The Organization and Control of an Evolving Interdependent Population
SSubmitted to the Journal of the Royal Society Interface (2014)
The Organization and Control of an Evolving Interdependent Population
Dervis C. Vural, ∗ Alexander Isakov, † and L. Mahadevan
2, 3, ‡ Department of Physics, University of Notre Dame Department of Physics, Harvard University School of Engineering and Applied Sciences, Department of Organismic and Evolutionary Biology, Harvard University
Starting with Darwin, biologists have asked how populations evolve from a low fitness state thatis evolutionarily stable to a high fitness state that is not. Specifically of interest is the emergenceof cooperation and multicellularity where the fitness of individuals often appears in conflict withthat of the population. Theories of social evolution and evolutionary game theory have produceda number of fruitful results employing two-state two-body frameworks. In this study we departfrom this tradition and instead consider a multi-player, multi-state evolutionary game, in which thefitness of an agent is determined by its relationship to an arbitrary number of other agents. We showthat populations organize themselves in one of four distinct phases of interdependence dependingon one parameter, selection strength. Some of these phases involve the formation of specializedlarge-scale structures. We then describe how the evolution of independence can be manipulatedthrough various external perturbations.
I. INTRODUCTION
Cooperative behavior, as exemplified by multicellularlife, seems to have evolved at least 25 times indepen-dently - once for plants, once or twice for animals, oncefor brown algae, and possibly several times for fungi,slime molds, and red algae [1]. On shorter time-scales,the social composition of eukaryotes such as
S. cerevisiae ,and biofilm forming bacteria such as
P. aeruginosa candramatically change in a brief period [2–5]. In a re-lated context, tumor formation is a rare example of thetransition, taking place in the reverse direction, from amulticellular to an essentially unicellular lifestyle. Inter-estingly, cancer cells end up cooperating by collectivelysecreting angiogenic factors, and it seems possible, atleast in principle, that there may even be cheaters (i.e.those who do not secrete the growth factors) among thiscollection of cooperating cheaters [6, 7].Evolutionary game theory provides excellent insightinto how altruistic and cooperative behavior can emergeto maximize the fitness of the group despite the appar-ent fitness advantage of cheating individuals [8, 9]. Inthe context of evolution of cooperation, these modelstypically investigate the outcome of repeated runs of theprisoner’s dilemma between pairs of agents that have twostrategies, cheating and cooperating. Variants of themodel include structured interactions, coupled popula-tions, coevolution, stored reputation, punishment, andpreferential or random partner choosing [10–18].However, real life is more complicated in a numberof ways. First, many actual games are massively multi-player [19–22]. The fitness of an organism may dependon its simultaneous relationship with multiple players.Second, biology allows for a much larger variety of inter-nal states beyond cooperating or defecting. For example,the genetic makeup of an organism may be suitable for ∗ Corresponding Author; [email protected] † Co-first author ‡ Corresponding Author; [email protected] cooperation with only an exclusive few, while some or-ganisms may be incapable of defecting or cooperating alltogether. Third, real social evolution leads to highly or-ganized dependence structures beyond the homogeneousmixtures or aggregates of cooperator-defector states thatwe often see in the majority of evolutionary game mod-els. From biochemical to societal scales, life organizesitself in highly complex arrangements of cliques, com-munities, cycles and hierarchies.Without compromising the simplicity and tractabil-ity offered by traditional evolutionary game theory, herewe propose an evolutionary model in which the fitnessof an agent is determined, not by the outcome of atwo-player two-state game, but instead a multi-playermulti-state one. Thus, our focus is not the coopera-tor / defector ratio in the population, but rather thelarge-scale structure of all exchanges; i.e. the interde-pendence between agents or groups of agents within agenetically heterogeneous population. We ask how in-dependent agents become interdependent through thesimple laws of evolution, whether positive selection is anecessary or a sufficient condition for the formation of in-terdependence, what kinds of interdependent structuresare stable / unstable, and how these structures and pro-cesses depend on evolutionary parameters. Accordingly,the present model offers a clear framework for classi-fying and categorizing different regimes of interdepen-dence, as well as allowing for careful control of evolu-tionary parameters that may be influencing recent non-intuitive empirical outcomes [23]. We determine whichkinds of external perturbations promote anti-sociality(e.g. in order to eradicate biofilms) and which otherkinds can inhibit anti-sociality (e.g. as to suppress orreverse tumor growth) by simulating the introduction ofselfish/altruistic strains into a population or the admin-istration of anti-sociality/sociality promoting drugs. Weevaluate the success rate of these evolutionary interven-tions as a function of the original population structure,drug dose, fraction of drug-resistant agents, and repro-duction speed of the target species. a r X i v : . [ q - b i o . P E ] N ov û ü Reproduction: Selection: Mutation:
FIG. 1.
Schematic of the model . The evolutionary algo-rithm is carried out by assuming that (1) fitness of a node ω ( x i ) is a monotonically increasing function of edge influx x i = bn i, in − cn i, out , where n i, in and n i, out are the numberof in and out edges for node i . (2) reproduction preserves allin-out relationships with r fittest nodes replacing the leastfittest nodes, and (3) a small number of edges are randomlyadded/removed every generation, with probability p (cid:28) II. MODEL
Our multi-state multi-player game can be best visu-alized as a network of N (cid:29) A to B indicates that A contributes to the fitness of B at the cost of its own.Unlike the typical evolutionary game theoretic modelswhere the state of a player i is binary (cooperator /cheater), here the player states ψ i are characterized byhigh-dimensional vectors, i.e. ψ i = { x , x , . . . , x N } with x j ∈ { , } indicating whether i provides a fitnessbenefit to j . The evolutionary dynamics is governed bythe following assumptions (Fig[1]): (1) The fitness ω ( x i ) of a node i is assumed to be amonotonically increasing function of received net benefit x i = bn i, in − cn i, out , where n i, in and n i, out are the numberof in and out edges for node i . The parameter β = b/c quantifies the benefit of an edge (to the receiver) relativeto its cost (to the provider). (2) Every generation, the r most fit nodes produceoffspring that replace the r least fit nodes. Reproductionpreserves all edge relationships of the parent, i.e. parentsand offspring connect to the same agents. (3) There is a small mutation probability p per gen-eration with which edges are added/removed randomly.In other words, we have a fitness-based selection rulekeeping the number of agents N constant. Our simula-tion code is available as an electronic supplement.Our model has four parameters, kept constantthroughout the course of evolution: Population size N ,mutation probability p , number selected for replacement r , and the relative benefit β . For every run we keep trackof the total number of edges E ( t ) as a function of genera-tion number t . E ( t ) is a measure of the interdependenceof the population as well as the average fitness (the latterfollows from (cid:104) ω (cid:105) = (cid:80) i ω i /N = ( β − E ( t ) /N , which canbe positive or negative depending on the value of β ). Inaddition, we study the community structures and geneticcomposition within the population, which are defined interms of the connectivity matrix C of the network. Weuse the convention that C ij = 1 if j depends on i , and 0 otherwise, and represent these by black and white pixelsin array plots. Our simulations were run for N = 200,partly due to computational constraints. Although thismight appear small, we note that the relevant degree offreedom here is the number of edge slots N = 4 × ,ensuring that the evolutionary transitions we report arenot accidental fluctuations. III. RESULTS
The evolutionary dynamics and final interdependencestates depend on the values of β and relative selectionpressure r/m , where m = N p is the expected number ofmutations per generation (which is equal to the numberof mutants if p (cid:28) r (cid:28) m, β > O [ r ] ∼ O [ m ] , β > r (cid:28) m, β <
1) and selective destructive( O [ r ] ∼ O [ m ] , β < β determines the changein average fitness per edge, d (cid:104) ω (cid:105) /dE = c ( β − /N , andtherefore one intuitively expects E ( t ) to decrease for β < β > β > β < b − c < r (cid:28) m ) cansuch deleterious edges accumulate, and we get a randominterdependence network with E = N /
2. As expected,increasing r/m causes the network to become sparse andfragmented, and all structure vanishes as O [ r ] ∼ O [ m ].We now overview the constructive regime β >
1, whichproduces distinct phases of complexity (Fig[2], panels 1-6). The long term behavior of E ( t ), which can be viewedas a proxy for average fitness as well as interdependenceand complexity, depends non-monotonically on selectivepressure r/m : For small values of r/m , the asymptoticvalue E ( t → ∞ ) decreases with r/m (Fig[2], panels 1-3).However if r/m exceeds a critical point we see suddentransitions between well-defined discrete levels (Fig[2],panels 4-6). As selective pressure is increased further, wesee increasingly larger fluctuations around these levels.We describe these asymptotic states in more detail byconnectivity matrices (Fig[3], top row) and phylogenetictrees (Fig[3] bottom row) for varying levels of r/m . Thephylogenetic trees are obtained by quantifying the sim-ilarity distance D ij = (cid:80) k | C ik − C jk | + (cid:80) k | C ki − C kj | between all pairs of nodes i and j . In other words, if i and j receive from and provide to the same nodes, theyare considered to be genetically related, consistent withour reproduction rule (cf. Fig[1]).While the destructive β < β > N u m b e r o f D e p e nd e n ce s , E ( t ) Number of Generations, t
FIG. 2.
Regimes of constructive evolution ( β > ) . Plotted is the number of edges E ( t ) as a function of time (generations) t , as the selection strength is increased from left to right r = 2 , , , , ,
100 while mutation probability is kept constant N p = 20. The dashed straight lines indicate the stable number of edges corresponding to an integer number k of equal-sized“bunches”, E = N (cid:0) − k (cid:1) . The dashed curved line is the outcome of the fully neutral simulation ( r = 0). For all panels N = 200, β = 1 . plex phases governed by the value of r/m (Fig[4]): Atransition from cooperation to competition between in-dividuals (Fig[2] panels 1-3) is followed by unstable inter-actions between individuals and “bunches” (Fig[2] panel4), followed by a transition from cooperation to compe-tition between “bunches” (Fig[2] panels 5-6). We definea bunch to be the opposite of a graph theoretical com-munity; a group of nodes that form denser connectionstowards other groups, than they do within (cf. latter twopanels in Fig[3]). Dense outwards connections and sparseintra-connections are the key qualitative characteristic ofa highly specialized system. For example, nearly all en-ergy spent by a heart muscle cell is directed at servingother tissues. The same holds true in a specialized soci-ety, e.g. a lawyer dedicates most of her effort defendingnon-lawyers. The interdependence structures we reportin the latter two panels of Fig[3], conform to these bio-logical and social examples of specialization.We now move towards an understanding of the controland manipulation of the evolution of interdependence,which is now experimentally possible (albeit with mixedsuccess) in biomedical and ecological settings. For exam-ple, the sociality of P. aeruginosa can be manipulatedby drugs that suppress the microbe’s production of acommon good (iron scavenging siderophores). Since themicrobes that are resistant to the drug will altruisticallycontinue to produce the expensive siderophores, they aretaken over by their selfish counterparts affected by thedrug [5, 24, 25]. As a result, the iron-deficient popu-lation can be easily annihilated by the host’s immunesystem [26]. Note that the evolutionary fate of the drug-resistant group would have been the opposite, had thedrug been an antibiotic instead of a quorum blocker. Onthe other hand, there have also been experiments yield-ing the exact opposite outcome, where the drug aggra-vates the infection instead of impairing it, presumablyby leveling the relative advantage of cheaters [23]. We will use our model to quantify these mixed outcomes.Social evolution is complex, and its manipulation andcontrol requires a detailed quantitative understanding ofthe evolutionary outcomes of varying initial states andsystem parameters.To manipulate the sociality of a highly interdependent, β > η ofthe population, and block a fraction γ of their outgoingconnections of those that are selected. Following thisperturbation, we track the evolution of the network andcheck if our perturbation causes the entire population tolose all connections (which, for β >
1, amounts to min-imal fitness). If E ( t ) drops to and remains at zero wecount it as a success and we determine the fraction ofsuccesses for every parameter value. If η (cid:28)
1, the per-turbation can be interpreted as an external introductionof a new strain/species, or a novel mutation which intro-duces a very small number of selfish individuals in thepopulation. If η (cid:39) γ can be interpreted as the dose of the drug,or the degree of “selfishness” of the newly introducedspecies/strain.Fig[5] displays the dependence of success as a func-tion of η (empty vs. closed plot markers correspondto η = 2% and 98%), initial population structure k (quantifying the number of bunches) and γ . We con-sider fast reproducing and slow reproducing species sep-arately, shown in the left and right panels respectively.It is important to distinguish between two very dif-ferent mechanisms that bring the population back toits pre-perturbed state. The first is determined by thetime required for interdependence to evolve anew from E = 0. The original factors causing the establishmentof cooperation in the first place is present regardless of FIG. 3.
Interdependence and genetic composition . Connectivity matrices (top) and their respective phylogenetic trees(bottom) show the dramatic difference in the final organization of the population caused by varying selective strength (leftto right, r = 0 , , , , β > C ij is marked by black ifindividual i provides fitness to j , and left white if there is no exchange. The tree linkages in the bottom are formed accordingto smallest intercluster dissimilarity, defined by the L norm. For all panels N = 200, N p = 20, β = 1 .
01 is kept constant.We see the onset of “bunch” (anti-community) formation even in the weak selection limit (compare panel 1 to 2). The numberand definition of bunches increases with higher selection strength (panel 4). In the strong selection limit bunches competewith each other, leading to size heterogeneity (panel 5). our perturbation, and the effect of even the strongestdrug ( η = 1 , γ = 1) is to simply reset the evolutionaryclock. The second mechanism is evolution through therepopulation of the drug-resistant fraction, which hap-pens much faster, on reproductive time-scales. To clearlydistinguish between these two mechanisms, we set p = 0in Fig[5]; using a finite p scales down all the successrates, but does not otherwise change the qualitative de-pendence on η , γ and k .We observe a number of interesting features in theresponse of the population to external perturbations.For η (cid:39) β for populations with few bunches: Forslow reproducing populations a moderate dose works aswell as, or better than, a strong one. For larger num-ber of bunches, and faster reproduction rates the non-monotonicity vanishes: The stronger the dose, the betterthe outcome. A second remarkable outcome is the de-gree to which a few individuals can make a difference:Targeting η = 2% of the population is as effective astargeting η = 98% of the population provided the drughas a high enough dose. This is because few selfish in-dividuals, as is the case in tumors or invasive species,can devastate an entire population. Finally, we observea very strong dependence of the success rate on the ini-tial community structure. With increasing k and r thisdifference vanishes. IV. DISCUSSION
The phase diagram for the evolution of interdepen-dence is shown in Fig[4] and compactly summarizes ourresults. Panel 1 shows the number of dependences, while Panel 2 quantifies their structure through “bunch mod-ularity”. We define the latter by exchanging 1 ↔ A. Cooperation Between Individuals
In the neutral regime ( r (cid:28) m ) additions and deletionsof edges are equally likely. Thus, if the population startsfully independent, E ( t ) increases until the network isfully randomized with N / E ( t ) ∼ N (1 − e − pNt ), indicated bythe dashed curve in all panels of Fig[2]), the connectivitymatrix shows the onset of community formation; the in-terdependence structure and genetic composition of thepopulation is far from random (compare Fig[3a, 3b]).As the selection strength is increased ( r < m but not r (cid:28) m ) the fluctuations in E ( t ) are amplified. This iscaused by the random formation of nodes for which thenumber of in-edges are different than out-edges. How-ever the system is self stabilizing (Fig[2], panels 1,2); e.g.when the fit defectors reproduce, they typically replacetheir unfit providers, which in turn reduces their ownfitness. Consequently they are taken over by the fairand fit nodes that dominate the r (cid:28) m population. In Number ofDependences (cid:31) (cid:31) (cid:31) B e n e f it Β Anti (cid:31)
CommunityModularity (cid:31) (cid:31) (cid:31) B e n e f it Β Bunch ModularityNumber ofDependences (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29)(cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28)
Benefit Β : (cid:28) (cid:29) (cid:29) (cid:30) (cid:30) (cid:31) (cid:31) B un c h M odu l a r it y (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29)(cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) Benefit Β : (cid:28) (cid:29) (cid:29) (cid:30) (cid:30) (cid:31) (cid:31) N u m b e r o f D e p e nd e n ce s FIG. 4.
Phases of interdependence . The phase dia-grams (top row) displays the asymptotic dependence num-ber E (left) or the bunch modularity (right) as a functionof relevant system parameters. Traversing both phase dia-grams (bottom row) in the horizontal direction clarifies thephase profile. Since the some of the phases are highly dy-namic, the E (left-top) is taken to be the minimum numberof dependences in a large time-window in the long-time limit. Fig[3] panel 3, two large reciprocating groups can be dis-tinguished. They are taken advantage by smaller scaleopportunistic sub-populations. It is also possible to seea smaller sub-cooperative group sustaining itself withina larger cooperative group.
B. Competition Between Individuals As r approaches r c ∼ m/ E ( t ) become comparable to E ( t ) itself. Herethe selective competition is just high enough to allow forsmall cooperative communities to form and grow at arate much higher than random chance, but also highenough for cheaters to spread over their providers inone step, beyond recovery (Fig[1], panel 3). Althoughregime A and B have similar destabilizing factors, theirre-stabilization is very different. The drops in E ( t ) inregime A can recover through re-population , over time scales ∼ /r . In contrast regime B exhibits system-sizelosses from which the only way to recover is re-mutation ,over longer time scales determined by ∼ /m , as thesmallest cooperative group requires two mutations.Comparing A to B reveals that higher selectionstrength in this case leads to lower fitness; had one mixedthe stronger-selected population B with the weaker-selected one A, the former would be driven to extinc-tion. The behavior of B is similar to that expected ina classical prisoner’s dilemma, which emerges from ourmodel as a special case - survival of the fittest producesthe globally least fit outcome. C. Formation of Specialized Bunches As r is increased above the critical point r c ∼ m/ C is sparse and random for r just below r c ,we start seeing metastable bunches at r > r c , the num-ber and stability of which increases with r . The suddenjump in the edge number we see in Fig[3], panels 4-5 isanalogous to that found in [30].For a very large window of selective strength O [ m ] ∼ r < O [ N ], we see that the system can only maintaincertain discrete values of E . These are the stable con-figurations corresponding to an integer number of equalsized bunches ( k ) given by the relation E k = N (cid:0) − k (cid:1) , k = 0 , , , . . . , k max (Fig[2] panel 4). The maximumnumber of bunches k max is determined by the mutationrate, k max ∼ N/m (i.e. so that in steady state thereis one mutation per bunch per time step). However,we have observed k transiently increasing to 50% higherthan this value. Note that the degree of interdependence(and hence the average fitness) in the strong selectionlimit well exceeds that in the weak selection limit.It is interesting that in the limit r, m ∼
1, we seestructures more complex than bunches. These includehierarchies (smaller bunches within a bunch), cycles (3or more groups providing to one other), and hierarchiesof cycles (cycles within a cycle). In this limit the dynam-ics of E ( t ) still exhibits discrete steps similar to Fig[2]panel 5, however with more possible metastable plateauscorresponding to unequal sized matrix blocks. D. Competition Between Bunches
With increasing r the fluctuation in the number ofedges around the stable k starts increasing, and we seedestructive competition similar to that near the phaseboundary of B; however, now the competition is betweenthe bunches rather than the individuals, which createssignificant size differences between them. These fluctu-ations can lead to one bunch replacing another, causing E k to make large transitions between different values of k . Despite the apparent noise (Fig[2] panel 6) the depen-dence structure remains in a highly ordered state withhigh reciprocity. As r is increased further we see that (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31)(cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30)(cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29)(cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28)(cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27)(cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) Number ofBunches, k (cid:31)(cid:28) (cid:30)(cid:27) (cid:29)(cid:26) Fraction of Edges Blocked (cid:31) Γ (cid:30) A nn i h il a ti on P r ob a b ilit y P ( Γ ) f o r S l o w R e p r odu ce r s (r = ) A nn i h il a ti on P r ob a b ilit y P ( Γ ) f o r F a s t R e p r odu ce r s (r = ) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31)(cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30) (cid:30)(cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29) (cid:29)(cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28) (cid:28)(cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27) (cid:27)(cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) (cid:26) Number ofBunches, k (cid:31)(cid:28) (cid:30)(cid:27) (cid:29)(cid:26) Fraction of Edges Blocked (cid:31) Γ (cid:30) FIG. 5.
Manipulation and Control of Social evolu-tion . Of the entire population (N=200) we mark η = 2%(empty markers) and η = 98% (filled markers) of the nodesas susceptible to our perturbation. We then block γ (hori-zontal axis) of the out edges of the susceptible population,and determine the probability that the strong cooperatingpopulation collapses into a weak non-cooperating one (i.e to E = 0). To clearly distinguish between the spread of the re-sistant subpopulation η , and novel mutations that occur afterour perturbation, we set p = 0 (finite p merely adds noise tothe curves). We observe that the effect of the drug stronglydepends on the community structure for slow reproducingspecies (left, r = 15) but not for fast reproducing ones (right, r = 60). The dashed curve shows how devastating even asmall number of antisocial individuals be for the whole pop-ulation, and suggests treatments where individual cells aretargeted. The strong dependence of annihilation probabil-ity on k for small k (left panel) might explain why quorumblockers are effective against some biofilms forming bacteriabut not others. Interestingly, for k = 2, P ( γ = 0 .
5) is sig-nificantly higher than P ( γ = 0 . competition between bunches cause fluctuations compa-rable to the size of bunches. i.e. a small bunch canincrease in size by spreading over others until anothermetastable structure re-evolves. V. CONCLUSIONS
We have constructed a simple model that allows usto study the co-evolution of self-replicating interdepen-dent structures, and reported multiple evolutionary tran-sitions as β and r are varied. This model is quite gen-eral and has very few assumptions - the fitness functionis only assumed to be an arbitrary increasing functionof x and there are only two relevant parameters govern-ing the dynamics (selection strength r/m and relativebenefit b/c ) since the population size N does not makea qualitative difference as long as both N p and r aremuch smaller than N . Furthermore, the value β doesnot make a qualitative difference apart from whether itis larger or smaller than unity, and no quantitative dif-ference if | − β | is smaller than 1 /N . Unlike the typicalsimplified models of evolutionary game theory we do notassume that an individual’s behavior is the same towards all others (although some individuals can end up in astate where they give to all and receive from all). In thisrespect the states allowed in this work is a generaliza-tion of the two-state models common in the literature.Thus, we hope that our model can serve as a guidingframework for understanding the emergence of sociality.Even in this simple case, we observe a number of sur-prising and important phenomena. First, we report thateven the weakest selection strengths ( m (cid:29) r ) can pro-duce interdependence structures that are far from ran-dom. Thus, assumptions regarding “random interdepen-dence” invoked by neutral evolutionary arguments maybe too strong [27–29]. Our second observation is thenatural emergence of specialized bunches and multi scalestructures from the simple laws of evolutionary dynam-ics. As we probe the response of the system to various se-lection strengths, we see regimes of random interdepen-dence, competition between nodes, cooperation betweennodes (bunches) and competition between bunches.Thirdly we report that the regime β > , r > E on selection strength can have im-portant implications in medicine. For example, biofilmpopulations may be induced into a less virulent non-cooperative state by decreasing the selective pressure,so that a cooperative film behaving as Fig[2e] evolvesinto an intermediate non-cooperative state behaving asFig[2c]. This may be experimentally verified in P. aerug-inosa by increasing the available iron while keeping theirpopulation constant by limiting their carbon source.Another remarkable result is the non-monotonic con-nection between anti-social drug dose and the successfulannihilation of cooperativeness. Indeed, the model ex-hibits a a “contagion” effect which allows the manipu-lation of a few individuals to have population-wide ef-fects. It has been noted that introducing several selfishmutants (or using an anti-social drug effective on a fewindividuals) may be far more effective than manipulatingan entire population [23] and is consistent with experi-mental observations [5, 24, 25].Finally, when mutation rate is set to zero we observethat the behavior of E ( t ) resembles that of classical pop-ulation dynamics. The dynamics between providers andreceivers becomes qualitatively similar to that betweenthe predators and prey of a Lotka-Volterra type system.Starting from a randomized connectivity matrix and set-ting p = 0, it is common for E ( t ) to reach a fixed valueand oscillate around it.Due to its generality and applicability, our model hasroom for many natural extensions. For example, the dis-tribution of parameters and fitness functions in a morerealistic model could include spatial, temporal and indi-vidual heterogeneity. Further, the quantities p , b and c can be dynamic as they are themselves, to an ex-tent, subject to evolutionary forces. This can lead toa very interesting set of potential future studies explor-ing connections between interdependence and evolvabil-ity/efficiency. Another factor not taken into accounthere is the possibility of the change in population sizedue to statistical fluctuations (e.g. due to a time depen-dent energy input, or infection/predation). Such exten-sions would be appropriate to address systems in ecol-ogy, structured biological population, and provide in-sight into complicated social trends. Acknowledgement: