An analysis of the class of gene regulatory functions implied by a biochemical model

Understanding the integrated behavior of genetic regulatory networks, in which genes regulate one another's activities via RNA and protein products, is emerging as a dominant problem in systems biology. One widely studied class of models of such networks includes genes whose expression values assume Boolean values (i.e., on or off). Design decisions in the development of Boolean network models of gene regulatory systems include the topology of the network (including the distribution of input- and output-connectivity) and the class of Boolean functions used by each gene (e.g., canalizing functions, post functions, etc.). For example, evidence from simulations suggests that biologically realistic dynamics can be produced by scale-free network topologies with canalizing Boolean functions. This work seeks further insights into the design of Boolean network models through the construction and analysis of a class of models that include more concrete biochemical mechanisms than the usual abstract model, including genes and gene products, dimerization, cis-binding sites, promoters and repressors. In this model, it is assumed that the system consists of N genes, with each gene producing one protein product. Proteins may form complexes such as dimers, trimers, etc. The model also includes cis-binding sites to which proteins may bind to form activators or repressors. Binding affinities are based on structural complementarity between proteins and binding sites, with molecular binding sites modeled by bit-strings. Biochemically plausible gene expression rules are used to derive a Boolean regulatory function for each gene in the system. The result is a network model in which both topological features and Boolean functions arise as emergent properties of the interactions of components at the biochemical level. A highly biased set of Boolean functions is observed in simulations of networks of various sizes, suggesting a new characterization of the subset of Boolean functions that are likely to appear in gene regulatory networks.     

Spread of two linked social norms on complex interaction networks

In this paper, we study the spread of social norms, such as rules and customs that are components of human cultures. We consider the spread of two social norms, which are linked through individual behaviors. Spreading social norms depend not only on the social network structure, but also on the learning system. We consider four social network structures: (1) complete mixing, in which each individual interacts with the others at random, (2) lattice, in which each individual interacts with its neighbors with some probability and with the others at random, (3) power-law network, in which a few influential people have more social contacts than the others, and (4) random graph network, in which the number of contacts follows a Poisson distribution. Using the lattice model, we also investigate the effect of the small-world phenomenon on the dynamics of social norms. In our models, each individual learns a social norm by trial and error (individual learning) and also imitates the other's social norm (social learning). We investigate how social network structure and learning systems affect the spread of two linked social norms. Our main results are: (1) Social learning does not lead to coexistence of social norms. Individual learning produces coexistence, and the dynamics of coexistence depend on which social norms are learned individually. (2) Social norms spread fastest in the power-law network model, followed by the random graph model, the complete mixing model, the two-dimensional lattice model and the one-dimensional lattice. (3) We see a "small world effect" in the one-dimensional model, but not in two dimensions.     

How to make a biological switch

Some biological regulatory systems must "remember" a state for long periods of time. A simple type of system that can accomplish this task is one in which two regulatory elements negatively regulate one another. For example, two repressor proteins might control one another's synthesis. Qualitative reasoning suggests that such a system will have two stable states, one in which the first element is "on" and the second "off", and another in which these states are reversed. Quantitative analysis shows that the existence of two stable steady states depends on the details of the system. Among other things, the shapes of functions describing the effect of one regulatory element on the other must meet certain criteria in order for two steady states to exist. Many biologically reasonable functions do not meet these criteria. In particular, repression that is well described by a Michaelis-Menten-type equation cannot lead to a working switch. However, functions describing positive cooperativity of binding, non-additive effects of multiple operator sites, or depletion of free repressor can lead to working switches.     

Codon usage in Aspergillus nidulans.

Summary Synonymous codon usage in genes from the ascomycete (filamentous) fungus Aspergillus nidulans has been investigated. A total of 45 gene sequences has been analysed. Multivariate statistical analysis has been used to identify a single major trend among genes. At one end of this trend are lowly expressed genes, whereas at the other extreme lie genes known or expected to be highly expressed. The major trend is from nearly random codon usage (in the lowly expressed genes) to codon usage that is highly biased towards a set of 19–20 optimal codons. The G+C content of the A. nidulans genome is close to 50%, indicating little overall mutational bias, and so the codon usage of lowly expressed genes is as expected in the absence of selection pressure at silent sites. Most of the optimal codons are C- or G-ending, making highly expressed genes more G+C-rich at silent sites.     

Positive circuits and d-dimensional spatial differentiation: application to the formation of sense organs in Drosophila

We discuss a rule proposed by the biologist Thomas according to which the possibility for a genetic network (represented by a signed directed graph called a regulatory graph) to have several stable states implies the existence of a positive circuit. This result is already known for different models, differential or discrete formalism, but always with a network of genes contained in a single cell. Thus, we can ask about the validity of this rule for a system containing several cells and with intercellular genetic interactions. In this paper, we consider the genetic interactions between several cells located on a d-dimensional lattice, i.e., each point of lattice represents a cell to which we associate the expression level of n genes contained in this cell. With this configuration, we show that the existence of a positive circuit is a necessary condition for a specific form of multistationarity, which naturally corresponds to spatial differentiation. We then illustrate this theorem through the example of the formation of sense organs in Drosophila.     

Statistical fluctuations in processes of gene expression regulation: consideration of the problem from point of view of statistical mechanics

The regulation of gene expression is a basic problem of biology. In some cases, the gene activity is regulated by specific binding of regulatory proteins to DNA. In terms of statistical mechanics, this binding is described as the process of adsorption of ligands on the one-dimensional lattice and has a probability nature. As a random physical process, the adsorption of regulatory proteins on DNA introduces a noise to the regulation of gene activity. We derived equations, which make it possible to estimate this noise in the case of the binding of the lac repressor to the operator and showed that these estimates correspond to experimental data. Many ligands are able to bind nonspecifically to DNA. Nonspecific binding is characterized by a lesser equilibrium constant but a greater number of binding sites on the DNA, as compared with specific binding. Relations are presented, which enable one to estimate the probability of the binding of a ligand on a specific site and on nonspecific sites on DNA. The competition between specific and nonspecific binding of regulatory proteins plays a great role in the regulation of gene activity. Similar to the one-dimensional "lattice gas" of particles, ligands adsorbed on DNA produce "one-dimensional" pressure on proteins located at the termini of free regions of DNA. This pressure, an analog of osmotic pressure, may be of importance in processes leading to changes in chromatin structure and activation of gene expression.     

A biological interpretation of transient anomalous subdiffusion. II. Reaction kinetics

Reaction kinetics in a cell or cell membrane is modeled in terms of the first passage time for a random walker at a random initial position to reach an immobile target site in the presence of a hierarchy of nonreactive binding sites. Monte Carlo calculations are carried out for the triangular, square, and cubic lattices. The mean capture time is expressed as the product of three factors: the analytical expression of Montroll for the capture time in a system with a single target and no binding sites; an exact expression for the mean escape time from the set of lattice points; and a correction factor for the number of targets present. The correction factor, obtained from Monte Carlo calculations, is between one and two. Trapping may contribute significantly to noise in reaction rates. The statistical distribution of capture times is obtained from Monte Carlo calculations and shows a crossover from power-law to exponential behavior. The distribution is analyzed using probability generating functions; this analysis resolves the contributions of the different sources of randomness to the distribution of capture times. This analysis predicts the distribution function for a lattice with perfect mixing; deviations reflect imperfect mixing in an ordinary random walk.     

A proposal for using the ensemble approach to understand genetic regulatory networks

Understanding the genetic regulatory network comprising genes, RNA, proteins and the network connections and dynamical control rules among them, is a major task of contemporary systems biology. I focus here on the use of the ensemble approach to find one or more well-defined ensembles of model networks whose statistical features match those of real cells and organisms. Such ensembles should help explain and predict features of real cells and organisms. More precisely, an ensemble of model networks is defined by constraints on the "wiring diagram" of regulatory interactions, and the "rules" governing the dynamical behavior of regulated components of the network. The ensemble consists of all networks consistent with those constraints. Here I discuss ensembles of random Boolean networks, scale free Boolean networks, "medusa" Boolean networks, continuous variable networks, and others. For each ensemble, M statistical features, such as the size distribution of avalanches in gene activity changes unleashed by transiently altering the activity of a single gene, the distribution in distances between gene activities on different cell types, and others, are measured. This creates an M-dimensional space, where each ensemble corresponds to a cluster of points or distributions. Using current and future experimental techniques, such as gene arrays, these M properties are to be measured for real cells and organisms, again yielding a cluster of points or distributions in the M-dimensional space. The procedure then finds ensembles close to those of real cells and organisms, and hill climbs to attempt to match the observed M features. Thus obtains one or more ensembles that should predict and explain many features of the regulatory networks in cells and organisms.     

Physical origins of protein superfamilies

In this work, we discovered a fundamental connection between selection for protein stability and emergence of preferred structures of proteins. Using a standard exact three-dimensional lattice model we evolve sequences starting from random ones and determine the exact native structure after each mutation. Acceptance of mutations is biased to select for stable proteins. We found that certain structures, "wonderfolds", are independently discovered numerous times as native states of stable proteins in many unrelated runs of selection. The strong dependence of lattice fold usage on the structural determinant of designability quantitatively reproduces uneven fold usage in natural proteins. Diversity of sequences that fold into wonderfold structures gives rise to superfamilies, i.e. sets of dissimilar sequences that fold into the same or very similar structures. The present work establishes a model of pre-biotic structure selection, which identifies dominant structural patterns emerging upon optimization of proteins for survival in a hot environment. Convergently discovered pre-biotic initial superfamilies with wonderfold structures could have served as a seed for subsequent biological evolution involving gene duplications and divergence.     

Dynamics of Intraguild Predation Systems with Intraspecific Competition

This paper considers intraguild predation (IGP) systems where species in the same community kill and eat each other and there is intraspecific competition in each species. The IGP systems are characterized by a lattice gas model, in which reaction between sites on the lattice occurs in a random and independent way. Global dynamics of the model with two species demonstrate mechanisms by which IGP leads to survival/extinction of species. It is shown that an intermediary level of predation promotes survival of species, while over-predation or under-predation could result in species extinction. An interesting result is that increasing intraspecific competition of one species can lead to extinction of one or both species, while increasing intraspecific competitions of both species would result in coexistence of species in facultative predation. Initial population densities of the species are also shown to play a role in persistence of the system. Then the analysis is extended to IGP systems with one species. Numerical simulations confirm and extend our results.     

Gene perturbation and intervention in probabilistic Boolean networks

MOTIVATION: A major objective of gene regulatory network modeling, in addition to gaining a deeper understanding of genetic regulation and control, is the development of computational tools for the identification and discovery of potential targets for therapeutic intervention in diseases such as cancer. We consider the general question of the potential effect of individual genes on the global dynamical network behavior, both from the view of random gene perturbation as well as intervention in order to elicit desired network behavior. RESULTS: Using a recently introduced class of models, called Probabilistic Boolean Networks (PBNs), this paper develops a model for random gene perturbations and derives an explicit formula for the transition probabilities in the new PBN model. This result provides a building block for performing simulations and deriving other results concerning network dynamics. An example is provided to show how the gene perturbation model can be used to compute long-term influences of genes on other genes. Following this, the problem of intervention is addressed via the development of several computational tools based on first-passage times in Markov chains. The consequence is a methodology for finding the best gene with which to intervene in order to most likely achieve desirable network behavior. The ideas are illustrated with several examples in which the goal is to induce the network to transition into a desired state, or set of states. The corresponding issue of avoiding undesirable states is also addressed. Finally, the paper turns to the important problem of assessing the effect of gene perturbations on long-run network behavior. A bound on the steady-state probabilities is derived in terms of the perturbation probability. The result demonstrates that states of the network that are more 'easily reachable' from other states are more stable in the presence of gene perturbations. Consequently, these are hypothesized to correspond to cellular functional states. AVAILABILITY: A library of functions written in MATLAB for simulating PBNs, constructing state-transition matrices, computing steady-state distributions, computing influences, modeling random gene perturbations, and finding optimal intervention targets, as described in this paper, is available on request from is@ieee.org.     

Pair-edge approximation for heterogeneous lattice population models

To increase the analytical tractability of lattice stochastic spatial population models, several approximations have been developed. The pair-edge approximation is a moment-closure method that is effective in predicting persistence criteria and invasion speeds on a homogeneous lattice. Here we evaluate the effectiveness of the pair-edge approximation on a spatially heterogeneous lattice in which some sites are unoccupiable, or "dead". This model has several possible interpretations, including a spatial SIS epidemic model, in which some sites are occupied by immobile host-species individuals while others are empty. We find that, as in the homogeneous model, the pair-edge approximation is significantly more accurate than the ordinary pair approximation in determining conditions for persistence. However, habitat heterogeneity decreases invasion speed more than is predicted by the pair-edge approximation, and the discrepancy increases with greater clustering of "dead" sites. The accuracy of the approximation validates the underlying heuristic picture of population spread and therefore provides qualitative insight into the dynamics of lattice models. Conversely, the situations where the approximation is less accurate reveals limitations of pair approximation in the presence of spatial heterogeneity.     

Constructing a gene semantic similarity network for the inference of disease genes

MOTIVATION: The inference of genes that are truly associated with inherited human diseases from a set of candidates resulting from genetic linkage studies has been one of the most challenging tasks in human genetics. Although several computational approaches have been proposed to prioritize candidate genes relying on protein-protein interaction (PPI) networks, these methods can usually cover less than half of known human genes. RESULTS: We propose to rely on the biological process domain of the gene ontology to construct a gene semantic similarity network and then use the network to infer disease genes. We show that the constructed network covers about 50% more genes than a typical PPI network. By analyzing the gene semantic similarity network with the PPI network, we show that gene pairs tend to have higher semantic similarity scores if the corresponding proteins are closer to each other in the PPI network. By analyzing the gene semantic similarity network with a phenotype similarity network, we show that semantic similarity scores of genes associated with similar diseases are significantly different from those of genes selected at random, and that genes with higher semantic similarity scores tend to be associated with diseases with higher phenotype similarity scores. We further use the gene semantic similarity network with a random walk with restart model to infer disease genes. Through a series of large-scale leave-one-out cross-validation experiments, we show that the gene semantic similarity network can achieve not only higher coverage but also higher accuracy than the PPI network in the inference of disease genes.