Drivers of structural features in gene regulatory networks: From biophysical constraints to biological function.

Living cells can maintain their internal states, react to changing environments, grow, differentiate, divide, etc. All these processes are tightly controlled by what can be called a regulatory program. The logic of the underlying control can sometimes be guessed at by examining the network of influences amongst genetic components. Some associated gene regulatory networks have been studied in prokaryotes and eukaryotes, unveiling various structural features ranging from broad distributions of out-degrees to recurrent "motifs", that is small subgraphs having a specific pattern of interactions. To understand what factors may be driving such structuring, a number of groups have introduced frameworks to model the dynamics of gene regulatory networks. In that context, we review here such in silico approaches and show how selection for phenotypes, i.e., network function, can shape network structure.

Edge usage, motifs, and regulatory logic for cell cycling genetic networks.

The cell cycle is a tightly controlled process, yet it shows marked differences across species. Which of its structural features follow solely from the ability to control gene expression? We tackle this question in silico by examining the ensemble of all regulatory networks which satisfy the constraint of producing a given sequence of gene expressions. We focus on three cell cycle profiles coming from baker's yeast, fission yeast, and mammals. First, we show that the networks in each of the ensembles use just a few interactions that are repeatedly reused as building blocks. Second, we find an enrichment in network motifs that is similar in the two yeast cell cycle systems investigated. These motifs do not have autonomous functions, yet they reveal a regulatory logic for cell cycling based on a feed-forward cascade of activating interactions.

Motifs emerge from function in model gene regulatory networks.

Gene regulatory networks allow the control of gene expression patterns in living cells. The study of network topology has revealed that certain subgraphs of interactions or "motifs" appear at anomalously high frequencies. We ask here whether this phenomenon may emerge because of the functions carried out by these networks. Given a framework for describing regulatory interactions and dynamics, we consider in the space of all regulatory networks those that have prescribed functional capabilities. Markov Chain Monte Carlo sampling is then used to determine how these functional networks lead to specific motif statistics in the interactions. In the case where the regulatory networks are constrained to exhibit multistability, we find a high frequency of gene pairs that are mutually inhibitory and self-activating. In contrast, networks constrained to have periodic gene expression patterns (mimicking for instance the cell cycle) have a high frequency of bifan-like motifs involving four genes with at least one activating and one inhibitory interaction.

Distribution of essential interactions in model gene regulatory networks under mutation-selection balance.

Gene regulatory networks typically have low in-degrees, whereby any given gene is regulated by few of the genes in the network. They also tend to have broad distributions for the out-degree. What mechanisms might be responsible for these degree distributions? Starting with an accepted framework of the binding of transcription factors to DNA, we consider a simple model of gene regulatory dynamics. There, we show that selection for a target expression pattern leads to the emergence of minimum connectivities compatible with the selective constraint. As a consequence, these gene networks have low in-degree and "functionality" is parsimonious, i.e., is concentrated on a sparse number of interactions as measured for instance by their essentiality. Furthermore, we find that mutations of the transcription factors drive the networks to have broad out-degrees. Finally, these classes of models are evolvable, i.e., significantly different genotypes can emerge gradually under mutation-selection balance.

Adaptive networks of trading agents.

Multiagent models have been used in many contexts to study generic collective behavior. Similarly, complex networks have become very popular because of the diversity of growth rules giving rise to scale-free behavior. Here we study adaptive networks where the agents trade "wealth" when they are linked together while links can appear and disappear according to the wealth of the corresponding agents; thus the agents influence the network dynamics and vice versa. Our framework generalizes a multiagent model of Bouchaud and Mézard [Physica A 282, 536 (2000)], and leads to a steady state with fluctuating connectivities. The system spontaneously self-organizes into a critical state where the wealth distribution has a fat tail and the network is scale free; in addition, network heterogeneities lead to enhanced wealth condensation.

Network of inherent structures in spin glasses: scaling and scale-free distributions.

The local minima (inherent structures) of a system and their associated transition links give rise to a network. Here we consider the topological and distance properties of such a network in the context of spin glasses. We use steepest descent dynamics, determining for each disorder sample the transition links appearing within a given barrier height. We find that differences between linked inherent structures are typically associated with local clusters of spins; we interpret this within a framework based on droplets in which the characteristic "length scale" grows with the barrier height. We also consider the network connectivity and the degrees of its nodes. Interestingly, for spin glasses based on random graphs, the degree distribution of the network of inherent structures exhibits a nontrivial scale-free tail.

From simple to complex networks: inherent structures, barriers, and valleys in the context of spin glasses.

Given discrete degrees of freedom (spins) on a graph interacting via an energy function, what can be said about the energy local minima and associated inherent structures? Using the lid algorithm in the context of a spin glass energy function, we investigate the properties of the energy landscape for a variety of graph topologies. First, we find that the multiplicity N(s) of the inherent structures generically has a log-normal distribution. In addition, the large volume limit of ln / differs from unity, except for the Sherrington-Kirkpatrick model. Second, we find simple scaling laws for the growth of the height of the energy barrier between the two degenerate ground states and the size of the associated valleys. For finite connectivity models, changing the topology of the underlying graph does not modify qualitatively the energy landscape, but at the quantitative level the models can differ substantially.

Perturbing general uncorrelated networks.

This paper is a direct continuation of an earlier work, where we studied Erdös-Rényi random graphs perturbed by an interaction Hamiltonian favoring the formation of short cycles. Here, we generalize these results. We keep the same interaction Hamiltonian but let it act on general graphs with uncorrelated nodes and an arbitrary given degree distribution. It is shown that the results obtained for Erdös-Rényi graphs are generic, at the qualitative level. However, scale-free graphs are an exception to this general rule and exhibit a singular behavior, studied thoroughly in this paper, both analytically and numerically.

Network transitivity and matrix models.

This paper is a step towards a systematic theory of the transitivity (clustering) phenomenon in random networks. A static framework is used, with adjacency matrix playing the role of the dynamical variable. Hence, our model is a matrix model, where matrices are random, but their elements take values 0 and 1 only. Confusion present in some papers where earlier attempts to incorporate transitivity in a similar framework have been made is hopefully dissipated. Inspired by more conventional matrix models, analytic techniques to develop a static model with nontrivial clustering are introduced. Computer simulations complete the analytic discussion.

Uncorrelated random networks.

We define a statistical ensemble of nondegenerate graphs, i.e., graphs without multiple-connections and self-connections between nodes. The node degree distribution is arbitrary, but the nodes are assumed to be uncorrelated. This completes our earlier publication [Phys. Rev. 64, 046118 (2001)] where trees and degenerate graphs were considered. An efficient algorithm generating nondegenerate graphs is constructed. The corresponding computer code is available on request. Finite-size effects in scale-free graphs, i.e., those where the tail of the degree distribution falls like n(-beta), are carefully studied. We find that in the absence of dynamical internode correlations the degree distribution is cut at a degree value scaling like N(gamma), with gamma=min[1/2,1/(beta-1)], where N is the total number of nodes. The consequence is that, independently of any specific model, the internode correlations seem to be a necessary ingredient of the physics of scale-free networks observed in nature.

Tree networks with causal structure.

A geometry of networks endowed with a causal structure is discussed using the conventional framework of the equilibrium statistical mechanics. The popular growing network models appear as particular causal models. We focus on a class of tree graphs, an analytically solvable case. General formulas are derived, describing the degree distribution, the ancestor-descendant correlation, and the probability that a randomly chosen node lives at a given geodesic distance from the root. It is shown that the Hausdorff dimension d(H) of the causal networks is generically infinite, in contrast to the maximally random trees where it is generically finite.

Statistical ensemble of scale-free random graphs.

A thorough discussion of the statistical ensemble of scale-free connected random tree graphs is presented. Methods borrowed from field theory are used to define the ensemble and to study analytically its properties. The ensemble is characterized by two global parameters, the fractal and the spectral dimensions, which are explicitly calculated. It is discussed in detail how the geometry of the graphs varies when the weights of the nodes are modified. The stability of the scale-free regime is also considered: when it breaks down, either a scale is spontaneously generated or else, a "singular" node appears and the graphs become crumpled. A new computer algorithm to generate these random graphs is proposed. Possible generalizations are also discussed. In particular, more general ensembles are defined along the same lines and the computer algorithm is extended to arbitrary (degenerate) scale-free random graphs.

Theoretical Study of Ethylene Oligomerization by an Organometallic Nickel Catalyst.

The mechanism for ethylene oligomerization by (acac)NiH has been studied using density functional theory (DFT). The transition states for chain propagation and chain termination were optimized and the related reaction barriers calculated. Several possible mechanisms were considered for the chain termination step. Chain termination by beta-hydrogen elimination was found to be energetically unfavorable, and is not likely to be important. Instead, beta-hydrogen transfer to the incoming ethylene unit seems to be operative. The most favorable beta-hydrogen transfer pathway has two transition states. The first leads from a weak pi-complex between an incoming ethylene unit and (acac)NiCH(2)CH(2)R to an intermediate in which the two olefins C(2)H(4) and H(2)CCHR both are strongly pi-complexed to the nickel hydride (acac)NiH. The second barrier takes the intermediate to another weak pi-complex between (acac)NiCH(2)CH(3) and H(2)C=CHR from which the oligomer H(2)C=CHR can be released and the catalyst (acac)NiCH(2)CH(3) regenerated. Due to the mechanism of chain termination, the actual catalyst is proposed to be (acac)NiCH(2)CH(3) whereas (acac)NiH serves as a precursor or precatalyst.