A Stochastic Model for the Development of. Bateson-Dobzhansky-Muller Incompatibilities that Incorporates. Protein Interaction Networks

Transcription

1 A Stochastic Model for the Development of Bateson-Dobzhansky-Muller Incompatibilities that Incorporates Protein Interaction Networks Garner Cochran,, Andrius Dagilis,, Karen MacPherson,, Kerry Seitz,, Peter Olofsson, Kevin Livingstone August 13, 011 Department of Mathematics, Trinity University, San Antonio, Texas, 781 Equal Contributors Department of Biology, Trinity University, San Antonio, Texas, 781 1

2 Running Head: BDM interactions and PPI networks Keywords: Bateson-Dobzhansky-Muller Interactions, Protein-Protein Interaction Networks, Reproductive Incompatibility, Speciation Author for Correspondence: Kevin Livingstone Department of Biology 1 Trinity Place San Antonio, TX 781 ( voice ( fax klivings@trinity.edu

3 ABSTRACT Speciation is characterized by the development of reproductive isolating barriers between diverging groups. Intrinsic post-zygotic barriers of the type envisioned by Bateson, Dobzhansky, and Muller are deleterious epistatic interactions among loci that reduce hybrid fitness, leading to reproductive isolation. The first formal population genetic model of the development of these barriers was published by Orr in 1995, and here we develop a more general model of this process by incorporating finite protein-protein interaction networks, which limit the probability of deleterious interactions in vivo. Our model shows that the development of deleterious interactions is limited by the density of the proteinprotein interaction network. We have confirmed our analytical predictions of the number of possible interactions given the number of allele substitutions by using simulations on the Saccharomyces cerevisiae protein-protein interaction network. These results allow us to define the rate at which deleterious interactions are expected to form, and hence the speciation rate, for any protein-protein interaction network. 3

4 INTRODUCTION A long-recognized hallmark of speciation is the development of intrinsic reproductive isolating barriers (RIB. As evolutionary principles were being reconciled with modern genetics, BATESON (1909, DOBZHANSKY (1937, and MULLER (194 independently derived genetic models that allowed for the development of these barriers in diverging lineages. All of these models describe how fixation of mutations at two or more loci in different populations could result in sterile or inviable hybrid offspring, without the mutations causing lowered fitness within either population. Briefly, these models start with an ancestral population with genotype aabb; in one population, the A allele arises and becomes fixed, and in the other population, B arises and is fixed. The resulting hybrid from the AAbb aabb cross would have genotype AaBb, and as A and B have never been tested together, they could behave epistatically to cause a deleterious incompatibility. The accumulation of such Bateson-Dobzhansky-Muller incompatibilities (BDMIs can cause permanent isolation, and hence speciation. Recent work in a variety of taxa has led to the characterization of BDMIs, including cloning of the loci involved (ANDERSON et al. 010; PRESGRAVES 010; RIESEBERG and BLACKMAN 010. Although the BDM model for the development of RIBs was widely accepted, few efforts were made to extend this theory through mathematical models until Orr s landmark paper (ORR 1995, which has subsequently been elaborated by many others (TURELLI and ORR 000; GAVRILETS 003; KONDRASHOV 003; FITZPATRICK 008; PALMER and FELDMAN 009; FIERST and HANSEN 010. In the basic Orr model, two diverging lineages fix new alleles at K loci between them, and each new allele has a probability p of causing a negative interaction with all the alleles in the other genome at loci where a substitution has previously been fixed. One of the main insights to come from this model is that the probability of speciation rises as a function of K, a phenomenon that has come to be known as the snowball effect. This snowballing of BDMIs has recently been described in both Drosophila (MATUTE et al. 010 and Solanum (MOYLE and NAKAZATO 010. We know, however, that not all genes in the genome interact with each other in a way that could lead to the possibility of BDMIs between them. In fact, we have learned through genomics and proteomics techniques that across all the proteins in an organism, the largest fraction of proteins is connected to a small number of other proteins, and very few proteins act as central hubs with a myriad of interactions (JEONG et al. 001; WAGNER 001. While Orr incorporated limited interaction networks in a later paper, this attempt was recognized as only a caricature of a web of interactions (ORR and TURELLI 001. In this work, we incorporate the structure of finite protein-protein interaction (PPI networks to create a more general model for the development of BDMIs. MATERIALS AND METHODS Model: The starting point for the model is a network of the interactions between loci present in the most recent 4

5 common ancestor of two diverging groups. Interactions here are defined broadly, and could be physical, genetic, or biochemical, so long as there exists a way for the loci to possibly cause a BDMI. We treat interaction networks as undirected graphs where each node of the graph represents a locus, and each existing interaction is denoted by an edge. These graphs have no parallel edges, and because they are undirected, there is no distinction between edges (a, b and (b, a. The process of developing BDMIs proceeds by randomly selecting nodes without replacement, which corresponds to mutations being fixed in either lineage and only allows one mutation per locus. A potential BDMI arises when both nodes connected by an edge are selected, and the number of potential BDMIs after K mutations is a random variable, X K. We considered three different protein interaction network types that we call the complete, biological, and disjoint networks (Figure 1. The parameters we use to describe the properties of each graph and the speciation process are N, the number of nodes in the graph; N E, the number of edges in the graph; and K, the number of substitutions occurring in either lineage since divergence. In the complete network (Figure 1A, every protein has an interaction with every other protein, and X K = ( K. In this formulation, speciation would occur as described in Orr s original model (ORR The second network we considered is termed the biological network (Figure 1B. PPI networks follow a power law (WAGNER 001, which translates to orders of magnitude variability in the number of edges across nodes in the network, which we reasoned would affect X K. The PPI network of Saccharomyces cerevisiae was investigated as a model interaction network for our simulation studies. We used the database of physical and genetic interactions in BioGrid Release.0.55 (STARK et al. 006, which contains 6018 nodes and 157,861 edges after removing duplicates and converting directed edges to undirected edges. While previous studies have shown that such networks can have high false positive and negative rates (RIVES and GALITSKI 003, to our knowledge S. cerevisiae has the most complete and reliable of the PPI networks, and thus would be the best model. The disjoint network (Figure 1C models speciation through reciprocal silencing of gene duplicates, a phenomenon seen in Arabidopsis (BIKARD et al. 009 and Oryza sativa (MIZUTA et al In this graph, the nodes represent pairs of gene duplications present in the ancestral genome, and these pairs are connected by an edge because an organism must have at least one functioning duplicate in order to survive. Reproductive isolation in this model occurs when an ancestral population with duplicated loci A1 and A splits, and in one population A1 is silenced through deletion or drift to give genotype A1 s A1 s AA, while in the second population the other duplicate is silenced, giving the genotype A1A1 A s A s. The hybrid of these two populations would be genotype A1A1 s AA s, which would be viable because it possesses a functional copy of both genes. A proportion of selfed progeny from this individual, however, would have genotype A1 s A1 s A s A s, and would therefore be inviable or sterile. In this model, small islands of the genome become isolated first, which eventually leads to complete isolation. Analytically, the disjoint graph represents a worst case scenario for speciation due to the fact that it has the lowest possible number of edges ( N in 5

6 a graph where all nodes have at least one connection. Simulation Methods: The expected number of potential BDMIs in the yeast network after K substitutions was examined by running simulations for values of K from 5 to 490, in increments of 5. In each simulation, K nodes were chosen at random with equal probability, and the number of edges between these K nodes, X K, was counted. After 1000 runs of the selection process, frequencies for each of the edge counts from 0 to ( K were calculated. Another 1000 runs were then performed, and the frequencies were recalculated using the combined data. If the frequencies for any bin changed by more than 0.1%, another 1000 runs were performed and the process was repeated until the frequencies converged on stable estimates. Analytical Methods: In Orr s model, the probability of speciation, S, is given by: K S = 1 (1 p n 1 = 1 (1 p (K (1 n=1 where K is the number of mutations in both lineages, and p is the probability of a deleterious interaction between two loci. In our model, if X K is the number of potential BDMIs after K mutations, the conditional probability of speciation is S = 1 (1 p X K ( which we note is a random variable, depending on the outcome of X K. An expression for the (unconditional probability of speciation is obtained by computing the expected value of S: E[S] = 1 (1 p j P (X K = j (3 j=1 Because this probability needs the distribution of X K, which may be difficult to express, we use a first order Taylor expansion to approximate E[S] as a function of the expected number of potential BDMIs, E[X K ], between the K selected loci: E[S] 1 (1 p E[X K] (4 For any network structure, we must thus find E[X K ], and a general way to do this given the variety of topologies is to use indicator functions. An indicator function I is a random variable that takes on values in {0,1}, determined by whether some event is considered a success, in which case I = 1. Let us enumerate the edges in the graph from 1 to N E, and for edge j let success be defined as both nodes connecting that edge having been selected, which yields I j = 1. Then X K = I j (5 j=1 6

7 and by additivity of expected values: E[X K ] = E[I j ] j=1 = 1 P (I j = 1 j=1 (6 where P (I j = 1 = = ( N K ( N K K(K 1 for all j. N(N 1 The total number of combinations of mutations in the yeast graph is ( ( N K, and N K is the number of combinations when two mutated nodes are taken out of the set. Thus K(K 1 E[X K ] = N E N(N 1 ( (7 K = α where α = N E, the density of the network. ( N Note that E[X K ] is not dependent on the specific distribution of edges, only on K, N E, and N, and consequently this equation can be easily applied to any network. In the case of the complete network, the exact value of X K is ( K, which corresponds to the original model (ORR Thus, the probability of speciation can be expressed as follows: E[S] 1 (1 p α(k (8 In order to more fully describe the trajectory of speciation, we can also use a first order Taylor expansion to provide the variance of S, for which we need V ar[x K ]. While we can find a general function directly relating α and E[X K ], the specific structure of a network impacts the variance of X K, which can be calculated as follows: ( V ar[x K ] = N E P (1 P + N s P 3 + (( NE N S P 4 ( NE P (9 where N E = the number of edges total, N S = the number of edge pairs that share a node, P = ( N K ( N K, P 3 = ( N 3 K 3 ( N K, 7

8 and P 4 = ( N 4 K 4 ( N K. For proof of this equation, see Appendix A. Since no edge pairs share a node in the disjoint model, the equation for the variance in this graph is reduced to the following: V ar[x K ] = N E P (1 P + (( NE P 4 ( NE P (10 In the complete graph the variance is naturally 0. We can use V ar[x K ] to calculate the variance of S = 1 (1 p X K using the first order Taylor expansion: V ar[s] V ar[x K ] (1 p E[X K] (log(1 p (11 RESULTS AND DISCUSSION When running simulations on the yeast PPI network, the minimum number of sets of K draws required to reach convergence was on the order of 10 6, and the number of sets required to reach stable frequencies increased exponentially with K. These simulations can be compared to the predictions of the analytical model. We can use Equation 7 and Equation 9 to calculate E[X K ] and V ar[x K ] after the selection of K random nodes in the yeast PPI network. Given the 6018 nodes and 157,861 edges, E[X K ] = (K and V ar[xk ] = (K ( K (K ( K 4. The simulation data demonstrate that as K increases, the mean number of edges in the sample increases approximately as α (K, as predicted from our analytical results (Figure. While even for low values of K this estimate is close, at about K = 50 and above, the simulation and analytical results are nearly identical. The fact that E[X K ] can be reliably estimated given only the network density, α, leads to the question of how sensitive this analysis is to the reliability of the PPI dataset, especially given that questions have been raised about high false positive and negative rates in these datasets (RIVES and GALITSKI 003. We note that changes in the number of edges versus changes in α are proportional, so false positive and negative rates are less of a concern when the number of edges (PPI in the dataset is large. In the case of the yeast data, adding in or taking away 10,000 edges only increases or decreases alpha by 6.3%. We can use this model to examine the dynamics of speciation in each of the three networks we have considered. We have shown that speciation is dependent on the density of the network, α and the probability of an interaction being deleterious, p. As stated before, the complete network is the finite representation of Orr s model, with α = 1. In a complete network comparable in size to the yeast PPI network, incompatibilities snowball and speciation proceeds at a very rapid rate for 10 5 < p < 10 (Figure 3A, leading to a cumulative probability of speciation of 1 after substitutions have occurred at 0% of the loci. The disjoint model represents the need for at least one functional member of a pair of gene duplicates in the genome. 8

9 We have modeled speciation in a disjoint network with 6018 loci, representing a genome comprised entirely of 3009 duplicated loci (Figure 3C; while this is clearly implausible biologically, it does allow for comparisons to the other networks. The disjoint network has α = 1 N 1, the lowest possible value for α in a network where each node has at least one edge, leading to the slowest predicted rate of speciation. Interestingly, this prediction is in contrast with observations of an intraspecific BDMI caused by reciprocal silencing of duplicates in Arabidopsis (BIKARD et al. 009 and Oryza (MIZUTA et al These results could be reconciled by considering that mutations causing complete inactivation of duplicates might be more frequent, and such mutations would have a p of 1 in these networks. Speciation in the biological network proceeds at rates intermediate to the complete and disjoint networks (Figure 3B. We can use this network as a null model to look at speciation dynamics and to refine parameter estimates given additional assumptions and data. As an example, we can use the model to examine p. True biological estimates for p would be extremely difficult to obtain, as multiple alleles would need to be created for both loci in an interaction and all pairwise combinations evaluated to determine the phenotypic effects, so models such as this offer us one way to gain insight into this parameter. To obtain a lower bound for p, we start by following Orr in defining the random variable K S, the mutation that causes reproductive isolation. In our model, the probability that K S > N is non-zero, which makes K S not well defined. It seems reasonable, however, to assume that speciation will occur before all loci in the genome have undergone substitution, which makes P (K S > N = (1 p N E negligible. In effect, a reasonable range for p can be bounded by N E alone. In the case of the yeast PPI network, if for example we set P (K S > N 0.001, we estimate that p We can further refine a range for p by combining this model with experimental data. Under the assumption that speciation occurs before all loci in the genome have undergone substitution, we can compute E[K S ] conditional on K S N: E[K S K S N] π αp (1 [for derivation see (ORR 1995]. We can apply this equation to laboratory speciation in S. cerevisiae, where a BDMI was seen after 500 generations of strong divergent selection and 17 confirmed allelic substitutions (ANDERSON et al Substituting 17 for K S gives an estimate of p 0.6 in this system. Repeated lab trials under similar conditions would give a mean value for K S, providing an upper bound for p. In addition, this model could also be used to consider BDMIs that arise through more complex interactions. Complex interactions have been seen to cause RIBs in Saccharomyces sensu stricto yeasts (KAO et al. 010 and Drosophila (CABOT et al. 1994; ORR and IRVING 001, and some well established models look at such interactions (GAVRILETS 003. The number of complex interactions in a PPI network is far more difficult to compute either analytically or through simulations, primarily because there is no good definition for what constitutes a complex interaction, but our 9

10 model does provide a framework for such investigations. Finally, we note that our model presents a hypothesis as to why taxa may have similar rates of speciation (SEPKOSKI It can be assumed that the protein interaction networks for related taxa are of a similar size and density, which would lead to similar speciation rates. While other factors undoubtedly play a role in these processes, similarities in PPI networks among related taxa could account for at least part of this phenomenon. APPENDIX A VARIANCE PROOF Proof. In order to find the variance an approach must be made which takes into account the structure of individual nodes. Recall that X K = I j and hence j=1 V ar[x K ] = V ar[i j ] + j=1 i j=1 Cov[I i, I j ] where V ar[i j ] = E[I j ] E[I j], and Cov[I i, I j ] = E[I i I j ] E[I i ]E[I j ]. Since I j {0, 1} it follows that E[I j ] = E[I j] = P (I j = 1 = P, so V ar[i j ] = P (1 P. For the covariance, E[I i ] = E[I j ] = P, and E[I i I j ] = P (I i = 1 and I j = 1, the latter probability being dependent on whether the pair of edges share a node, or if they are disjoint. If they share a node, then P (I i = 1 and I j = 1 = ( N 3 K 3 ( N K = P 3 If they do not share a node, then P (I i = 1 and I j = 1 = ( N 4 K 4 ( N K = P 4 Using the notation from the expected value, i j=1 Cov[I i, I j ] = i<j=1 ( Cov[I i, I j ] = (N S P 3 + N D P 4 ( NE P where N S is the number of edge pairs which share a node, and N D is the number of edge pairs that do not share a node. Therefore ( V ar[x K ] = N E P (1 P + N s P 3 + (( NE N S P 4 ( NE P 10

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12 Acknowledgments: We gratefully acknowledge financial support from the Howard Hughes Medical Institute through a grant to Trinity University and the NSF through grant UBM REFERENCES ANDERSON, J. B., J. FUNT, D. A. THOMPSON, S. PRABHU, A. SOCHA, et al., 010 Determinants of Divergent Adaptation and Dobzhansky-Muller Interaction in Experimental Yeast Populations. Current Biology 0: BATESON, W., 1909 Heredity and variation in modern lights, In Darwin and Modern Science, editor, A. C. Seward. Cambridge University Press. BIKARD, D., D. PATEL, C. LE METTÉ, V. GIORGI, C. CAMILLERI, et al., 009 Divergent Evolution of Duplicate Genes Leads to Genetic Incompatibilities Within A. thaliana. Science 33: CABOT, E. L., A. W. DAVIS, N. A. JOHNSON, and C.-I. WU, 1994 Genetics of Reproductive Isolation in the Drosophila simulans Clade: Complex Epistasis Underlying Hybrid Male Sterility. Genetics 137: DOBZHANSKY, T., 1937 Genetic Nature of Species Differences. American Naturalist 71: FIERST, J. L., and T. F. HANSEN, 010 Genetic architecture and postzygotic reproductive isolation: Evolution of bateson-dobzhansky-muller incompatibilities in a polygenic model. Evolution 64: FITZPATRICK, B. M., 008 Hybrid Dysfunction: Population Genetic and Quantitative Genetic Perspectives. American Naturalist 171: GAVRILETS, S., 003 Perspective: Models of Speciation: What Have We Learned in 40 Years? Evolution 57: JEONG, H., S. P. MASON, A. L. BARABÁSI, and Z. N. OLTVAI, 001 Lethality and Centrality in Protein Networks. Nature 411: 41. KAO, K. C., K. SCHWARTZ, and G. SHERLOCK, 010 A Genome-Wide Analysis Reveals No Nuclear Dobzhansky- Muller Pairs of Determinants of Speciation Between S. cerevisiae and S. paradoxus, but Suggests More Complex Incompatibilities. PLoS Genetics 6: e KONDRASHOV, A. S., 003 Accumulation of Dobzhansky-Muller Incompatibilities Within a Spatially Structured Population. Evolution 57: MATUTE, D. R., I. A. BUTLER, D. A. TURISSINI, and J. A. COYNE, 010 A Test of the Snowball Theory for the Rate of Evolution of Hybrid Incompatibilities. Science 39: MIZUTA, Y., Y. HARUSHIMA, and N. KURATA, 010 Rice Pollen Hybrid Incompatibility Caused by Reciprocal Gene Loss of Duplicated Genes. Proceedings of the National Academy of Sciences USA 107: MOYLE, L. C., and T. NAKAZATO, 010 Hybrid Incompatibility "Snowballs" Between Solanum Species. Science 1

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14 FIGURE 1. EXAMPLE GRAPHS CONSIDERED IN THIS STUDY. (A The complete graph, where all nodes connect to all other nodes. (B The biological graph models networks where some nodes have many connections while others have few. (C The disjoint graph, where each node connects to only one other node. All graphs are of size N = 10. FIGURE. THE MEAN NUMBER OF INTERACTIONS, X K, AFTER K MUTATIONS. Simulation results for the Saccharomyces cerevisiae network (dots with 1 standard deviation error bars are compared to analytical model predictions for the mean (bold line and standard deviation (dashed lines. FIGURE 3. CUMULATIVE SPECIATION PROBABILITY CURVES FOR THE THREE DIFFERENT NET- WORKS. Curves for the complete (A, biological (B, and disjoint (C networks are shown. In all plots, N = 6018 and p = 10 (solid, 10 3 (dots, 10 4 (dashed, 10 5 (dash-dot 14

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