infection. Microbes Infect . 2006, 20 For Peer Review 21 8, 791-800. 22 [14] Maynard Smith, J., Evolution and the Theory of Games, Cambridge University 23 24 Press, 1982. 25 26 [15] Hofbauer, J., Sigmund, K., Evolutionary Games and Population Dynamics , 27 28 Cambridge University Press,1998. 29 [16] Nowak, M. A., Sasaki, A., Taylor, C., Fudenberg, D., Emergence of cooperation 30 31 and evolutionary stability in finite populations. Nature 2004, 428, 646-650. 32 33 [17] Schuster, S., Kreft, J. U., Schroeter, A., Pfeiffer, T., Use of game-theoretical 34 35 methods in biochemistry and biophysics. J. Biol. Phys. 2008, 34, 1-17. 36 [18] Perc, M., Szolnoki, A., Coevolutionary games - a mini review. BioSystems 2010, 37 38 99, 109-125. 39 40 [19] Hauert, C., Doebeli, M., Spatial structure often inhibits the evolution of 41 42 cooperation in the snowdrift game. Nature 2004, 428, 643-646. 43 44 [20] Tomassini, M., Pestelacci, E., Luthi, L., Mutual trust and cooperation in the 45 evolutionary hawks-doves game. Biosystems 2010, 99, 50-59. 46 47 [21] Kreft, J. U., Conflicts of interest in biofilms. Biofilms 2004, 1, 265-276. 48 49 [22] Pfeiffer, T., Schuster, S., Game-theoretical approaches to studying the evolution 50 51 of biochemical systems. Trends Biochem. Sci. 2005, 30, 20-25. 52 [23] Aledo, J. C., Perez-Claros, J. A., Esteban del Valle, A., Switching between 53 54 cooperation and competition in the use of extracellular glucose. J. Mol. Evol. 55 56 2007, 65, 328-339. 57 58 [24] Cohen, S., Dror, G,, Ruppin, E., Feature selection via coalitional game theory. 59 60 Neural Comput. 2007, 19, 1939-1961.

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17 1 2 3 4 5 [25] Axelrod, R., The Evolution of Cooperation, Basic Books, 1984. 6 [26] Hauert, C., Effects of space in 2 x 2 games. Intern. J. Bifurc. Chaos 2002, 12, 7 8 1531-1548. 9 10 [27] Doebeli, M., Hauert, C., Models of cooperation based on the Prisoner's Dilemma 11 12 and the snowdrift game. Ecol. Lett . 2005, 8, 748-766. 13 [28] Myerson, R. B., Game Theory: Analysis of Conflict , Harvard University Press, 14 15 1991. 16 17 [29] Hauert, C., Spatial effects in social dilemmas. J. Theor. Biol. 2006, 240, 627- 18 19 636. 20 For Peer Review 21 [30] Gore, J., Youk, H., van Oudenaarden, A., Snowdrift game dynamics and 22 facultative cheating in yeast. Nature 2009, 459, 253-256. 23 24 [31] Kovarova-Kovar, K., Egli, T., Growth kinetics of suspended microbial cells: 25 26 from single-substrate-controlled growth to mixed-substrate kinetics. Microbiol. 27 28 Mol. Biol. Rev. 1998, 62, 646-666. 29 [32] Travisano, M., Velicer, G. J., Strategies of microbial cheater control. Trends 30 31 Microbiol. 2004, 12, 72-78. 32 33 [33] Maclean, R.C., Brandon, C., Stable public goods cooperation and dynamic social 34 35 interactions in yeast. J. Evol. Biol . 2008, 21, 1836-1843. 36 [34] Pfeiffer, T., Schuster, S., Bonhoeffer, S., Cooperation and competition in the 37 38 evolution of ATP-producing pathways. Science 2001, 292, 504-507. 39 40 [35] Velicer, G. J., Social strife in the microbial world. Trends Microbiol. 2003, 11, 41 42 330-337. 43 44 [36] Mattison, R. G., Harayama, S., The predatory soil flagellate Heteromita globosa 45 stimulates toluene biodegradation by a Pseudomonas sp. FEMS Microbiol. Lett . 46 47 2001, 194, 39-45. 48 49 [37] Badotti, F., Batista, A. S., Stambuk, B. U., Sucrose active transport and 50 51 fermentation by Saccharomyces cerevisiae . Braz. Arch. Biol. Technol . 2006, 49 52 Special, 115-123. 53 54 [38] Beg, Q. K., Vazquez, A., Ernst, J., de Menezes, M. A., Bar-Joseph, Z., Barabási, 55 56 A. L., Oltvai, Z. N., Intracellular crowding defines the mode and sequence of 57 58 substrate uptake by Escherichia coli and constrains its metabolic activity. Proc. 59 60 Natl. Acad. Sci. U.S.A. 2007, 104, 12663-12668.

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18 1 2 3 4 5 [39] Slade, K. M., Baker, R., Chua, M., Thompson, N. L., Pielak, G. J., Effects of 6 recombinant protein expression on green fluorescent protein diffusion in 7 8 Escherichia coli . Biochemistry 2009, 48, 5083-5089. 9 10 11 12 13 14 15 16 17 18 19 20 For Peer Review 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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19 1 2 3 4 5 Figure legends 6 7 8 Figure 1. Interplay between microbial cells upon secretion of exoenzymes. Cheater 9 10 cells (denoted by ‘Cheat’) do not produce exoenzymes (incised circles) and benefit from 11 12 the growth substrate (monomers) released by the enzyme secreted by cooperating cells 13 14 (‘Coop’). The initial substrate is indicated by dimers (e.g. sucrose), even though it may 15 consist of polymers such as cellulose. Besides this “desirable” substrate, other substrates 16 17 may be present. A possibly present periplasmic space is represented by the double 18 19 envelope. Note that the concentration of the growth substrate is higher near cooperating 20 For Peer Review 21 cells in this spatially heterogeneous system. 22 23 24 Figure 2. Relative fitness of defectors as a function of total cell density according to 25 26 Eqs. (1) and (4). ρ’, density at which the fitness of defectors and cooperators is equal 27 -1 28 and both have frequency 0.5 in the population. Parameter values: 'max = 0.4 h , 0 = 29 2 30 0.2 h-1, c = 0.1 h-1, K = 2 mM, δ = 1 mM , q = 8 x 10 −9 mM m . This leads to ρ’ = 10 7.97 . 31 32 33 34 Figure 3. 3D plot of the equilibrium frequency of cooperators, x* , vs. the total cell 35 density, ρ , and the cost of cooperation, c. The white plateau corresponds to the case of 36 37 pure cooperation (harmony game, high-benefit case). For c ≥ c2 (black plane), always 38 39 the Prisoner’s Dilemma results irrespective of cell density (low-benefit case, cf. Eq. 40 41 (11)). The area between these planes corresponds to coexistence (Snowdrift game, 42 43 intermediate-benefit case). For c < c2 and large population densities, the frequency of 44 cooperators tends to zero (Eq. (10)), so that practically pure defection occurs. 45 46 Accordingly, the black colour at the bottom of the surface representing the snowdrift 47 48 game is the same as for the Prisoner’s Dilemma game. Parameter values are the same as 49 50 in Fig. 2 except the varying costs. 51 52 53 Figure 4. Plot of the exoenzyme production (assumed to be proportional to equilibrium 54 55 density of cooperators, x*) vs. the total equilibrium cell density, ρ. ρ’’, threshold 56 57 between snowdrift and harmony games (Eq. (12)). For further explanations, see text. 58 8.02 59 Parameter values are as in Fig. 2. This leads to ρ’’ = 10 . 60

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