Evolving protein interaction networks through gene duplication

Activity networks determine project performance

Projects are characterised by activity networks with a critical path, a sequence of activities from start to end, that must be finished on time to complete the project on time. Watching over the critical path is the project manager's strategy to ensure timely project completion. This intense focus on a single path contrasts the broader complex structure of the activity network, and is due to our poor understanding on how that structure influences this critical path. Here, we use a generative model and detailed data from 77 real world projects (+ $10 bn total budget) to demonstrate how this network structure forces us to look beyond the critical path. We introduce a duplication-split model of project schedules that yields (i) identical power-law in- and-out degree distributions and (ii) a vanishing fraction of critical path activities with schedule size. These predictions are corroborated in real projects. We demonstrate that the incidence of delayed activities in real projects is consistent with the expectation from percolation theory in complex networks. We conclude that delay propagation in project schedules is a network property and it is not confined to the critical path.

Early effects of gene duplication on the robustness and phenotypic variability of gene regulatory networks

Background

Research on gene duplication is abundant and comes from a wide range of approaches, from high-throughput analyses and experimental evolution to bioinformatics and theoretical models. Notwithstanding, a consensus is still lacking regarding evolutionary mechanisms involved in evolution through gene duplication as well as the conditions that affect them. We argue that a better understanding of evolution through gene duplication requires considering explicitly that genes do not act in isolation. It demands studying how the perturbation that gene duplication implies percolates through the web of gene interactions. Due to evolution’s contingent nature, the paths that lead to the final fate of duplicates must depend strongly on the early stages of gene duplication, before gene copies have accumulated distinctive changes.

Methods

Here we use a widely-known model of gene regulatory networks to study how gene duplication affects network behavior in early stages. Such networks comprise sets of genes that cross-regulate. They organize gene activity creating the gene expression patterns that give cells their phenotypic properties. We focus on how duplication affects two evolutionarily relevant properties of gene regulatory networks: mitigation of the effect of new mutations and access to new phenotypic variants through mutation.

Results

Among other observations, we find that those networks that are better at maintaining the original phenotype after duplication are usually also better at buffering the effect of single interaction mutations and that duplication tends to enhance further this ability. Moreover, the effect of mutations after duplication depends on both the kind of mutation and genes involved in it. We also found that those phenotypes that had easier access through mutation before duplication had higher chances of remaining accessible through new mutations after duplication.

Conclusion

Our results support that gene duplication often mitigates the impact of new mutations and that this effect is not merely due to changes in the number of genes. The work that we put forward helps to identify conditions under which gene duplication may enhance evolvability and robustness to mutations.

Structure of networks that evolve under a combination of growth and contraction

We present analytical results for the emerging structure of networks that evolve via a combination of growth (by node addition and random attachment) and contraction (by random node deletion). To this end we consider a network model in which at each time step a node addition and random attachment step takes place with probability P_{add} and a random node deletion step takes place with probability P_{del}=1-P_{add}. The balance between the growth and contraction processes is captured by the parameter η=P_{add}-P_{del}. The case of pure network growth is described by η=1. In the case that 0<η<1, the rate of node addition exceeds the rate of node deletion and the overall process is of network growth. In the opposite case, where -1<η<0, the overall process is of network contraction, while in the special case of η=0 the expected size of the network remains fixed, apart from fluctuations. Using the master equation and the generating function formalism, we obtain a closed-form expression for the time-dependent degree distribution P_{t}(k). The degree distribution P_{t}(k) includes a term that depends on the initial degree distribution P_{0}(k), which decays as time evolves, and an asymptotic distribution P_{st}(k) which is independent of the initial condition. In the case of pure network growth (η=1), the asymptotic distribution P_{st}(k) follows an exponential distribution, while for -1<η<1 it consists of a sum of Poisson-like terms and exhibits a Poisson-like tail. In the case of overall network growth (0<η<1) the degree distribution P_{t}(k) eventually converges to P_{st}(k). In the case of overall network contraction (-1<η<0) we identify two different regimes. For -1/3<η<0 the degree distribution P_{t}(k) quickly converges towards P_{st}(k). In contrast, for -1<η<-1/3 the convergence of P_{t}(k) is initially very slow and it gets closer to P_{st}(k) only shortly before the network vanishes. Thus, the model exhibits three phase transitions: a structural transition between two functional forms of P_{st}(k) at η=1, a transition between an overall growth and overall contraction at η=0, and a dynamical transition between fast and slow convergence towards P_{st}(k) at η=-1/3. The analytical results are found to be in very good agreement with the results obtained from computer simulations.

Feature importance network reveals novel functional relationships between biological features in Arabidopsis thaliana

Understanding how the different cellular components are working together to form a living cell requires multidisciplinary approaches combining molecular and computational biology. Machine learning shows great potential in life sciences, as it can find novel relationships between biological features. Here, we constructed a dataset of 11,801 gene features for 31,522 Arabidopsis thaliana genes and developed a machine learning workflow to identify linked features. The detected linked features are visualised as a Feature Important Network (FIN), which can be mined to reveal a variety of novel biological insights pertaining to gene function. We demonstrate how FIN can be used to generate novel insights into gene function. To make this network easily accessible to the scientific community, we present the FINder database, available at finder.plant.tools.

Exponential synchronization for variable-order fractional discontinuous complex dynamical networks with short memory via impulsive control

This paper considers the exponential synchronization issue for variable-order fractional complex dynamical networks (FCDNs) with short memory and derivative couplings via the impulsive control scheme, where dynamical nodes are modeled to be discontinuous. Firstly, the mathematics model with respect to variable-order fractional systems with short memory is established under the impulsive controller, in which the impulse strength is not only determined by the impulse control gain, but also the order of the control systems. Secondly, the exponential stability criterion for variable-order fractional systems with short memory is developed. Thirdly, the hybrid controller, which consists of the impulsive coupling controller and the discontinuous feedback controller, is designed to realize the synchronization objective. In addition, by constructing Lyapunov functional and applying inequality analysis techniques, the synchronization conditions are achieved in terms of linear matrix inequalities (LMIs). Finally, two simulation examples are performed to verify the effectiveness of the developed synchronization scheme and the theoretical outcomes.

Scaling up real networks by geometric branching growth

Real networks often grow through the sequential addition of new nodes that connect to older ones in the graph. However, many real systems evolve through the branching of fundamental units, whether those be scientific fields, countries, or species. Here, we provide empirical evidence for self-similar growth of network structure in the evolution of real systems-the journal-citation network and the world trade web-and present the geometric branching growth model, which predicts this evolution and explains the symmetries observed. The model produces multiscale unfolding of a network in a sequence of scaled-up replicas preserving network features, including clustering and community structure, at all scales. Practical applications in real instances include the tuning of network size for best response to external influence and finite-size scaling to assess critical behavior under random link failures.

Revisiting Parameter Estimation in Biological Networks: Influence of Symmetries

Graph models often give us a deeper understanding of real-world networks. In the case of biological networks they help in predicting the evolution and history of biomolecule interactions, provided we map properly real networks into the corresponding graph models. In this paper, we show that for biological graph models many of the existing parameter estimation techniques overlook the critical property of graph symmetry (also known formally as graph automorphisms), thus the estimated parameters give statistically insignificant results concerning the observed network. To demonstrate it and to develop accurate estimation procedures, we focus on the biologically inspired duplication-divergence model, and the up-to-date data of protein-protein interactions of seven species including human and yeast. Using exact recurrence relations of some prominent graph statistics, we devise a parameter estimation technique that provides the right order of symmetries and uses phylogenetically old proteins as the choice of seed graph nodes. We also find that our results are consistent with the ones obtained from maximum likelihood estimation (MLE). However, the MLE approach is significantly slower than our methods in practice.

A review of protein-protein interaction network alignment: From pathway comparison to global alignment

Network alignment provides a comprehensive way to discover the similar parts between molecular systems of different species based on topological and biological similarity. With such a strong basis, one can do comparative studies at a systems level in the field of computational biology. In this survey paper, we focus on protein-protein interaction networks and review some representative algorithms for network alignment in the past two decades as well as the state-of-the-art aligners. We also introduce the most popular evaluation measures in the literature to benchmark the performance of these approaches. Finally, we address several future challenges and the possible ways to conquer the existing problems of biological network alignment.

Power-law distribution of degree-degree distance: A better representation of the scale-free property of complex networks

Whether real-world complex networks are scale free or not has long been controversial. Recently, in Broido and Clauset [A. D. Broido, A. Clauset, Nat. Commun. 10, 1017 (2019)], it was claimed that the degree distributions of real-world networks are rarely power law under statistical tests. Here, we attempt to address this issue by defining a fundamental property possessed by each link, the degree-degree distance, the distribution of which also shows signs of being power law by our empirical study. Surprisingly, although full-range statistical tests show that degree distributions are not often power law in real-world networks, we find that in more than half of the cases the degree-degree distance distributions can still be described by power laws. To explain these findings, we introduce a bidirectional preferential selection model where the link configuration is a randomly weighted, two-way selection process. The model does not always produce solid power-law distributions but predicts that the degree-degree distance distribution exhibits stronger power-law behavior than the degree distribution of a finite-size network, especially when the network is dense. We test the strength of our model and its predictive power by examining how real-world networks evolve into an overly dense stage and how the corresponding distributions change. We propose that being scale free is a property of a complex network that should be determined by its underlying mechanism (e.g., preferential attachment) rather than by apparent distribution statistics of finite size. We thus conclude that the degree-degree distance distribution better represents the scale-free property of a complex network.

Evolving complexity: how tinkering shapes cells, software and ecological networks

A common trait of complex systems is that they can be represented by means of a network of interacting parts. It is, in fact, the network organization (more than the parts) that largely conditions most higher-level properties, which are not reducible to the properties of the individual parts. Can the topological organization of these webs provide some insight into their evolutionary origins? Both biological and artificial networks share some common architectural traits. They are often heterogeneous and sparse, and most exhibit different types of correlations, such as nestedness, modularity or hierarchical patterns. These properties have often been attributed to the selection of functionally meaningful traits. However, a proper formulation of generative network models suggests a rather different picture. Against the standard selection-optimization argument, some networks reveal the inevitable generation of complex patterns resulting from reuse and can be modelled using duplication-rewiring rules lacking functionality. These give rise to the observed heterogeneous, scale-free and modular architectures. Here, we examine the evidence for tinkering in cellular, technological and ecological webs and its impact in shaping their architecture. Our analysis suggests a serious consideration of the role played by selection as the origin of network topology. Instead, we suggest that the amplification processes associated with reuse might shape these graphs at the topological level. In biological systems, selection forces would take advantage of emergent patterns. This article is part of the theme issue 'Unifying the essential concepts of biological networks: biological insights and philosophical foundations'.

Flexible model selection for mechanistic network models

Network models are applied across many domains where data can be represented as a network. Two prominent paradigms for modelling networks are statistical models (probabilistic models for the observed network) and mechanistic models (models for network growth and/or evolution). Mechanistic models are better suited for incorporating domain knowledge, to study effects of interventions (such as changes to specific mechanisms) and to forward simulate, but they typically have intractable likelihoods. As such, and in a stark contrast to statistical models, there is a relative dearth of research on model selection for such models despite the otherwise large body of extant work. In this article, we propose a simulator-based procedure for mechanistic network model selection that borrows aspects from Approximate Bayesian Computation along with a means to quantify the uncertainty in the selected model. To select the most suitable network model, we consider and assess the performance of several learning algorithms, most notably the so-called Super Learner, which makes our framework less sensitive to the choice of a particular learning algorithm. Our approach takes advantage of the ease to forward simulate from mechanistic network models to circumvent their intractable likelihoods. The overall process is flexible and widely applicable. Our simulation results demonstrate the approach's ability to accurately discriminate between competing mechanistic models. Finally, we showcase our approach with a protein-protein interaction network model from the literature for yeast (Saccharomyces cerevisiae).

Conservation and divergence of ancestral AGAMOUS/SEEDSTICK subfamily genes from the basal angiosperm Magnolia wufengensis

AGAMOUS/SEEDSTICK (AG/STK) subfamily genes play crucial roles in the reproductive development of plants. However, most of our current knowledge of AG/STK subfamily genes is restricted to core eudicots and grasses, and the knowledge of ancestral exon-intron structures, expression patterns, protein-protein interaction patterns and functions of AG/STK subfamily genes remains unclear. To determine these, we isolated AG/STK subfamily genes (MawuAG1, MawuAG2 and MawuSTK) from a woody basal angiosperm Magnolia wufengensis (Magnoliaceae). MawuSTK arose from the gene duplication event occurring before the diversification of extant angiosperms, and MawuAG1 and MawuAG2 may result from a gene duplication event occurring before the divergence of Magnoliaceae and Lauraceae. Gene duplication led to apparent diversification in their expression and interaction patterns. It revealed that expression in both stamens and carpels likely represents the ancestral expression profiles of AG lineage genes, and expression of STK-like genes in stamens may have been lost soon after the appearance of the STK lineage. Moreover, AG/STK subfamily proteins may have immediately established interactions with the SEPALLATA (SEP) subfamily proteins following the emergence of the SEP subfamily; however, their interactions with the APETALA1/FRUITFULL subfamily proteins or themselves differ from those found in monocots and basal and core eudicots. MawuAG1 plays highly conserved roles in the determinacy of stamen, carpel and ovule identity, while gene duplication contributed to the functional diversification of MawuAG2 and MawuSTK. In addition, we investigated the evolutionary history of exon-intron structural changes of the AG/STK subfamily, and a novel splice-acceptor mode (GUU-AU) and the convergent evolution of N-terminal extension in the euAG and PLE subclades were revealed for the first time. These results further advance our understanding of ancestral AG/STK subfamily genes in terms of phylogeny, exon-intron structures, expression and interaction patterns, and functions, and provide strong evidence for the significance of gene duplication in the expansion and evolution of the AG/STK subfamily.

A Bootstrap Method for Goodness of Fit and Model Selection with a Single Observed Network

Network models are applied in numerous domains where data arise from systems of interactions among pairs of actors. Both statistical and mechanistic network models are increasingly capable of capturing various dependencies among these actors. Yet, these dependencies pose statistical challenges for analyzing such data, especially when the data set comprises only a single observation of one network, often leading to intractable likelihoods regardless of the modeling paradigm and limiting the application of existing statistical methods for networks. We explore a subsampling bootstrap procedure to serve as the basis for goodness of fit and model selection with a single observed network that circumvents the intractability of such likelihoods. Our approach is based on flexible resampling distributions formed from the single observed network, allowing for more nuanced and higher dimensional comparisons than point estimates of quantities of interest. We include worked examples for model selection, with simulation, and assessment of goodness of fit, with duplication-divergence model fits for yeast (S.cerevisiae) protein-protein interaction data from the literature. The proposed approach produces a flexible resampling distribution that can be based on any network statistics of one's choosing and can be employed for both statistical and mechanistic network models.

A novel evolutionary model for constructing gene coexpression networks with comprehensive features

BACKGROUND

Uncovering the evolutionary principles of gene coexpression network is important for our understanding of the network topological property of new genes. However, most existing evolutionary models only considered the evolution of duplication genes and only based on the degree of genes, ignoring the other key topological properties. The evolutionary mechanism by which how are new genes integrated into the ancestral networks are not yet to be comprehensively characterized. Herein, based on the human ribonucleic acid-sequencing (RNA-seq) data, we develop a new evolutionary model of gene coexpression network which considers the evolutionary process of both duplication genes and de novo genes.

RESULTS

Based on the human RNA-seq data, we construct a gene coexpression network consisting of 8061 genes and 638624 links. We find that there are 1394 duplication genes and 126 de novo genes in the network. Then based on human gene age data, we reproduce the evolutionary process of this gene coexpression network and develop a new evolutionary model. We find that the generation rates of duplication genes and de novo genes are approximately 3.58/Myr (Myr=Million year) and 0.31/Myr, respectively. Based on the average degree and coreness of parent genes, we find that the gene duplication is a random process. Eventually duplication genes only inherit 12.89% connections from their parent genes and the retained connections have a smaller edge betweenness. Moreover, we find that both duplication genes and de novo genes prefer to develop new interactions with genes which have a large degree and a large coreness. Our proposed model can generate an evolutionary network when the number of newly added genes or the length of evolutionary time is known.

CONCLUSIONS

Gene duplication and de novo genes are two dominant evolutionary forces in shaping the coexpression network. Both duplication genes and de novo genes develop new interactions through a "rich-gets-richer" mechanism in terms of degree and coreness. This mechanism leads to the scale-free property and hierarchical architecture of biomolecular network. The proposed model is able to construct a gene coexpression network with comprehensive biological characteristics.

Identification of influential invaders in evolutionary populations

The identification of the most influential nodes has been a vibrant subject of research across the whole of network science. Here we map this problem to structured evolutionary populations, where strategies and the interaction network are both subject to change over time based on social inheritance. We study cooperative communities, which cheaters can invade because they avoid the cost of contributions that are associated with cooperation. The question that we seek to answer is at which nodes cheaters invade most successfully. We propose the weighted degree decomposition to identify and rank the most influential invaders. More specifically, we distinguish two kinds of ranking based on the weighted degree decomposition. We show that a ranking strategy based on negative-weighted degree allows to successfully identify the most influential invaders in the case of weak selection, while a ranking strategy based on positive-weighted degree performs better when the selection is strong. Our research thus reveals how to identify the most influential invaders based on statistical measures in dynamically evolving cooperative communities.

Gene duplication led to divergence of expression patterns, protein-protein interaction patterns and floral development functions of AGL6-like genes in the basal angiosperm Magnolia wufengensis (Magnoliaceae)

The MADS-box family genes play critical roles in the regulation of growth and development of flowering plants. AGAMOUS-LIKE 6 (AGL6)-like genes are one of the most enigmatic subfamilies of the MADS-box family because of highly variable expression patterns and ambiguous functions, which have long puzzled researchers. A lot of AGL6 homologs have been identified from gymnosperms and angiosperms. However, only a few have been characterized, especially for basal angiosperm taxa. Magnolia wufengensis is a woody basal angiosperm from the family Magnoliaceae. In the current study, the phylogenesis, expression and protein-protein interaction (PPI) patterns, and functions of two AGL6 homologs from M. wufengensis, MawuAGL6-1 and MawuAGL6-2, were analyzed. Phylogenetic analysis indicated that the two AGL6 duplicates may have arisen by gene duplication before the divergence of Magnoliaceae and Lauraceae, with the diversification of their expression and PPI patterns after gene duplication. Functional analysis revealed that, in addition to common functions in accelerating flowering, MawuAGL6-1 might be responsible for flower meristem determinacy, while MawuAGL6-2 is preferentially recruited to regulate tepal morphogenesis. These findings further advance our understanding of the evolution of phylogenesis, expression, interaction and functions of AGL6 lineage genes from basal angiosperms, as well as the entire AGL6 lineage genes, and the significance of AGL6 lineage genes in the evolution and biological diversity.

Topological evolution of coexpression networks by new gene integration maintains the hierarchical and modular structures in human ancestors

We analyze the global structure and evolution of human gene coexpression networks driven by new gene integration. When the Pearson correlation coefficient is greater than or equal to 0.5, we find that the coexpression network consists of 334 small components and one "giant" connected subnet comprising of 6317 interacting genes. This network shows the properties of power-law degree distribution and small-world. The average clustering coefficient of younger genes is larger than that of the elderly genes (0.6685 vs. 0.5762). Particularly, we find that the younger genes with a larger degree also show a property of hierarchical architecture. The younger genes play an important role in the overall pivotability of the network and this network contains few redundant duplicate genes. Moreover, we find that gene duplication and orphan genes are two dominant evolutionary forces in shaping this network. Both the duplicate genes and orphan genes develop new links through a "rich-gets-richer" mechanism. With the gradual integration of new genes into the ancestral network, most of the topological structure features of the network would gradually increase. However, the exponent of degree distribution and modularity coefficient of the whole network do not change significantly, which implies that the evolution of coexpression networks maintains the hierarchical and modular structures in human ancestors.

Scale-free networks are rare

Real-world networks are often claimed to be scale free, meaning that the fraction of nodes with degree k follows a power law k-α, a pattern with broad implications for the structure and dynamics of complex systems. However, the universality of scale-free networks remains controversial. Here, we organize different definitions of scale-free networks and construct a severe test of their empirical prevalence using state-of-the-art statistical tools applied to nearly 1000 social, biological, technological, transportation, and information networks. Across these networks, we find robust evidence that strongly scale-free structure is empirically rare, while for most networks, log-normal distributions fit the data as well or better than power laws. Furthermore, social networks are at best weakly scale free, while a handful of technological and biological networks appear strongly scale free. These findings highlight the structural diversity of real-world networks and the need for new theoretical explanations of these non-scale-free patterns.

Zipf's Law, unbounded complexity and open-ended evolution

A major problem for evolutionary theory is understanding the so-called open-ended nature of evolutionary change, from its definition to its origins. Open-ended evolution (OEE) refers to the unbounded increase in complexity that seems to characterize evolution on multiple scales. This property seems to be a characteristic feature of biological and technological evolution and is strongly tied to the generative potential associated with combinatorics, which allows the system to grow and expand their available state spaces. Interestingly, many complex systems presumably displaying OEE, from language to proteins, share a common statistical property: the presence of Zipf's Law. Given an inventory of basic items (such as words or protein domains) required to build more complex structures (sentences or proteins) Zipf's Law tells us that most of these elements are rare whereas a few of them are extremely common. Using algorithmic information theory, in this paper we provide a fundamental definition for open-endedness, which can be understood as postulates. Its statistical counterpart, based on standard Shannon information theory, has the structure of a variational problem which is shown to lead to Zipf's Law as the expected consequence of an evolutionary process displaying OEE. We further explore the problem of information conservation through an OEE process and we conclude that statistical information (standard Shannon information) is not conserved, resulting in the paradoxical situation in which the increase of information content has the effect of erasing itself. We prove that this paradox is solved if we consider non-statistical forms of information. This last result implies that standard information theory may not be a suitable theoretical framework to explore the persistence and increase of the information content in OEE systems.

Partial-Nodes-Based State Estimation for Complex Networks With Unbounded Distributed Delays

In this brief, the new problem of partial-nodes-based (PNB) state estimation problem is investigated for a class of complex network with unbounded distributed delays and energy-bounded measurement noises. The main novelty lies in that the states of the complex network are estimated through measurement outputs of a fraction of the network nodes. Such fraction of the nodes is determined by either the practical availability or the computational necessity. The PNB state estimator is designed such that the error dynamics of the network state estimation is exponentially ultimately bounded in the presence of measurement errors. Sufficient conditions are established to ensure the existence of the PNB state estimators and then the explicit expression of the gain matrices of such estimators is characterized. When the network measurements are free of noises, the main results specialize to the case of exponential stability for error dynamics. Numerical examples are presented to verify the theoretical results.

Network Analysis as a Grand Unifier in Biomedical Data Science

Biomedical data scientists study many types of networks, ranging from those formed by neurons to those created by molecular interactions. People often criticize these networks as uninterpretable diagrams termed hairballs; however, here we show that molecular biological networks can be interpreted in several straightforward ways. First, we can break down a network into smaller components, focusing on individual pathways and modules. Second, we can compute global statistics describing the network as a whole. Third, we can compare networks. These comparisons can be within the same context (e.g., between two gene regulatory networks) or cross-disciplinary (e.g., between regulatory networks and governmental hierarchies). The latter comparisons can transfer a formalism, such as that for Markov chains, from one context to another or relate our intuitions in a familiar setting (e.g., social networks) to the relatively unfamiliar molecular context. Finally, key aspects of molecular networks are dynamics and evolution, i.e., how they evolve over time and how genetic variants affect them. By studying the relationships between variants in networks, we can begin to interpret many common diseases, such as cancer and heart disease.

A Preferential Attachment Paradox: How Preferential Attachment Combines with Growth to Produce Networks with Log-normal In-degree Distributions

Every network scientist knows that preferential attachment combines with growth to produce networks with power-law in-degree distributions. How, then, is it possible for the network of American Physical Society journal collection citations to enjoy a log-normal citation distribution when it was found to have grown in accordance with preferential attachment? This anomalous result, which we exalt as the preferential attachment paradox, has remained unexplained since the physicist Sidney Redner first made light of it over a decade ago. Here we propose a resolution. The chief source of the mischief, we contend, lies in Redner having relied on a measurement procedure bereft of the accuracy required to distinguish preferential attachment from another form of attachment that is consistent with a log-normal in-degree distribution. There was a high-accuracy measurement procedure in use at the time, but it would have have been difficult to use it to shed light on the paradox, due to the presence of a systematic error inducing design flaw. In recent years the design flaw had been recognised and corrected. We show that the bringing of the newly corrected measurement procedure to bear on the data leads to a resolution of the paradox.

The Emergence of Canalization and Evolvability in an Open-Ended, Interactive Evolutionary System

Many believe that an essential component for the discovery of the tremendous diversity in natural organisms was the evolution of evolvability, whereby evolution speeds up its ability to innovate by generating a more adaptive pool of offspring. One hypothesized mechanism for evolvability is developmental canalization, wherein certain dimensions of variation become more likely to be traversed and others are prevented from being explored (e.g., offspring tend to have similar-size legs, and mutations affect the length of both legs, not each leg individually). While ubiquitous in nature, canalization is rarely reported in computational simulations of evolution, which deprives us of in silico examples of canalization to study and raises the question of which conditions give rise to this form of evolvability. Answering this question would shed light on why such evolvability emerged naturally, and it could accelerate engineering efforts to harness evolution to solve important engineering challenges. In this article, we reveal a unique system in which canalization did emerge in computational evolution. We document that genomes entrench certain dimensions of variation that were frequently explored during their evolutionary history. The genetic representation of these organisms also evolved to be more modular and hierarchical than expected by chance, and we show that these organizational properties correlate with increased fitness. Interestingly, the type of computational evolutionary experiment that produced this evolvability was very different from traditional digital evolution in that there was no objective, suggesting that open-ended, divergent evolutionary processes may be necessary for the evolution of evolvability.

Mean field analysis of algorithms for scale-free networks in molecular biology

The sampling of scale-free networks in Molecular Biology is usually achieved by growing networks from a seed using recursive algorithms with elementary moves which include the addition and deletion of nodes and bonds. These algorithms include the Barabási-Albert algorithm. Later algorithms, such as the Duplication-Divergence algorithm, the Solé algorithm and the iSite algorithm, were inspired by biological processes underlying the evolution of protein networks, and the networks they produce differ essentially from networks grown by the Barabási-Albert algorithm. In this paper the mean field analysis of these algorithms is reconsidered, and extended to variant and modified implementations of the algorithms. The degree sequences of scale-free networks decay according to a powerlaw distribution, namely P(k) ∼ k^−γ, where γ is a scaling exponent. We derive mean field expressions for γ, and test these by numerical simulations. Generally, good agreement is obtained. We also found that some algorithms do not produce scale-free networks (for example some variant Barabási-Albert and Solé networks).

Puzzles in modern biology. V. Why are genomes overwired?

Many factors affect eukaryotic gene expression. Transcription factors, histone codes, DNA folding, and noncoding RNA modulate expression. Those factors interact in large, broadly connected regulatory control networks. An engineer following classical principles of control theory would design a simpler regulatory network. Why are genomes overwired? Neutrality or enhanced robustness may lead to the accumulation of additional factors that complicate network architecture. Dynamics progresses like a ratchet. New factors get added. Genomes adapt to the additional complexity. The newly added factors can no longer be removed without significant loss of fitness. Alternatively, highly wired genomes may be more malleable. In large networks, most genomic variants tend to have a relatively small effect on gene expression and trait values. Many small effects lead to a smooth gradient, in which traits may change steadily with respect to underlying regulatory changes. A smooth gradient may provide a continuous path from a starting point up to the highest peak of performance. A potential path of increasing performance promotes adaptability and learning. Genomes gain by the inductive process of natural selection, a trial and error learning algorithm that discovers general solutions for adapting to environmental challenge. Similarly, deeply and densely connected computational networks gain by various inductive trial and error learning procedures, in which the networks learn to reduce the errors in sequential trials. Overwiring alters the geometry of induction by smoothing the gradient along the inductive pathways of improving performance. Those overwiring benefits for induction apply to both natural biological networks and artificial deep learning networks.

The Beneficial Role of Mobility for the Emergence of Innovation

Innovation is a key ingredient for the evolution of several systems, including social and biological ones. Focused investigations and lateral thinking may lead to innovation, as well as serendipity and other random discovery processes. Some individuals are talented at proposing innovation (say innovators), while others at deeply exploring proposed novelties, at getting further insights on a theory, or at developing products, services, and so on (say developers). This separation in terms of innovators and developers raises an issue of paramount importance: under which conditions a system is able to maintain innovators? According to a simple model, this work investigates the evolutionary dynamics that characterize the emergence of innovation. In particular, we consider a population of innovators and developers, in which agents form small groups whose composition is crucial for their payoff. The latter depends on the heterogeneity of the formed groups, on the amount of innovators they include, and on an award-factor that represents the policy of the system for promoting innovation. Under the hypothesis that a "mobility" effect may support the emergence of innovation, we compare the equilibria reached by our population in different cases. Results confirm the beneficial role of "mobility", and the emergence of further interesting phenomena.

Protein interaction evolution from promiscuity to specificity with reduced flexibility in an increasingly complex network

A key question regarding protein evolution is how proteins adapt to the dynamic environment in which they function and how in turn their evolution shapes the protein interaction network. We used extant and resurrected ancestral plant MADS-domain transcription factors to understand how SEPALLATA3, a protein with hub and glue properties, evolved and takes part in network organization. Although the density of dimeric interactions was saturated in the network, many new interactions became mediated by SEPALLATA3 after a whole genome triplication event. By swapping SEPALLATA3 and its ancestors between dimeric networks of different ages, we found that the protein lost the capacity of promiscuous interaction and acquired specificity in evolution. This was accompanied with constraints on conformations through proline residue accumulation, which made the protein less flexible. SHORT VEGETATIVE PHASE on the other hand (non-hub) was able to gain protein-protein interactions due to a C-terminal domain insertion, allowing for a larger interaction interface. These findings illustrate that protein interaction evolution occurs at the level of conformational dynamics, when the binding mechanism concerns an induced fit or conformational selection. Proteins can evolve towards increased specificity with reduced flexibility when the complexity of the protein interaction network requires specificity.

Cluster Synchronization on Multiple Nonlinearly Coupled Dynamical Subnetworks of Complex Networks With Nonidentical Nodes

In this paper, cluster synchronization on multiple nonlinearly coupled dynamical subnetworks of complex networks with nonidentical nodes and stochastic perturbations is studied. Based on the general leader-follower's model, an improved network structure model that consists of multiple pairs of matching subnetworks, each of which includes a leaders' subnetwork and a followers' subnetwork, is proposed. Moreover, the dynamical behaviors of the nodes belonging to the same pair of matching subnetworks are identical, while the ones belonging to different pairs of unmatched subnetworks are nonidentical. In this new setting, the aim is to design some suitable adaptive pinning controllers on the chosen nodes of each followers' subnetwork, such that the nodes in each subnetwork can be exponentially synchronized onto their reference state. Then, some cluster synchronization criteria for multiple nonlinearly coupled dynamical subnetworks of complex networks are established, and a pinning control scheme that the nodes with very large or low degrees are good candidates for applying pinning controllers is presented. Suitable adaptive update laws are used to deal with the unknown feedback gains between the pinned nodes and their leaders. Finally, several numerical simulations are given to demonstrate the effectiveness and applicability of the proposed approach.

Alignment of PPI Networks Using Semantic Similarity for Conserved Protein Complex Prediction

Network alignment is a computational technique to identify topological similarity of graph data by mapping link patterns. In bioinformatics, network alignment algorithms have been applied to protein-protein interaction (PPI) networks to discover evolutionarily conserved substructures at the system level. In particular, local network alignment of PPI networks searches for conserved functional components between species and predicts unknown protein complexes and signaling pathways. In this article, we present a novel approach of local network alignment by semantic mapping. While most previous methods find protein matches between species by sequence homology, our approach uses semantic similarity. Given Gene Ontology (GO) and its annotation data, we estimate functional closeness between two proteins by measuring their semantic similarity. We adopted a new semantic similarity measure, simVICD, which has the best performance for PPI validation and functional match. We tested alignment between the PPI networks of well-studied yeast protein complexes and the genome-wide PPI network of human in order to predict human protein complexes. The experimental results demonstrate that our approach has higher accuracy in protein complex prediction than graph clustering algorithms, and higher efficiency than previous network alignment algorithms.

Evolution of herbivore-induced early defense signaling was shaped by genome-wide duplications in Nicotiana

Herbivore-induced defenses are widespread, rapidly evolving and relevant for plant fitness. Such induced defenses are often mediated by early defense signaling (EDS) rapidly activated by the perception of herbivore associated elicitors (HAE) that includes transient accumulations of jasmonic acid (JA). Analyzing 60 HAE-induced leaf transcriptomes from closely-related Nicotiana species revealed a key gene co-expression network (M4 module) which is co-activated with the HAE-induced JA accumulations but is elicited independently of JA, as revealed in plants silenced in JA signaling. Functional annotations of the M4 module were consistent with roles in EDS and a newly identified hub gene of the M4 module (NaLRRK1) mediates a negative feedback loop with JA signaling. Phylogenomic analysis revealed preferential gene retention after genome-wide duplications shaped the evolution of HAE-induced EDS in Nicotiana. These results highlight the importance of genome-wide duplications in the evolution of adaptive traits in plants.

Joint estimation of preferential attachment and node fitness in growing complex networks

Complex network growth across diverse fields of science is hypothesized to be driven in the main by a combination of preferential attachment and node fitness processes. For measuring the respective influences of these processes, previous approaches make strong and untested assumptions on the functional forms of either the preferential attachment function or fitness function or both. We introduce a Bayesian statistical method called PAFit to estimate preferential attachment and node fitness without imposing such functional constraints that works by maximizing a log-likelihood function with suitably added regularization terms. We use PAFit to investigate the interplay between preferential attachment and node fitness processes in a Facebook wall-post network. While we uncover evidence for both preferential attachment and node fitness, thus validating the hypothesis that these processes together drive complex network evolution, we also find that node fitness plays the bigger role in determining the degree of a node. This is the first validation of its kind on real-world network data. But surprisingly the rate of preferential attachment is found to deviate from the conventional log-linear form when node fitness is taken into account. The proposed method is implemented in the R package PAFit.

Global Alignment of Protein-Protein Interaction Networks: A Survey

In this paper, we survey algorithms that perform global alignment of networks or graphs. Global network alignment aligns two or more given networks to find the best mapping from nodes in one network to nodes in other networks. Since graphs are a common method of data representation, graph alignment has become important with many significant applications. Protein-protein interactions can be modeled as networks and aligning these networks of protein interactions has many applications in biological research. In this survey, we review algorithms for global pairwise alignment highlighting various proposed approaches, and classify them based on their methodology. Evaluation metrics that are used to measure the quality of the resulting alignments are also surveyed. We discuss and present a comparison between selected aligners on the same datasets and evaluate using the same evaluation metrics. Finally, a quick overview of the most popular databases of protein interaction networks is presented focusing on datasets that have been used recently.

Social inheritance can explain the structure of animal social networks

The social network structure of animal populations has major implications for survival, reproductive success, sexual selection and pathogen transmission of individuals. But as of yet, no general theory of social network structure exists that can explain the diversity of social networks observed in nature, and serve as a null model for detecting species and population-specific factors. Here we propose a simple and generally applicable model of social network structure. We consider the emergence of network structure as a result of social inheritance, in which newborns are likely to bond with maternal contacts, and via forming bonds randomly. We compare model output with data from several species, showing that it can generate networks with properties such as those observed in real social systems. Our model demonstrates that important observed properties of social networks, including heritability of network position or assortative associations, can be understood as consequences of social inheritance.

Competition between global and local online social networks

The overwhelming success of online social networks, the key actors in the Web 2.0 cosmos, has reshaped human interactions globally. To help understand the fundamental mechanisms which determine the fate of online social networks at the system level, we describe the digital world as a complex ecosystem of interacting networks. In this paper, we study the impact of heterogeneity in network fitnesses on the competition between an international network, such as Facebook, and local services. The higher fitness of international networks is induced by their ability to attract users from all over the world, which can then establish social interactions without the limitations of local networks. In other words, inter-country social ties lead to increased fitness of the international network. To study the competition between an international network and local ones, we construct a 1:1000 scale model of the digital world, consisting of the 80 countries with the most Internet users. Under certain conditions, this leads to the extinction of local networks; whereas under different conditions, local networks can persist and even dominate completely. In particular, our model suggests that, with the parameters that best reproduce the empirical overtake of Facebook, this overtake could have not taken place with a significant probability.

Delay decomposition approach to [Formula: see text] filtering analysis of genetic oscillator networks with time-varying delays

In this paper, the [Formula: see text] filtering problem is treated for N coupled genetic oscillator networks with time-varying delays and extrinsic molecular noises. Each individual genetic oscillator is a complex dynamical network that represents the genetic oscillations in terms of complicated biological functions with inner or outer couplings denote the biochemical interactions of mRNAs, proteins and other small molecules. Throughout the paper, first, by constructing appropriate delay decomposition dependent Lyapunov-Krasovskii functional combined with reciprocal convex approach, improved delay-dependent sufficient conditions are obtained to ensure the asymptotic stability of the filtering error system with a prescribed [Formula: see text] performance. Second, based on the above analysis, the existence of the designed [Formula: see text] filters are established in terms of linear matrix inequalities with Kronecker product. Finally, numerical examples including a coupled Goodwin oscillator model are inferred to illustrate the effectiveness and less conservatism of the proposed techniques.

Cross-Disciplinary Network Comparison: Matchmaking Between Hairballs

Biological systems are complex. In particular, the interactions between molecular components often form dense networks that, more often than not, are criticized for being inscrutable 'hairballs'. We argue that one way of untangling these hairballs is through cross-disciplinary network comparison-leveraging advances in other disciplines to obtain new biological insights. In some cases, such comparisons enable the direct transfer of mathematical formalism between disciplines, precisely describing the abstract associations between entities and allowing us to apply a variety of sophisticated formalisms to biology. In cases where the detailed structure of the network does not permit the transfer of complete formalisms between disciplines, comparison of mechanistic interactions in systems for which we have significant day-to-day experience can provide analogies for interpreting relatively more abstruse biological networks. Here, we illustrate how these comparisons benefit the field with a few specific examples related to network growth, organizational hierarchies, and the evolution of adaptive systems.

Binding interface change and cryptic variation in the evolution of protein-protein interactions

Background

Physical interactions between proteins are essential for almost all biological functions and systems. To understand the evolution of function it is therefore important to understand the evolution of molecular interactions. Of key importance is the evolution of binding specificity, the set of interactions made by a protein, since change in specificity can lead to “rewiring” of interaction networks. Unfortunately, the interfaces through which proteins interact are complex, typically containing many amino-acid residues that collectively must contribute to binding specificity as well as binding affinity, structural integrity of the interface and solubility in the unbound state.

Results

In order to study the relationship between interface composition and binding specificity, we make use of paralogous pairs of yeast proteins. Immediately after duplication these paralogues will have identical sequences and protein products that make an identical set of interactions. As the sequences diverge, we can correlate amino-acid change in the interface with any change in the specificity of binding. We show that change in interface regions correlates only weakly with change in specificity, and many variants in interfaces are functionally equivalent. We show that many of the residue replacements within interfaces are silent with respect to their contribution to binding specificity.

Conclusions

We conclude that such functionally-equivalent change has the potential to contribute to evolutionary plasticity in interfaces by creating cryptic variation, which in turn may provide the raw material for functional innovation and coevolution.

Electronic supplementary material

The online version of this article (doi:10.1186/s12862-016-0608-1) contains supplementary material, which is available to authorized users.

Network simulation reveals significant contribution of network motifs to the age-dependency of yeast protein-protein interaction networks

Advances in proteomic technologies combined with sophisticated computing and modeling methods have generated an unprecedented amount of high-throughput data for system-scale analysis. As a result, the study of protein-protein interaction (PPI) networks has garnered much attention in recent years. One of the most fundamental problems in studying PPI networks is to understand how their architecture originated and evolved to their current state. By investigating how proteins of different ages are connected in the yeast PPI networks, one can deduce their expansion procedure in evolution and how the ancient primitive network expanded and evolved. Studies have shown that proteins are often connected to other proteins of a similar age, suggesting a high degree of age preference between interacting proteins. Though several theories have been proposed to explain this phenomenon, none of them considered protein-clusters as a contributing factor. Here we first investigate the age-dependency of the proteins from the perspective of network motifs. Our analysis confirms that proteins of the same age groups tend to form interacting network motifs; furthermore, those proteins within motifs tend to be within protein complexes and the interactions among them largely contribute to the observed age preference in the yeast PPI networks. In light of these results, we describe a new modeling approach, based on "network motifs", whereby topologically connected protein clusters in the network are treated as single evolutionary units. Instead of modeling single proteins, our approach models the connections and evolutionary relationships of multiple related protein clusters or "network motifs" that are collectively integrated into an existing PPI network. Through simulation studies, we found that the "network motif" modeling approach can capture yeast PPI network properties better than if individual proteins were considered to be the simplest evolutionary units. Our approach provides a fresh perspective on modeling the evolution of yeast PPI networks, specifically that PPI networks may have a much higher age-dependency of interaction density than had been previously envisioned.

Mean field theory for biology inspired duplication-divergence network model

The duplication-divergence network model is generally thought to incorporate key ingredients underlying the growth and evolution of protein-protein interaction networks. Properties of the model have been elucidated through numerous simulation studies. However, a comprehensive theoretical study of the model is lacking. Here, we derived analytic expressions for quantities describing key characteristics of the network-the average degree, the degree distribution, the clustering coefficient, and the neighbor connectivity-in the mean-field, large-N limit of an extended version of the model, duplication-divergence complemented with heterodimerization and addition. We carried out extensive simulations and verified excellent agreement between simulation and theory except for one partial case. All four quantities obeyed power-laws even at moderate network size ( N∼10(4)), except the degree distribution, which had an additional exponential factor observed to obey power-law. It is shown that our network model can lead to the emergence of scale-free property and hierarchical modularity simultaneously, reproducing the important topological properties of real protein-protein interaction networks.

Asymptotic Properties of a Random Graph with Duplications

We deal with a random graph model evolving in discrete time steps by duplicating and deleting the edges of randomly chosen vertices. We prove the existence of an almost surely asymptotic degree distribution, with stretched exponential decay; more precisely, the proportion of vertices of degreedtends to some positive numbercd&gt; 0 almost surely as the number of steps goes to ∞, andcd~ (eπ)1/2d1/4e-2√dholds asd→ ∞.

Asymptotic Properties of a Random Graph with Duplications

We deal with a random graph model evolving in discrete time steps by duplicating and deleting the edges of randomly chosen vertices. We prove the existence of an almost surely asymptotic degree distribution, with stretched exponential decay; more precisely, the proportion of vertices of degree d tends to some positive number cd > 0 almost surely as the number of steps goes to ∞, and cd ~ (eπ)1/2d1/4e-2√d holds as d → ∞.