Immune networks modeled by replicator equations

In order to evaluate the role of idiotypic networks in the operation of the immune system a number of mathematical models have been formulated. Here we examine a class of B-cell models in which cell proliferation is governed by a non-negative, unimodal, symmetric response function f(h), where the field h summarizes the effect of the network on a single clone. We show that by transforming into relative concentrations, the B-cell network equations can be brought into a form that closely resembles the replicator equation. We then show that when the total number of clones in a network is conserved, the dynamics of the network can be represented by the dynamics of a replicator equation. The number of equilibria and their stability are then characterized using methods developed for the study of second-order replicator equations. Analogies with standard Lotka-Volterra equations are also indicated. A particularly interesting result of our analysis is the fact that even though the immune network equations are not second-order, the number and stability of their equilibria can be obtained by a superposition of second-order replicator systems. As a consequence, the problem of finding all of the equilibrium points of the nonlinear network equations can be reduced to solving linear equations.     

Information Flow Analysis of Interactome Networks

Recent studies of cellular networks have revealed modular organizations of genes and proteins. For example, in interactome networks, a module refers to a group of interacting proteins that form molecular complexes and/or biochemical pathways and together mediate a biological process. However, it is still poorly understood how biological information is transmitted between different modules. We have developed information flow analysis, a new computational approach that identifies proteins central to the transmission of biological information throughout the network. In the information flow analysis, we represent an interactome network as an electrical circuit, where interactions are modeled as resistors and proteins as interconnecting junctions. Construing the propagation of biological signals as flow of electrical current, our method calculates an information flow score for every protein. Unlike previous metrics of network centrality such as degree or betweenness that only consider topological features, our approach incorporates confidence scores of protein–protein interactions and automatically considers all possible paths in a network when evaluating the importance of each protein. We apply our method to the interactome networks of Saccharomyces cerevisiae and Caenorhabditis elegans. We find that the likelihood of observing lethality and pleiotropy when a protein is eliminated is positively correlated with the protein's information flow score. Even among proteins of low degree or low betweenness, high information scores serve as a strong predictor of loss-of-function lethality or pleiotropy. The correlation between information flow scores and phenotypes supports our hypothesis that the proteins of high information flow reside in central positions in interactome networks. We also show that the ranks of information flow scores are more consistent than that of betweenness when a large amount of noisy data is added to an interactome. Finally, we combine gene expression data with interaction data in C. elegans and construct an interactome network for muscle-specific genes. We find that genes that rank high in terms of information flow in the muscle interactome network but not in the entire network tend to play important roles in muscle function. This framework for studying tissue-specific networks by the information flow model can be applied to other tissues and other organisms as well.     

On equilibrium properties of evolutionary multi-player games with random payoff matrices

The analysis of equilibrium points in biological dynamical systems has been of great interest in a variety of mathematical approaches to biology, such as population genetics, theoretical ecology or evolutionary game theory. The maximal number of equilibria and their classification based on stability have been the primary subjects of these studies, for example in the context of two-player games with multiple strategies. Herein, we address a different question using evolutionary game theory as a tool. If the payoff matrices are drawn randomly from an arbitrary distribution, what are the probabilities of observing a certain number of (stable) equilibria? We extend the domain of previous results for the two-player framework, which corresponds to a single diploid locus in population genetics, by addressing the full complexity of multi-player games with multiple strategies. In closing, we discuss an application and illustrate how previous results on the number of equilibria, such as the famous Feldman-Karlin conjecture on the maximal number of isolated fixed points in a viability selection model, can be obtained as special cases of our results based on multi-player evolutionary games. We also show how the probability of realizing a certain number of equilibria changes as we increase the number of players and number of strategies.     

Blood flow switching among pulmonary capillaries is decreased during high hematocrit.

Pulmonary capillary perfusion within a single alveolar wall continually switches among segments, even when large-vessel hemodynamics are constant. The mechanism is unknown. We hypothesize that the continually varying size of plasma gaps between individual red blood cells affects the likelihood of capillary segment closure and the probability of cells changing directions at the next capillary junction. We assumed that an increase in hematocrit would decrease the average distance between red blood cells, thereby decreasing the switching at each capillary junction. To test this idea, we observed 26 individual alveolar capillary networks by using videomicroscopy of excised canine lung lobes that were perfused first at normal hematocrit (31-43%) and then at increased hematocrit (51-62%). The number of switches decreased by 38% during increased hematocrit (P < 0.01). These results support the idea that a substantial part of flow switching among pulmonary capillaries is caused by the particulate nature of blood passing through a complex network of tubes with continuously varying hematocrit.     

Epidemic threshold and network structure: The interplay of probability of transmission and of persistence in small-size directed networks

A growing number of studies are investigating the effect of contact structure on the dynamics of epidemics in large-scale complex networks. Whether findings thus obtained apply also to networks of small size, and thus to many real-world biological applications, is still an open question. We use numerical simulations of disease spread in directed networks of 100 individual nodes with a constant number of links. We show that, no matter the type of network structure (local, small-world, random and scale-free), there is a linear threshold determined by the probability of infection transmission between connected nodes and the probability of infection persistence in an infected node. The threshold is significantly lower for scale-free networks compared to local, random and small-world ones only if super-connected nodes have a higher number of links both to and from other nodes. The starting point, the node at which the epidemic starts, does not affect the threshold conditions, but has a marked influence on the final size of the epidemic in all kinds of network. There is evidence that contact structure has an influence on the average final size of an epidemic across all starting nodes, with significantly lower values in scale-free networks at equilibrium. Simulations in scale-free networks show a distinctive time-series pattern, which, if found in a real epidemic, can be used to infer the underlying network structure. The findings have relevance also for meta-population ecology and species conservation.     

Numerical Simulation of Blood Flow Through Microvascular Capillary Networks

A numerical method is implemented for computing blood flow through a branching microvascular capillary network. The simulations follow the motion of individual red blood cells as they enter the network from an arterial entrance point with a specified tube hematocrit, while simultaneously updating the nodal capillary pressures. Poiseuille’s law is used to describe flow in the capillary segments with an effective viscosity that depends on the number of cells residing inside each segment. The relative apparent viscosity is available from previous computational studies of individual red blood cell motion. Simulations are performed for a tree-like capillary network consisting of bifurcating segments. The results reveal that the probability of directional cell motion at a bifurcation (phase separation) may have an important effect on the statistical measures of the cell residence time and scattering of the tube hematocrit across the network. Blood cells act as regulators of the flow rate through the network branches by increasing the effective viscosity when the flow rate is high and decreasing the effective viscosity when the flow rate is low. Comparison with simulations based on conventional models of blood flow regarded as a continuum indicates that the latter underestimates the variance of the hematocrit across the vascular tree.     

Spatial correlations in attribute communities

Community detection is an important tool for exploring and classifying the properties of large complex networks and should be of great help for spatial networks. Indeed, in addition to their location, nodes in spatial networks can have attributes such as the language for individuals, or any other socio-economical feature that we would like to identify in communities. We discuss in this paper a crucial aspect which was not considered in previous studies which is the possible existence of correlations between space and attributes. Introducing a simple toy model in which both space and node attributes are considered, we discuss the effect of space-attribute correlations on the results of various community detection methods proposed for spatial networks in this paper and in previous studies. When space is irrelevant, our model is equivalent to the stochastic block model which has been shown to display a detectability-non detectability transition. In the regime where space dominates the link formation process, most methods can fail to recover the communities, an effect which is particularly marked when space-attributes correlations are strong. In this latter case, community detection methods which remove the spatial component of the network can miss a large part of the community structure and can lead to incorrect results.     

Equilibrium properties of a multi-locus, haploid-selection, symmetric-viability model

Under haploid selection, a multi-locus, diallelic, two-niche Levene (1953) model is studied. Viability coefficients with symmetrically opposing directional selection in each niche are assumed, and with a further simplification that the most and least favored haplotype in each niche shares no alleles in common, and that the selection coefficients monotonically increase or decrease with the number of alleles shared. This model always admits a fully polymorphic symmetric equilibrium, which may or may not be stable.We show that a stable symmetric equilibrium can become unstable via either a supercritical or subcritical pitchfork bifurcation. In the supercritical bifurcation, the symmetric equilibrium bifurcates to a pair of stable fully polymorphic asymmetric equilibria; in the subcritical bifurcation, the symmetric equilibrium bifurcates to a pair of unstable fully polymorphic asymmetric equilibria, which then connect to either another pair of stable fully polymorphic asymmetric equilibria through saddle-node bifurcations, or to a pair of monomorphic equilibria through transcritical bifurcations. As many as three fully polymorphic stable equilibria can coexist, and jump bifurcations can occur between these equilibria when model parameters are varied.In our Levene model, increasing recombination can act to either increase or decrease the genetic diversity of a population. By generating more hybrid offspring from the mating of purebreds, recombination can act to increase genetic diversity provided the symmetric equilibrium remains stable. But by destabilizing the symmetric equilibrium, recombination can ultimately act to decrease genetic diversity.     

Reconciling taxonomy and phylogenetic inference: formalism and algorithms for describing discord and inferring taxonomic roots

Background

Understanding the evolution of biological networks can provide insight into how their modular structure arises and how they are affected by environmental changes. One approach to studying the evolution of these networks is to reconstruct plausible common ancestors of present-day networks, allowing us to analyze how the topological properties change over time and to posit mechanisms that drive the networks?? evolution. Further, putative ancestral networks can be used to help solve other difficult problems in computational biology, such as network alignment.

Results

We introduce a combinatorial framework for encoding network histories, and we give a fast procedure that, given a set of gene duplication histories, in practice finds network histories with close to the minimum number of interaction gain or loss events to explain the observed present-day networks. In contrast to previous studies, our method does not require knowing the relative ordering of unrelated duplication events. Results on simulated histories and real biological networks both suggest that common ancestral networks can be accurately reconstructed using this parsimony approach. A software package implementing our method is available under the Apache 2.0 license at http://cbcb.umd.edu/kingsford-group/parana.

Conclusions

Our parsimony-based approach to ancestral network reconstruction is both efficient and accurate. We show that considering a larger set of potential ancestral interactions by not assuming a relative ordering of unrelated duplication events can lead to improved ancestral network inference.     

Wave speed in excitable random networks with spatially constrained connections

Very fast oscillations (VFO) in neocortex are widely observed before epileptic seizures, and there is growing evidence that they are caused by networks of pyramidal neurons connected by gap junctions between their axons. We are motivated by the spatio-temporal waves of activity recorded using electrocorticography (ECoG), and study the speed of activity propagation through a network of neurons axonally coupled by gap junctions. We simulate wave propagation by excitable cellular automata (CA) on random (Erdös-Rényi) networks of special type, with spatially constrained connections. From the cellular automaton model, we derive a mean field theory to predict wave propagation. The governing equation resolved by the Fisher-Kolmogorov PDE fails to describe wave speed. A new (hyperbolic) PDE is suggested, which provides adequate wave speed that saturates with network degree , in agreement with intuitive expectations and CA simulations. We further show that the maximum length of connection is a much better predictor of the wave speed than the mean length. When tested in networks with various degree distributions, wave speeds are found to strongly depend on the ratio of network moments rather than on mean degree , which is explained by general network theory. The wave speeds are strikingly similar in a diverse set of networks, including regular, Poisson, exponential and power law distributions, supporting our theory for various network topologies. Our results suggest practical predictions for networks of electrically coupled neurons, and our mean field method can be readily applied for a wide class of similar problems, such as spread of epidemics through spatial networks.     

Analysis of gene network robustness based on saturated fixed point attractors

The analysis of gene network robustness to noise and mutation is important for fundamental and practical reasons. Robustness refers to the stability of the equilibrium expression state of a gene network to variations of the initial expression state and network topology. Numerical simulation of these variations is commonly used for the assessment of robustness. Since there exists a great number of possible gene network topologies and initial states, even millions of simulations may be still too small to give reliable results. When the initial and equilibrium expression states are restricted to being saturated (i.e., their elements can only take values 1 or −1 corresponding to maximum activation and maximum repression of genes), an analytical gene network robustness assessment is possible. We present this analytical treatment based on determination of the saturated fixed point attractors for sigmoidal function models. The analysis can determine (a) for a given network, which and how many saturated equilibrium states exist and which and how many saturated initial states converge to each of these saturated equilibrium states and (b) for a given saturated equilibrium state or a given pair of saturated equilibrium and initial states, which and how many gene networks, referred to as viable, share this saturated equilibrium state or the pair of saturated equilibrium and initial states. We also show that the viable networks sharing a given saturated equilibrium state must follow certain patterns. These capabilities of the analytical treatment make it possible to properly define and accurately determine robustness to noise and mutation for gene networks. Previous network research conclusions drawn from performing millions of simulations follow directly from the results of our analytical treatment. Furthermore, the analytical results provide criteria for the identification of model validity and suggest modified models of gene network dynamics. The yeast cell-cycle network is used as an illustration of the practical application of this analytical treatment.     

Network Scale Modeling of Lymph Transport and Its Effective Pumping Parameters

The lymphatic system is an open-ended network of vessels that run in parallel to the blood circulation system. These vessels are present in almost all of the tissues of the body to remove excess fluid. Similar to blood vessels, lymphatic vessels are found in branched arrangements. Due to the complexity of experiments on lymphatic networks and the difficulty to control the important functional parameters in these setups, computational modeling becomes an effective and essential means of understanding lymphatic network pumping dynamics. Here we aimed to determine the effect of pumping coordination in branched network structures on the regulation of lymph flow. Lymphatic vessel networks were created by building upon our previous lumped-parameter model of lymphangions in series. In our network model, each vessel is itself divided into multiple lymphangions by lymphatic valves that help maintain forward flow. Vessel junctions are modeled by equating the pressures and balancing mass flows. Our results demonstrated that a 1.5 s rest-period between contractions optimizes the flow rate. A time delay between contractions of lymphangions at the junction of branches provided an advantage over synchronous pumping, but additional time delays within individual vessels only increased the flow rate for adverse pressure differences greater than 10.5 cmH₂O. Additionally, we quantified the pumping capability of the system under increasing levels of steady transmural pressure and outflow pressure for different network sizes. We observed that peak flow rates normally occurred under transmural pressures between 2 to 4 cmH₂O (for multiple pressure differences and network sizes). Networks with 10 lymphangions per vessel had the highest pumping capability under a wide range of adverse pressure differences. For favorable pressure differences, pumping was more efficient with fewer lymphangions. These findings are valuable for translating experimental measurements from the single lymphangion level to tissue and organ scales.     

Using indirect protein-protein interactions for protein complex prediction

Protein complexes are fundamental for understanding principles of cellular organizations. As the sizes of protein-protein interaction (PPI) networks are increasing, accurate and fast protein complex prediction from these PPI networks can serve as a guide for biological experiments to discover novel protein complexes. However, it is not easy to predict protein complexes from PPI networks, especially in situations where the PPI network is noisy and still incomplete. Here, we study the use of indirect interactions between level-2 neighbors (level-2 interactions) for protein complex prediction. We know from previous work that proteins which do not interact but share interaction partners (level-2 neighbors) often share biological functions. We have proposed a method in which all direct and indirect interactions are first weighted using topological weight (FS-Weight), which estimates the strength of functional association. Interactions with low weight are removed from the network, while level-2 interactions with high weight are introduced into the interaction network. Existing clustering algorithms can then be applied to this modified network. We have also proposed a novel algorithm that searches for cliques in the modified network, and merge cliques to form clusters using a "partial clique merging" method. Experiments show that (1) the use of indirect interactions and topological weight to augment protein-protein interactions can be used to improve the precision of clusters predicted by various existing clustering algorithms; and (2) our complex-finding algorithm performs very well on interaction networks modified in this way. Since no other information except the original PPI network is used, our approach would be very useful for protein complex prediction, especially for prediction of novel protein complexes.     

Modelling periodic oscillation in gene regulatory networks by cyclic feedback systems

In this paper, we develop a new methodology to analyze and design periodic oscillators of biological networks, in particular gene regulatory networks with multiple genes, proteins and time delays, by using negative cyclic feedback systems. We show that negative cyclic feedback networks have no stable equilibria but stable periodic orbits when certain conditions are satisfied. Specifically, we first prove the basic properties of the biological networks composed of cyclic feedback loops, and then extend our results to general cyclic feedback network with less restriction, thereby making our theoretical analysis and design of oscillators easy to implement, even for large-scale systems. Finally, we use one circadian network formed by a period protein (PER) and per mRNA, and one biologically plausible synthetic gene network, to demonstrate the theoretical results. Since there is less restriction on the network structure, the results of this paper can be expected to apply to a wide variety of areas on modelling, analyzing and designing of biological systems.     

How is the equilibrium of continuous strategy game different from that of discrete strategy game?

Cooperation in the prisoner's dilemma (PD) played on various networks has been explained by so-called network reciprocity. Most of the previous studies presumed that players can offer either cooperation (C) or defection (D). This discrete strategy seems unrealistic in the real world, since actual provisions might not be discrete, but rather continuous. This paper studies the differences between continuous and discrete strategies in two aspects under the condition that the payoff function of the former is a linear interpolation of the payoff matrix of the latter. The first part of this paper proves theoretically that for two-player games, continuous and discrete strategies have different equilibria and game dynamics in a well-mixed but finite population. The second part, conducting a series of numerical experiments, reveals that such differences become considerably large in the case of PD games on networks. Furthermore, it shows, using the Wilcoxon sign-rank test, that continuous and discrete strategy games are statistically significantly different in terms of equilibria. Intensive discussion by comparing these two kinds of games elucidates that describing a strategy as a real number blunts D strategy invasion to C clusters on a network in the early stage of evolution. Thus, network reciprocity is enhanced by the continuous strategy.     

Local and collective transitions in sparsely-interacting ecological communities

Interactions in natural communities can be highly heterogeneous, with any given species interacting appreciably with only some of the others, a situation commonly represented by sparse interaction networks. We study the consequences of sparse competitive interactions, in a theoretical model of a community assembled from a species pool. We find that communities can be in a number of different regimes, depending on the interaction strength. When interactions are strong, the network of coexisting species breaks up into small subgraphs, while for weaker interactions these graphs are larger and more complex, eventually encompassing all species. This process is driven by the emergence of new allowed subgraphs as interaction strength decreases, leading to sharp changes in diversity and other community properties, and at weaker interactions to two distinct collective transitions: a percolation transition, and a transition between having a unique equilibrium and having multiple alternative equilibria. Understanding community structure is thus made up of two parts: first, finding which subgraphs are allowed at a given interaction strength, and secondly, a discrete problem of matching these structures over the entire community. In a shift from the focus of many previous theories, these different regimes can be traversed by modifying the interaction strength alone, without need for heterogeneity in either interaction strengths or the number of competitors per species.     

Statistics of weighted brain networks reveal hierarchical organization and Gaussian degree distribution

Whole brain weighted connectivity networks were extracted from high resolution diffusion MRI data of 14 healthy volunteers. A statistically robust technique was proposed for the removal of questionable connections. Unlike most previous studies our methods are completely adapted for networks with arbitrary weights. Conventional statistics of these weighted networks were computed and found to be comparable to existing reports. After a robust fitting procedure using multiple parametric distributions it was found that the weighted node degree of our networks is best described by the normal distribution, in contrast to previous reports which have proposed heavy tailed distributions. We show that post-processing of the connectivity weights, such as thresholding, can influence the weighted degree asymptotics. The clustering coefficients were found to be distributed either as gamma or power-law distribution, depending on the formula used. We proposed a new hierarchical graph clustering approach, which revealed that the brain network is divided into a regular base-2 hierarchical tree. Connections within and across this hierarchy were found to be uncommonly ordered. The combined weight of our results supports a hierarchically ordered view of the brain, whose connections have heavy tails, but whose weighted node degrees are comparable.     

Dynamics to Equilibrium in Network Games: Individual Behavior and Global Response

Various social contexts can be depicted as games of strategic interactions on networks, where an individual’s welfare depends on both her and her partners’ actions. Whereas much attention has been devoted to Bayes-Nash equilibria in such games, here we look at strategic interactions from an evolutionary perspective. To this end, we present the results of a numerical simulations program for these games, which allows us to find out whether Nash equilibria are accessible by adaptation of player strategies, and in general to identify the attractors of the evolution. Simulations allow us to go beyond a global characterization of the cooperativeness at equilibrium and probe into individual behavior. We find that when players imitate each other, evolution does not reach Nash equilibria and, worse, leads to very unfavorable states in terms of welfare. On the contrary, when players update their behavior rationally, they self-organize into a rich variety of Nash equilibria, where individual behavior and payoffs are shaped by the nature of the game, the social network’s structure and the players’ position within the network. Our results allow to assess the validity of mean-field approaches we use to describe the dynamics of these games. Interestingly, our dynamically-found equilibria generally do not coincide with (but show qualitatively the same features of) those resulting from theoretical predictions in the context of one-shot games under incomplete information.     

Numerical bifurcation analysis of distance-dependent on-center off-surround shunting neural networks

 On-center off-surround shunting neural networks are often applied as models for content-addressable memory (CAM), the equilibria being the stored memories. One important demand of biological plausible CAMs is that they function under a broad range of parameters, since several parameters vary due to postnatal maturation or learning. Ellias, Cohen and Grossberg have put much effort into showing the stability properties of several configurations of on-center off-surround shunting neural networks. In this article we present numerical bifurcation analysis of distance-dependent on-center off-surround shunting neural networks with fixed external input. We varied four parameters that may be subject to postnatal maturation: the range of both excitatory and inhibitory connections and the strength of both inhibitory and excitatory connections. These analyses show that fold bifurcations occur in the equilibrium behavior of the network by variation of all four parameters. The most important result is that the number of activation peaks in the equilibrium behavior varies from one to many if the range of inhibitory connections is decreased. Moreover, under a broad range of the parameters the stability of the network is maintained. The examined network is implemented in an ART network, Exact ART, where it functions as the classification layer F2. The stability of the ART network with the F2-field in different dynamic regimes is maintained and the behavior is functional in Exact ART. Through a bifurcation the learning behavior of Exact ART may even change from forming local representations to forming distributed representations. Received: 23 January 1996 / Accepted in revised form: 1 July 1996     

A ternary logic model for recurrent neuromime networks with delay

 In contrast to popular recurrent artificial neural network (RANN) models, biological neural networks have unsymmetric structures and incorporate significant delays as a result of axonal propagation. Consequently, biologically inspired neural network models are more accurately described by nonlinear differential-delay equations rather than nonlinear ordinary differential equations (ODEs), and the standard techniques for studying the dynamics of RANNs are wholly inadequate for these models. This paper develops a ternary-logic based method for analyzing these networks. Key to the technique is the realization that a nonzero delay produces a bounded stability region. This result significantly simplifies the construction of sufficient conditions for characterizing the network equilibria. If the network gain is large enough, each equilibrium can be classified as either asymptotically stable or unstable. To illustrate the analysis technique, the swim central pattern generator (CPG) of the sea slug Tritonia diomedea is examined. For wide range of reasonable parameter values, the ternary analysis shows that none of the network equilibria are stable, and thus the network must oscillate. The results show that complex synaptic dynamics are not necessary for pattern generation. Received: 15 June 1994/Accepted in revised form: 10 February 1995