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Slide 1: ARTICLE IN PRESS Journal of Theoretical Biology 250 (2008) 560–568 www.elsevier.com/locate/yjtbi Simple quasispecies models for the survival-of-the-flattest effect: The role of space Josep Sardanyesa, Santiago F. Elenab, Ricard V. Solea,c,Ã ´ ´ a Complex Systems Lab (ICREA-UPF), Barcelona Biomedical Research Park (PRBB-GRIB), Dr. Aiguader 88, 08003 Barcelona, Spain b ´ `ncia, Spain Instituto de Biologıa Molecular y Celular de Plantas, CSIC-UPV, 46022 Vale c Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA Received 29 May 2007; received in revised form 27 September 2007; accepted 25 October 2007 Available online 30 October 2007 Abstract The survival-of-the-flattest effect postulates that under high mutation rates natural selection does not necessarily favor the faster replicators. Under such conditions, genotypes which are robust against deleterious mutational effects may be favored instead, even at the cost of a slower replication. This tantalizing hypothesis has been recently proved using digital organisms, subviral RNA plant pathogens (viroids), and an animal RNA virus. In this work we study a simple theoretical system composed by two competing quasispecies which are located at two widely different fitness landscapes that represent, respectively, a fit and a flat quasispecies. The fit quasispecies is characterized by high replication rate and low mutational robustness, whereas the flat quasispecies is characterized by low replication rate but high mutational robustness. By using a mean field model, in silico simulations with digital replicons and a two-dimensional spatial model given by a stochastic cellular automata (CA), we predict the presence of an absorbing first-order phase transition with critical slowing down between selection for replication speed and selection for mutational robustness, where the surpassing of a critical mutation rate involves the outcompetition of the fit quasispecies by the flat one. Furthermore, it is shown that space, which involves a lower critical mutation rate, broadens the conditions at which the survival-of-the-flattest may occur. r 2007 Elsevier Ltd. All rights reserved. Keywords: Survival-of-the-flattest; Quasispecies spatial dynamics; Robustness; Virus evolution; Critical phenomena; Absorbing first-order phase transitions 1. Introduction tions in which the frequency of the wildtype and of each mutant genotype not only depends on their replication The quasispecies theory of molecular evolution, origin- rates but also on their constant genesis by mutation from ally developed by Eigen and collaborators (Eigen et al., genotypes which are close in genotypic space. In a constant 1988; Eigen and Schuster, 1979) has become the standard environment and in the absence of other external theoretical framework used to model the evolution of RNA perturbations, this distribution of genotypes is known as viruses (Domingo and Holland, 1997; Domingo, 2002). the quasispecies (Eigen et al., 1988). One of the keystones of theoretical quasispecies, shared by The existence of a population structure in quasispecies RNA viruses, is that replication fidelity is so low that the strongly affects the way selection acts, because the number of mutant offspring generated in a population may evolutionary success of individual genomes does not exceed the number of offspring identical to the parental depend anymore on their own replication rate but also genotype. This gives rise to highly polymorphic popula- on the average growth rate of the quasispecies they belong to. Fast replicating genomes that produce low-fitness ÃCorresponding author. Complex Systems Lab (ICREA-UPF), Barce- offspring can be outcompeted by slow replicating genomes provided the latter inhabits a region of sequence space lona Biomedical Research Park (PRBB-GRIB), Dr. Aiguader 88, 08003 Barcelona, Spain. Tel.:+34 933160532; fax: +34 933160550. characterized by high neutrality and connectivity (Schuster ´ E-mail address: ricard.sole@upf.edu (R.V. Sole). and Swetina, 1988; van Nimwegen et al., 1999; Wilke, 0022-5193/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2007.10.027

Slide 2: ARTICLE IN PRESS ´s J. Sardanye et al. / Journal of Theoretical Biology 250 (2008) 560–568 561 2001b). This phenomenon has been dubbed as the that after infection, the Human immunodeficiency virus type quasispecies effect (van Nimwegen et al., 1999; Wilke, 1 (HIV-1) is able to establish well-differentiated popula- 2001a) or more recently as the survival-of-the-flattest tions which are organ-specific and show limited gene flow (Wilke et al., 2001) in clear reference to Darwin’s ´ among them (Borderı´ a et al., 2007; Sanjuan et al., 2004). In survival-of-the-fittest concept. Indeed, authors who have these cases, organ-specificity appears not as a consequence casted doubts about the relevance of the quasispecies of founder events but as a consequence of differences in the model to real viruses based their criticism in the fact that adaptive constraints imposed by heterogeneous cell types the quasispecies effect was never observed in vivo (Holmes and the existence of fitness tradeoffs across organs. In the and Moya, 2002; Jenkins et al., 2001). However, two recent case of plant viruses, spatial structure appears at two experiments give strong support to the validity of the different levels. First, it has been shown that different quasispecies effect for real viral populations. In the first leaves can be infected with different viral subpopulations, experiment, two viroids (small circular RNAs that infect which, for the case of perennial plants, may further plants and do not encode for any protein) populations were differentiate into different branches and sub-branches, as allowed to compete at increasing mutation rates (Codoner ˜ it has been shown, for example, for Plum pox virus (PPV) et al., 2006). At low mutation rate the faster but genetically (Jridi et al., 2006). In this case, population structure likely homogeneous replicator outcompeted the slower but highly arises as a consequence of the strong bottlenecks associated polymorphic one, as expected from the survival-of-the- with the systemic movement of viruses from source to sink fittest effect. However, the result of the competition was leaves (Hall et al., 2001; Li and Roossinck, 2004; Sacristan ´ reversed at high mutation rate and the slower replicator et al., 2003). Second, within a given infected leaf, that for won the competition by taking advantage of its larger the sake of simplicity can be considered as a two- ´ mutational robustness. In the second study (Sanjuan et al., dimensional space, populations initiated at different 2007), two populations of Vesicular stomatitis virus (VSV) infectious foci do not overlap after confluence but exclude that differed in their replication rates and robustness each other, generating a patched distribution of genotypes competed at increasing concentrations of chemical muta- (Dietrich and Maiss, 2003) (see Fig. 1). gens. Below a certain concentration of mutagens, the fittest In the present report we analyze the dynamics of two VSV outcompeted the flattest one. Above this critical competing quasispecies by using a simple mean field model, concentration, the competition result was reverted and the in silico simulations with digital replicons, and a stochastic flattest VSV systematically displaced the fittest one. cellular automata (CA) model, which allows to simulate One peculiarity of viral infections that is usually ignored the competition process explicitly considering the potential by most quasispecies models is the existence of a spatial effect of physical space. In all these approaches, the fit population structure. Only a few studies have analyzed the quasispecies has a fast replication rate but is surrounded by error–threshold transition in a spatial context (Altemeyer genotypes which are strongly deleterious and thus muta- and McCaskill, 2001; Toyabe and Sano, 2005). It is well tions always have a strong negative effect on the average known that spatial dynamics can deeply change the population fitness. On the contrary, the flat one has a lower outcome of competition even under the absence of selection replication rate but is located at a neutral and highly ´ (Sole and Bascompte, 2006). Within infected hosts, viruses connected region of sequence space, and thus mutations do not behave as a single well-stirred population but as a exert a mild impact on viral fitness. In particular, we are collection of subpopulations that colonize and reproduce interested in studying the effect of increasing mutation rate on different compartments constituted by different tissues on the outcome of the competition process, paying special and organs. For example, it has been extensively shown attention to the role of spatial structure. In short, our Fig. 1. Spatial distribution of two PPV variants infecting the same leaf of N. benthamiana. Each variant was labeled with a different fluorescent protein (red and green). Coinfected shall appear as yellow fluorescent. The right panel is a 5-fold magnification of the left one, and shows that only the line of cells at the confluence edge can be infected by the two PPV variants (taken from Dietrich and Maiss, 2003).

Slide 3: ARTICLE IN PRESS 562 ´s J. Sardanye et al. / Journal of Theoretical Biology 250 (2008) 560–568 results suggest the existence of a critical mutation rate at which an absorbing first-order phase transition with slowing down between selection for fast replication and Fitness selection for mutational robustness takes place. Beyond the critical mutation rate, the flat quasispecies outcompetes the fit one. In the vicinity of this transition both quasispecies can coexist during an extremely long time. Moreover, space (1) is shown to broaden the conditions at which the survival- of-the-flattest can be observed. The results of the in silico simulations are in complete agreement with the mean field analysis. (2) 2. Quasispecies mean field model As a first approach to the analysis of the competition dynamics between two quasispecies located at two different fitness landscapes, a mean field model has been used. The fx Q (0) x0 y0 (0) fy Q model considers a perfectly mixed system with a fit quasispecies, x, and a flat one, y. Under the assumption P (0) fx (1 Q) (0) fy (1 Q) (1) fy (1 Q) i ðxi þ yi Þ ¼ 1, the model is defined by the following set of ODEs: (1) x1 y1 (1) fy Q dx0 fx ¼ f ð0Þ Qx0 À x0 Fðx; yÞ À x0 , x (1) dt Fig. 2. The fit (1) and the flat (2) quasispecies are located at two different dx1 fitness landscapes. At low mutation rate (A) populations are centered on ¼ f ð0Þ ð1 À QÞx0 þ f ð1Þ x1 À x1 Fðx; yÞ À x1 , x x (2) the peak; at high mutation rate (B) populations drift on the landscape dt away from the peak. The lower diagram provides a schematic representa- dy0 tion of the quasispecies competition model (Section 2). Dashed arrows ¼ f ð0Þ Qy0 þ f ð1Þ y1 ð1 À QÞ À y0 Fðx; yÞ À y0 , y y (3) indicate the competition among both quasispecies. dt dy1 ¼ f ð1Þ Qy1 þ f ð0Þ y0 ð1 À QÞ À y1 Fðx; yÞ À y1 , y y (4) m ¼ 1 À Q. The qualitative behavior of Eqs. (1)–(4) can be dt studied by means of linear stability analysis of the P where the term Fðx; yÞ ¼ i¼0;1 ½ðf ðiÞ À Þxi þ ðf ðiÞ À Þyi Š, x y equilibrium points. Note that the concentration equili- indicates the outflow (i.e., excess production) defining a brium for both flat quasispecies is the same i.e., yà ¼ yà , 0 1 constant population (CP) constraint. Here x0 denotes the because we have assumed that their self-replication rates relative concentration of the master sequence and x1 the are equal, thus both Eqs. (3)–(4) becoming symmetric. concentration of all the mutant sequences, which are Hence, in order to simplify our model we can reduce the grouped in an average sequence different form the master system (1)–(4)P a three-dimensional dynamical system by to one. The self-replication rate for the master sequence is considering _ _ i yi ¼ y and y0 þ y1 ¼ y, thus having given by f ð0Þ , while the self-replication constant for the _ y ¼ f y y À yFðx; yÞ À y. It can be shown that this reduced x other sequences (grouped in x1 ) is f ð1Þ . Here, as a system has four fixed points, given by Pà ð0Þ, Pà ð0; 0; 1Þ, x consequence of the assumed fitness landscape, the master Pà ð0; 1; 0Þ and Pà ðxà ; 1 À xà ; 0Þ, with 0 0 sequence x0 replicates faster than the pool of mutants x1 f ð0Þ Q À f ð1Þ x x and thus f ð0Þ bf ð1Þ . Both variables y0 and y1 indicate the xà ¼ 0 . (5) x x f ð0Þ À f ð1Þ x x relative concentration of the flat quasispecies, which is a densely connected one because it is located in a flat fitness For the sake of simplicity, linear stability analysis is done landscape and thus backward mutations are allowed to by setting  ¼ 0. It is easy to show that the fixed point Pà ð0Þ happen (see Fig. 2). Finally,  denotes the decay rate for the is always unstable when self-replication rates are higher replicators, which is defined to be the same for both than zero, since the eigenvalues obtained from the Jacobi quasispecies. Note that our model allows us to study two matrix, Lð0Þ, are lð1Þ ¼ f ð0Þ Q, lð2Þ ¼ f ð1Þ and lð3Þ ¼ f y . Thus x x different scenarios considering both no decay ( ¼ 0) and all these eigenvalues will be positive or zero (when self- decay (a0). replication rates are zero). Let us now analyze the most Hereafter we will assume that f ð0Þ ¼ f ð1Þ ¼ f y of ð0Þ . In all y y x interesting case, that is, the survival-of-the-flattest scenario, the equations Q denotes the average quality factor of self- corresponding to the fixed point Pà ð0; 0; 1Þ. The stability of replication, which we assume to be the same for all the this point is obtained from the eigenvalues of Lð0; 0; 1Þ, sequences, thus being the mutation rate, m, defined as which are given by lð1Þ ¼ f ð0Þ Q À f y , lð2Þ ¼ f ð1Þ À f y , and x x

Slide 4: ARTICLE IN PRESS ´s J. Sardanye et al. / Journal of Theoretical Biology 250 (2008) 560–568 563 by lð3Þ ¼ Àf y . Note that lð2;3Þ o0, since it is assumed that f y 4f ð1Þ . From lð1Þ we can derive the critical mutation rate x 1.5 0.6 which makes this fixed point becoming stable i.e., lð1Þ o0, Equilibrium concentration 0.4 thus indicating that the flat quasispecies will be the 0.2 surviving one. Such a critical condition is defined to be x1 1 0 fy 0.6 0.7 0.8 0.9 mc ¼ 1 À . (6) f ð0Þ x y1 0.5 Hence, the survival-of-the-flattest will take place provided y0 m4mc . The stability properties for the third fixed point x0 Pà ð0; 1; 0Þ, are obtained from the eigenvalues of Lð0; 1; 0Þ, 0 which are given by lð1Þ ¼ f ð0Þ Q À f ð1Þ , lð2Þ ¼ Àf ð1Þ , and by x x x 0 0.2 0.4 0.6 0.8 1 lð3Þ ¼ f y À f ð1Þ . Note that lð2Þ is always negative, and lð3Þ is x µ=1–Q always positive because f y 4f ð1Þ , being such a point a x 0.6 saddle (since lð3Þ involves an unstable subspace). Moreover, 0.6 x1 x1 Concentration lð1Þ is positive when Q4f ð1Þ =f ð0Þ . As f ð0Þ bf ð1Þ , lð1Þ will also x x x x 0.4 x0 0.4 x0 y0,1 be typically positive. The stability of the last fixed point is analyzed from Lðxà ; 1 À xà ; 0Þ. The eigenvalues for this case 0 0 0.2 0.2 y0,1 are: lð1Þ ¼ Àf ð0Þ Q, lð2Þ ¼ f y À f ð0Þ Q, and lð3Þ ¼ f ð1Þ À f ð0Þ Q. x x x x 0 0 Note that lð1Þ is always negative. Hence, this point will be 1 10 100 1 10 100 time time stable if f ð0Þ Q4f y and f ð0Þ Q4f ð1Þ . x x x The temporal dynamics of Eqs. (1)–(4) has also been Fig. 3. (A) Equilibrium concentrations at increasing mutation rate with numerically studied using the standard fourth-order f x ¼ 1, f x ¼ 0:05, f ð0Þ ¼ f ð1Þ ¼ 0:3, and  ¼ 0. The inset shows an ð0Þ ð1Þ y y Runge–Kutta method with a constant time step dt ¼ 0:1 enlarged view of the sharp transition point indicated with the arrow. Here x0 , x1 , y0 and y1 are represented as open circles, closed circles, filled and (assuming f ð0Þ ¼ 1bf ð1Þ ¼ 0:05, f ð0Þ ¼ f ð1Þ ¼ 0:3of ð0Þ , and x x y y x open triangles, respectively. Here the initial conditions were: x0 ð0Þ ¼ 0:35,  ¼ 0). Numerical results are in agreement with linear x1 ð0Þ ¼ 0:2, y0 ð0Þ ¼ 0:15, y1 ð0Þ ¼ 0:3. The time dynamics associated to the stability analysis. In Fig. 3A we compute the equilibrium survival-of-the-fittest and to the-survival-of-the-flattest are shown in (B) concentration (after discarding the transient period) at and (C) (x0 : thick solid line; x1 : thin solid line; y0 : thick dashed line; and increasing mutation rates. The inset clearly shows that such y1 : thin dashed line), with same parameter values as in (A) with Q ¼ 0:4 (B), and Q ¼ 0:25 (C). Here, for both scenarios, we use: x0 ð0Þ ¼ 0:5, a concentration decreases for the fit quasispecies (x0 ), and x1 ð0Þ ¼ 0, y0 ð0Þ ¼ y1 ð0Þ ¼ 0:25. increases for the pool of mutants (x1 ). Once the critical mutation is overcome, the fit quasispecies becomes extinct approach has been previously used to study viral dynamics and the flat one achieves a non-trivial stationary concen- and evolution (Codoner et al., 2006; Sole et al., 1999, ˜ ´ tration. In the same figure we also show the time dynamics 2006). Let us define two populations of digital replicons, of the system below (Fig. 3B) and above (Fig. 3C) the critical mutation rate. Fig. 3A indicates the presence of a given by a fit quasispecies, SF , and a flat one, Sf , both of i i sharp change in the qualitative dynamics once the critical them defined as populations composed of binary strings mutation rate is overcome. Such a result can be interpreted named SF ;f ¼ ðSF ;f ; . . . ; S F ;f Þ, with SF ;f 2 f0; 1g, being n the i i1 in ij as an absorbing first-order phase transition among the sequence length, where i ¼ 1; . . . ; N, being N the maximum survival-of-the-fittest and the survival-of-the-flattest sce- population number of sequences (we use n ¼ 32 and narios, in agreement with the study of Wilke (2001b). The N ¼ 500). The fit quasispecies is formed by the master extinction time of these quasispecies hyperbolically di- sequence (all-ones string) and its entire mutant spectrum verges in the vicinity of the critical mutation point, being (all the other 2n À 1 sequences). The master sequence is extremely long very near the phase transition point (results assumed to self-replicate at the maximum probability (i.e., not shown). This actually corresponds to a phenomenon of rm ¼ 1). The pool of mutants self-replicates with a lower critical slowing down in the vicinity of the phase transition. probability, rp ¼ 10À3 , because we also assume that the fit quasispecies is located at a sharp fitness landscape and 3. In silico bit string model mutations have large deleterious effects. On the contrary, the flat quasispecies corresponds to a robust neutral Next, we extended the previous mean field model by network in which mutations do not affect self-replication considering two competing populations of quasispecies in rates. Thus, all possible sequence combinations for the flat the form of binary digital replicators, thus considering the quasispecies self-replicate at rf ¼ 0:1. whole mutant spectrum of the quasispecies, within a As initial conditions we randomly inoculate the popula- multidimensional sequence space. This computational tion with master sequences of the fit quasispecies and with

Slide 5: ARTICLE IN PRESS 564 ´s J. Sardanye et al. / Journal of Theoretical Biology 250 (2008) 560–568 random flat quasispecies. At each generation in the bit will be copied with error is algorithm the following set of rules was repeated N times: m ¼ 1 À ð1 À mb Þn . (7) According to expression (7), the critical mutation rate 1. A string from the population was randomly chosen and obtained from mc ¼ 0:072 is mc ¼ 0:908. The value predicted b replicated according to the above-mentioned probabil- with the mean field model (i.e., expression (6)), using the self- ities. replication probabilities as replication rates, is given by 2. Replication takes place by replacing one of the strings in mc ¼ 0:9. Note that this mutation value perfectly matches the population (also taken at random) say Sjai , by a with the one analytically derived from the mean field model. copy of Si . The self-replication mechanism presents Fig. 4 also shows three examples of the time evolution of the error at rates mb , per bit and replication cycle, sequences used in the diagram. Specifically, in Fig. 4A we respectively. display the scenario below the critical per-bit mutation, where the master sequence (solid thick line) achieves a higher Fig. 4 shows the average equilibrium concentration as a stationary concentration than the mutant strings (from top function of the per-bit mutation probability for three to down we can follow the time evolution from the first to different types of sequences: the master string; some the fifth neighbor in sequence space). Fig. 4B shows the time sequences of the mutant spectrum of the fit quasispecies dynamics near the critical mutation. For this particular run, (i.e., the five nearest hypercubic orthant neighbors given by the flat quasispecies was able to outcompete the fit one. Note the sequences differing up to five bits from the master one); that the master sequence and the strings forming the pool and the flat quasispecies (represented with the normalized of mutants become extinct at approximately 900 generations. sum of all the sequences belonging to this quasispecies) at If mutation probability is increased the same outcome is increasing mutation probabilities. From this diagram, the found although the fit quasispecies becomes extinct earlier critical mutation probability is shown to be mb % 0:072. In (see Fig. 4C). order to compare this critical mutation rate with the one obtained with the mean field model we must take into 4. Bit string spatial model account that mb is the per-bit mutation probability. The probability of error-free copy of the master sequence is We finally develop a spatially extended stochastic given by ð1 À mb Þn , and thus the probability that at least one model including the whole quasispecies structure of both survival of the fittest 100 master sequence SF(1) Equilibrium concentrations 1 SF(2) Concentration survival of the flattest 10-1 SF(3) 0.5 SF(4) 10-2 SF(5) 0 0 200 400 10-3 generations 0 0.05 0.1 µb 0.6 1 Concentration Concentration 0.3 0.5 0 0 0 250 500 0 500 1000 generations generations Fig. 4. Normalized equilibrium concentrations (in linear-log plot) as a function of per-bit mutation probability, mb , with rp ¼ 10À3 and rf ¼ 0:1 (after discarding 5000 generations and averaging over 10 independent runs). The curves below the master sequence correspond (from top to bottom) to the mutant sequences of the fit quasispecies differing up to five bits from the master sequence. Note that beyond mc % 0:072, the flat quasispecies (thick line) is b able to outcompete the fit one. The arrows indicate the per-bit mutation probabilities used to display the time dynamics of both quasispecies: (A) survival- of-the-fittest-scenario with mb ¼ 0:023; (B) very near the critical mutation with mb ¼ 0:07; and (C) survival-of-the-flattest scenario with mb ¼ 0:078. In all the time plots: master sequence (solid thick line), mutants differing up to five bits from the master sequence (solid thin lines), and the normalized sum of the sequences belonging to the flat quasispecies (dashed thick line).

Slide 6: ARTICLE IN PRESS ´s J. Sardanye et al. / Journal of Theoretical Biology 250 (2008) 560–568 565 competing quasispecies. We define a L  L state space Survival of the fittest 1 OðLÞ 2 Z2 , with zero-flux boundary conditions (simulating, master sequence for example, the bounded system of plant leaves). Each B quasispecies (the fit, SF ¼ ðsF ; . . . ; sF Þ, and the flat, i i1 in 0.8 D Sf ¼ ðsf ; . . . ; sf Þ) have a potential population composed Mean concentration i i1 in C of jHn j (with H¼ 2 and n ¼ 16) binary strings which 0.6 Survival of the flattest A define the states of the automaton and correspond to the whole spectrum of strings living in the vertices of two 16th- 0.4 dimensional Boolean hypercubes. We choose, at each generation t, L  L random cells to ensure an average updating of all the lattice cells per generation. Assuming 0.2 that only a single bit string can occupy a lattice cell and a Moore neighborhood for replication events, we apply two 0 state transition rules (with same probability), according to 0 0.02 0.04 0.06 0.08 0.1 µb Fig. 5. Quasispecies populations at increasing per-bit mutation rates for 1. Self-replication: If the cell is occupied by a sequence of the spatial model, with L ¼ 128, f pool ¼ 0:05, f m ¼ 1, f flat ¼ 0:1 and  ¼ the fit quasispecies, SF , and the neighbor site is empty, i 0:01 (each data point is the mean concentration Æ standard deviations SF , replicates with probability f m 2 ½0; 1Š if it is the i averaged over five replicas after discarding t ¼ 2  105 generations). The master sequence, SF (i.e., all-ones string), according to flat quasispecies (considering the whole population of strings) is indicated m with the empty circles and the dashed line. Below the master sequence we the next reactions: show (from top to bottom) the strings differing up to five bits from the f m ð1Àmb Þn master sequence. The capital letters indicate the mutation probabilities SF þ f ÀÀÀ ! 2SF , m À m used in Fig. 6. f m W ij SF þ f À ! SF þ SF . m ÀÀ m jam If the cell is occupied by a deleterious mutant of the fit The results show an approximate linear decay in the quasispecies, it replicates to an empty neighbor site with concentration of the master sequence of the fit quasispecies probability f pool 2 ½0; 1Š, according to the following pair as mutation rate increases. The flat quasispecies can of reactions: outcompete the fit one beyond mb % 0:079 (see Fig. 5). The spatio-temporal dynamics associated to four different f pool ð1Àmb Þn SF þ f ÀÀÀÀÀ 2SF , ! iam mutation probabilities is shown in Fig. 6. The first one iam f pool W ij (Fig. 6A) shows the dynamics with low per-bit mutation SF þ f ÀÀÀ ! SF þ SF . iam À iam jai rate ðmb ¼ 0:0075Þ. As mutation rate is low, the master sequence of the fit quasispecies (solid arrow) rapidly If the cell is occupied by a string of the flat quasispecies, spreads, invading the space and outcompeting the flat it replicates towards an empty neighbor with probability replicator (dashed arrow). The time evolution for the f flat 2 ½0; 1Š, according to mutant sequences differing in 1 (up) and in 2 (down) bits f flat ð1Àmb Þn from the master sequence of the fit quasispecies are also Sf þ f ÀÀÀÀÀ 2Sf , i ! i shown. These are actually the only sequences found in the f flat W ij mutant spectrum since mutation is very low. Sf þ f ÀÀÀ ! Sf þ Sf . i À i jai If mutation rate is increased up to mb ¼ 0:0375, the master sequence of the fit quasispecies grows slowly, and 2. Decay: If the cell is occupied by a string, it decays with the mutant spectrum contains a higher diversity of probability  2 ½0; 1Š, leaving a free space according to sequences (see Fig. 6B). The mutant sequences belonging  to the fit quasispecies (from top to down) differing in up to Si À f. ! five bits from the master sequence are specifically shown. Actually, the sequences differing in more than six zeros We use: f m ¼ 1, f pool ¼ 0:05, f flat ¼ 0:1 and  ¼ 0:01. from the master sequence are not found in the pool of The terms W ij correspond to the probabilities of erroneous mutants (results not shown). The third case (Fig. 6C) replication and are givenP by W ij ¼ ð1 À mb ÞnÀd H ½Si ;Sj Š Á shows the dynamics with a mutation rate near to the md H ½Si ;Sj Š , being d H ½Si ; Sj Š ¼ n jsk À sk j, the Hamming k¼1 i j critical value. Here, the fit quasispecies asymptotically distance between the two sequences. As initial conditions outcompetes the flat one. Note that the diversity of two patches were inoculated with all-one sequences for sequences of the mutant spectrum increases (for this case both populations (see Fig. 6 middle). we find sequences differing in 1–9 zeros from the master We investigate the effect of increasing mutation rate in sequence). As we are near the transition point, the time that the population structure of both competing quasispecies. the fit quasispecies spends in outcompeting the flat one is

Slide 7: ARTICLE IN PRESS 566 ´s J. Sardanye et al. / Journal of Theoretical Biology 250 (2008) 560–568 Fig. 6. Spatio-temporal dynamics for the CA model with same probabilities as in Fig. 5, and: (A) mb ¼ 0:0075; (B) mb ¼ 0:0375; (C) mb ¼ 0:06; and (D) mb ¼ 0:1. For time dynamics (in linear-log scale): flat quasispecies (dashed arrow), master sequence of the fit quasispecies (solid arrow), pool of deleterious mutants of the fit quasispecies (the other time trajectories). In the snapshots obtained from the time series we show: flattest strings (white); empty cells (black); fit quasispecies (gray gradient represented in the middle of the figure). Here the upper gray band corresponds to the master sequence and the rest of colors (from top to bottom) to the sequences differing from 1 to 16 bits from the master sequence. As initial conditions (middle) we inoculated the lattice with two patches containing all-one sequences for both flat and fit quasispecies. longer, in agreement with the previous models which spatially mixed case, from Eq. (6): showed the critical slowing down phenomenon. The dynamics with mb ¼ 0:1 are shown in Fig. 6D. Here the f y ðmc Þ ¼ f ð0Þ ð1 À mc Þ, x per-bit mutation value is above the critical mutation probability and thus the flat quasispecies outcompetes the which will be maximal for mc ¼ 0 and zero at mc ¼ 1 (as fit one. Now all possible sequences of the mutant spectrum expected). The spatial model also follows a linear relation, appear, and the master sequence of the fit quasispecies but it now decays as (small solid arrow) rapidly extincts (approximately after t ¼ 2200 generations). f S ðmc Þ ¼ f ð0Þ ð1 À að; nÞmc Þ, y x In Fig. 7 we represent the critical mutation value for all the models analyzed in this work. Interestingly, the values where að; nÞ is some (to be analyzed elsewhere) function of of mc obtained for the CA model, in which space is spatial constraints, decay and—presumably—genome explicitly incorporated, are systematically below the values length. This deviation between the spatial and non-spatial obtained for those models that do not consider spatial counterparts of the string model seems consistent with structure (i.e., mean field and bit-string models), strongly previous work by Altemeyer and McCaskill (2001) on error suggesting that replication on a spatially constrained threshold for spatially extended quasispecies (see also scenario rather than in a well-stirred environment enlarges Toyabe and Sano, 2005). In their study, these authors the parameter space at which the survival-of-the-flattest showed that decreased diffusion in a given spatial domain may occur. The f y ðmc ) function is easily derived from the leads to a decrease in the error threshold mutation. Here a

Slide 8: ARTICLE IN PRESS ´s J. Sardanye et al. / Journal of Theoretical Biology 250 (2008) 560–568 567 1 slowing down between selection for fast replication and selection for robustness takes place. In previous work, the survival of the flattest numerical value of mc was obtained as the solution of a quadratic expression (Wilke et al., 2001) and, hence, it was Self replication rate flat quasispecies not straightforward to visualize the effect that the different parameters involved in the expression (replication rate and robustness) had on mc . Our estimator of mc , however, has a much simpler expression since only depends on the ratio 0.5 between the self-replication rates of the flat quasispecies, f y , and the master sequence (i.e., the mutation-free genotypes), f ð0Þ , of the fit quasispecies. The larger the x difference in replication rates between these two quasis- pecies, the smaller the value of mc and therefore, the easier to observe the survival-of-the-flattest effect. We have also extended the analysis of these theoretical survival of the fittest quasispecies performing in silico simulations with binary 0 0 0.2 0.4 0.6 0.8 1 replicons, thus potentially taking into account the whole µ mutant spectrum of these quasispecies. The results of these simulations are in complete agreement with the mean field Fig. 7. Critical mutation rates as a function of self-replication rate of the model. In general, the conclusions drawn from the CA flat quasispecies. We show the values of mc predicted by the mean field ð0Þ model are in excellent agreement with the mean field model (thick solid line) with f x ¼ 1 and  ¼ 0. For the in silico model (white circles, with rm ¼ 1 and rp ¼ 0:05), we plot the value of the lower analysis and with the in silico simulations, although the mutation probability involving (after 3000 generations) a whole popula- spatial structure is shown to broaden the conditions at tion composed by flattest strings. For the CA model (triangles) we which the survival-of-the-flattest may occur. This result is compute (using Eq. (7)), the lower mutation probability involving the of clear relevance for the experimental data supporting the outcompetition of the fit quasispecies by the flat one with f m ¼ 1, applicability of the survival-of-the-flattest hypothesis to f pool ¼ 0:05,  ¼ 0:05 and L ¼ 128 (the same critical mutation values are found for L ¼ 40, 80, and 200 (results not shown)). Dashed and thin solid viral populations and in particular to the observations of lines correspond, respectively, to linear regressions for the critical Codoner et al. (2006) with viroids and by extension to plant ˜ mutation values for the bit string model of Section 3 and for the CA viruses. Plant viruses replicate and systematically move model. within infected plants forming spatially structured patchy populations rather than a single well-mixed population (Dietrich and Maiss, 2003; Hall et al., 2001; Jridi et al., related phenomenon is being observed within the context 2006). Different leaves may not necessarily be infected with of competing quasispecies. the same virus genotype and, even within a single infected leaf, different genotypes may not coinfect the same cell and 5. Discussion thus, not compete each other intracellularly. This being the case, a flat quasispecies may increase its frequency in the In the present work we have explored the dynamics of metapopulation by avoiding direct competition with the fit competition between two quasispecies located at two quasispecies even at low mutation rate. widely different fitness landscapes that correspond, respec- Antiviral drugs with mutagenic action (e.g., nucleoside tively, to a high-fitness and low-robustness (the fittest analogues like ribavirin) have been widely used, in landscape) and to a low-fitness and high-robustness (the combination with other drugs or interferons, as antiviral flattest landscape) scenarios. The effect of different therapies against HIV-1 and Hepatitis C virus. However, a mutation rates on the outcome of the competition between classic observation is that drug-resistant virus readily these two quasispecies has been explored. A mean field emerged upon treatment (Gao et al., 1992; Keulen et al., model, in silico simulations with binary digital replicators 1999; Larder and Kemp, 1989; Pfeiffer and Kirkegaard, and a spatially explicit model given by a stochastic two- 2003). In general, these resistant mutants present mutations dimensional cellular automata (CA) have been employed. at the polymerase gene which enhance fidelity at the cost of All these approaches show, in agreement with previous lowering replication down. In addition to this mechanism ´ studies (Codoner et al., 2006; Sanjuan et al., 2007; Wilke ˜ of resistance, flat quasispecies might provide an alternative et al., 2001; Wilke, 2001a), that under high mutation rates way of escaping from mutagenic therapies. In this sense, the flat quasispecies are advantaged, and thus selection ´ Sanjuan et al. (2007) have recently shown robust popula- positively acts on robustness more strongly than on tions of VSV, an animal RNA virus, that were able of replication speed. outcompeting the fittest wildtype in the presence of high The mean field analysis has allowed us to derive an concentrations of mutagens. These findings, along with our analytical solution for the critical mutation rate, mc , at theoretical results, have clear implications for the evolution which an absorbing first-order phase transition with critical of drug-resistant viral strains in vivo. When the mutagen

Slide 9: ARTICLE IN PRESS 568 ´s J. Sardanye et al. / Journal of Theoretical Biology 250 (2008) 560–568 has a high concentration across host tissues, robust Holmes, E.C., Moya, A., 2002. Is the quasispecies concept relevant to quasispecies can be selected and dominate the viral RNA viruses? J. Virol. 76, 460–465. Jenkins, G.M., Worobey, M., Woelk, C.H., Holmes, E.C., 2001. Evidence population. After suppression of the mutagenic therapy, for the non-quasispecies evolution of RNA viruses. Mol. Biol. Evol. wildtype would come back owing to its replicative 18, 987–994. advantage and, thus shall increase its frequency in the Jridi, C., Martin, J.F., Mareie-Jeanne, V., Labonne, G., Blanc, S., 2006. population. However, if flat quasispecies are spatially Distinct viral populations differentiate and evolve independently in a confined into reservoirs, then our results show that the single perennial host plant. J. Virol. 80, 2349–2357. time required to eliminate them from the population will be Keulen, W., van Wijk, A., Schuurman, R., Berkhout, B., Boucher, C.A., 1999. Increased polymerase fidelity of lamivudine-resistant HIV-1 much longer than expected for the case of a well-mixed variants does not limit their evolutionary potential. AIDS 13, single population. 1343–1349. Larder, B.A., Kemp, S.D., 1989. Multiple mutations in HIV-1 reverse Acknowledgments transcriptase confer high-level resistance to zidovudine (AZT). Science 246, 1155–1158. Li, H., Roossinck, M.J., 2004. Genetic bottlenecks reduce population This work has been supported by an E.U. PACE Project variation in an experimental RNA virus population. J. Virol. 78, ´ grant to J. Sardanyes within the 6th Framework Program 10582–10587. under Contract FP6-002035 (Programmable Artificial Cell Pfeiffer, J.K., Kirkegaard, K.P., 2003. A single mutation in poliovirus ` Evolution) and by the Santa Fe Institute. Work in Valencia RNA-dependent RNA polymerase confers resistance to mutagenic nucleotide analogs via increased fidelity. Proc. Natl Acad. Sci. USA was supported by Grant BFU2006-14819-C02-01/BMC 100, 7289–7294. from the Spanish MEC-FEDER and by the EMBO Young ´ Sacristan, S., Malpica, J.M., Fraile, A., Garcı´ a-Arenal, F., 2003. Investigator Program. Estimation of population bottlenecks during systemic movement of Tobacco mosaic virus in tobacco plants. J. Virol. 77, 9906–9911. References ´ Sanjuan, R., Codoner, F.M., Moya, A., Elena, S.F., 2004. Natural ˜ selection and the organ-specific differentiation of HIV-1 V3 hypervari- Altemeyer, S., McCaskill, J.S., 2001. Error threshold for spatially resolved able region. Evolution 58, 1185–1194. ´ ´ Sanjuan, R., Cuevas, J.M., Furio, V., Holmes, E.C., Moya, A., 2007. evolution in the quasispecies model. Phys. Rev. Lett. 86 (25), 5819–5822. Selection for robustness in mutagenized RNA viruses. PLoS Genet 3 ´ Borderı´ a, A.V., Codoner, F.M., Sanjuan, R., 2007. Selection promotes (6), e93. ˜ organ compartmentalization in HIV-1: evidence from gag and pol Schuster, P., Swetina, J., 1988. Stationary mutant distributions and evolutionary optimization. Bull. Math. Biol. 50, 635–660. genes. Evolution 61, 272–279. ` ´ ´ Sole, R.V., Bascompte, J., 2006. Self-organization in complex ecosystems. Codoner, F.M., Daros, J.A., Sole, R.V., Elena, S.F., 2006. The fittest ˜ versus the flattest: experimental confirmation of the quasispecies effect Monographs in Population Biology. Princeton University Press, Princeton. with subviral pathogens. PLOS Pathog. 2(12):e136. doi: 10.1371/ ´ ´ Sole, R.V., Ferrer, R., Gonzalez-Garcı´ a, I., Quer, J., Domingo, E., 1999. journal.ppat.0020136. Dietrich, C., Maiss, E., 2003. Fluorescent labeling reveals spatial Red Queen dynamics, competition and critical points in a model of separation of potyvirus populations in mixed infected ‘‘Nicotiana RNA virus quasispecies. J. Theor. Biol. 240 (3), 353–359. ´ ´ Sole, R.V., Sardanyes, J., Dı´ ez, J., Mas, A., 2006. Information catastrophe benthamiana’’ plants. J. Gen. Virol. 84, 2871–2876. Domingo, E., 2002. Quasispecies theory in virology. J. Virol. 76, 463–465. in RNA viruses through replication thresholds. J. Theor. Biol. 240 (3), Domingo, E., Holland, J.J., 1997. RNA virus mutations and fitness for 353–359. survival. Annu. Rev. Microbiol. 51, 151–178. Toyabe, S., Sano, M., 2005. Spatial supression of error catastrophe in a growing pattern. Physica D 203, 1–8. Eigen, M., Schuster, P., 1979. The Hypercycle. A Principle of Natural Self- Organization. Springer, Berlin. van Nimwegen, E., Crutchfield, J.P., Huynen, M., 1999. Neutral evolu- Eigen, M., McCaskill, J., Schuster, P., 1988. Molecular quasispecies.