Session D35: Focus Session: Biological Networks

Synthetic Gene Networks: \textit{De novo constructs -- in numero} descriptions

Uncovering the structure and function of gene regulatory networks has become one of the central challenges of the post-genomic era. Theoretical models of protein-DNA feedback loops and gene regulatory networks have long been proposed, and recently, certain qualitative features of such models have been experimentally corroborated. This talk will focus on model and experimental results that demonstrate how a naturally occurring gene network can be used as a ``parts list'' for synthetic network design. The model formulation leads to computational and analytical approaches relevant to nonlinear dynamics and statistical physics, and the utility of such a formulation will be demonstrated through the consideration of specific design criteria for several novel genetic devices. Fluctuations originating from small molecule-number effects will be discussed in the context of model predictions, and the experimental validation of these stochastic effects underscores the importance of internal noise in gene expression. The underlying methodology highlights the utility of engineering-based methods in the design of synthetic gene regulatory networks.   

Origin of Modularity in Recombination Evolution

Modularity is a well-known phenomenon in biology. Modularity implies a hierarchical character, and is manifested in both phenotypic and genotypic levels. A module is defined, in general, as a component which operates relatively independently of other components of the system. The independence is in both the structural and functional levels. How does modularity originate? Evolvability is a selectable trait and modularity enhances evolvability. Thus, under conditions that select for evolvability, we expect to see the emergence of modularity. We used a spin-glass model to simulate the evolution of genomes. This model captures the interactions between amino acids or epistasis between genes. The evolutions include both sequence evolution and structure evolution. The environment changes and recombination plays an important role in evolution. We will present our result of the emergence of modularity, a symmetry breaking of the system. We will present the dependence of modularity on the amplitude and frequency of environment changing. The crucial role of recombination in the emergence of modularity will be discussed as well.   

Dynamic network analysis of protein interactions

Network approaches have recently become a popular tool to study complex systems such as cellular metabolism and protein interactions. A substantial number of analyses of the protein interaction network (PIN) of the yeast {\it Saccharomyces cerevisiae} have considered this network as a static entity, not taking the network's dynamic nature into account. Here, we examine the time-variation of gene regulation superimposed on the PIN by defining mRNA expression profiles throughout the cell cycle as node weights. To characterize these network dynamics, we have both developed a set of novel network measures as well as studied previously published measures for weighted networks. We expect that our approach will provide a deeper understanding of protein regulation during the cell cycle.   

Growth induced instability in metabolic networks

Networks of molecular interactions are essential for mass, energy, and information transport into and within cells. Thus, understanding the emergent physical properties of various network architectures is of fundamental interest in biology. One such architecture, product-feedback inhibition is widely used in~the regulation of biosynthetic pathways of all organisms. Importantly, these biosynthetic pathways are often coupled both by the use of a common substrate and by stoichiometric utilization of their products for cell growth. We analyze networks having~the following three essential features: all branches start from a common substrate, the product of each branch inhibits the first dedicated step towards its synthesis, and all products are essential for growth. We show that such a coupled network can have at most one steady state. However, the network~may be~unstable about this steady state, even if the branches are individually stable. In the unstable region, the network exhibits limit-cycle oscillations which arise via a Hopf bifurcation. In the oscillating regime, a two-branch coupled network can be mapped to a three-species frustrated system. Our results highlight new design principles essential for realizing robust biosynthetic pathways.   

Emergent Criticality from Co-evolution in Random Boolean Networks

The co-evolution of network topology and dynamics is studied in an evolutionary Boolean network model that is a ``coarse-grained" model of a gene regulatory network. We find that a critical state emerges spontaneously from the interplay between topology and dynamics when the network is updated by a rule that rewires its internal connections based on the activities of nodes and changes the dynamical functions. The final evolved state is shown to be critical and independent of initial conditions. The network appears to be driven to a random Boolean network with uniform in-degree of 2 in the large network limit. However, for biologically realized network sizes, significant finite-size effects are observed including a broad in-degree distribution and an average in-degree connection between 2 and 3. These results may be important for explaining the formation of heterogeneous topology in real gene regulatory networks. Detailed work is discussed in the paper Phys. Rev. E \textbf{74}, 041910 (2006).   

Protein-Protein interaction networks: why static MpK model works and preferential attachment does not

Various approaches have been proposed to explain the observed scale free structure $p(k) \sim k^{-\gamma}$ of protein-protein interaction networks. We argue that the preferential attachment coming from gene duplication[1] is questionable. A static ``MpK'' model produces the scale free structure via computer simulations[2] for unexplained reasons. On the other hand, it was analytically proved[3] that deterministic threshold models produce scale free networks (with $\gamma\equiv 2$) if fitness distributions are exponential. We study the static MpK model further and find the above analytical proof applicable with extensions, and $\gamma$ dependent on the threshold parameter. This work not only predicts the dependence of $\gamma$ on protein concentrations, but also provides a generic mechanism of scale free networks. The clustering coefficient distribution in the model is interpreted by a simple picture. \newline \newline [1] A.-L. Barab\'asi and Z. N. Oltvai, Nature Reviews Genetics \textbf{5}, 101 (2004). \newline [2] E. J. Deeds, O. Ashenberg, E. I. Shakhnovich, Proc. Natl. Acad. Sci. USA \textbf{103}, 311 (2006). \newline [3] G. Caldarelli, A. Capocci, P. De Los Rios, and M. A. Mu\~noz, Phys. Rev. Lett. \textbf{89}, 258702 (2002).   

A closer look at activity in metabolic networks

Single-cell organisms are assumed to optimize growth under specific conditions. Using flux balance analysis, it is possible to estimate the number of reactions that are utilized (active) by the metabolism in random and optimal metabolic states. Here we investigate the mechanisms that determine the number of active reactions mathematically and compare them to those of real organisms.   

Stochastic Chemical Kinetics in Biochemical Reaction Networks.

We used various analytic and numerical methods to elucidate complex dynamics in stochastic signal transduction. We demonstrate that the commonly used linear noise approximation to solving the chemical master equation fails when the number of proteins becomes too low. Consequently, we developed a new analytical approximation to the solution of the master equation, based on the generating function approach, which works in a much wider range of protein number fluctuations. We show that in a linear signaling pathway, a reaction rate at a node could be tuned so the node acts either as a noise amplifier or as a noise filter. For more complex cascades, we mapped the stochastic chemical kinetics master equation into a quantum field theoretical problem, which we solved using the variational principle. We demonstrate stochastic resonance in signal transmission in enzymatic cascades with and without feedback loops.   

Noise propagation in combined cellular control motifs

A cell's ability to respond robustly to noisy stimuli critically depends on the structure of its regulation and control circuitry, as well as kinetic parameters. While kinetic parameters take a wide range of values, there is markedly less variation in the basic network building blocks. We have explored the functional implications of several motif-combinations, investigating their information processing properties. Adopting a spectral-analysis approach, we study how circuit topology affects the propagation or attenuation of intrinsic and extrinsic noise. Finally, we discuss possible fitness benefits of the different circuit topologies, relating design principles to evolutionary selection.   

Stochastic effects in reaction networks

Experiments that yield information about single cells make clear that intrinsic noise in reactions involving low copy numbers of molecules can have important functional consequences. Although it is typically assumed that noise introduces isotropic fluctuations about a mean, this need not be the case. Within the Langevin framework, we develop ``rules of thumb'' for understanding the impact of noise on systems of reactions. We show analytically how increases in either the magnitude or correlation time of fluctuations can give rise to amplifications and bifurcations. As an example, we consider the enzymatic cycle studied by Goldbeter and Koshland. Fluctuations in the total number of enzyme for the forward reaction have been shown to amplify the concentration of the modified substrate and can even create additional peaks in its distribution. We show how our results lead to a transparent physical interpretation of these observations, and we clarify how ultrasensitivity, amplification, bifurcation, and stochastic focusing relate to each other.   

Universal patterns in the behavior of complex systems: from relaxation in fractal networks to distribution of income

The study of relaxation is an active area of research in the fields of dielectric, mechanical and optical spectroscopy, which is insufficiently developed for the case of complex systems. It has been established that the relaxation of many systems deviates markedly from classical Debye dispersion function (in the frequency domain) or from pure exponential decay (in the time domain), but the exact ways in which these deviations occur and their significance are still debated issues. Here we propose that a fractal-tree network appropriately describes the relaxation pathway in a variety of complex systems and predicts coupled (or hierarchical) as well as uncoupled (parallel) relaxation processes. This approach has been originally introduced for description of dielectric relaxation in Cantorian trees in biology. Upon adequate generalization this approach sheds new light on a variety of processes, ranging from kinetics of protein-ligand rebinding through distribution of income in populations of humans.   

Understanding Dynamic Patterns of NF-$\kappa $B Signaling: Derivation and Analysis of a Minimal Model through Sensitivity Analysis

Understanding the pleiotropism of NF-$\kappa $B signal transduction is a challenge of clear medical importance and systems biology. Current mathematical modeling frameworks for NF-$\kappa $B signal transduction, though limited to a small signaling module located in a downstream of IKK, heavily rely on the parameterizations and the numerical studies of ODE models and doubtless lack intuitive explanations about underlying mechanisms of the dynamic patterns of the NF-$\kappa $B signaling. Here we present a systematic way to derive a minimal model from an up-to-dated and detailed NF-$\kappa $B signaling network by means of sensitivity analysis. Using analysis of the minimal model, we predict a dose-response curve shape, existence of Hopf-bifurcation, and underlying mechanisms of all possible dynamic patterns of NF-$\kappa $B signaling. Simulating the detailed ODE model for NF-$\kappa $B signaling network with large sets of the parameter values that are sampled from the biologically feasible parameter space, we present an ensemble of all possible dynamic patterns of NF-$\kappa $B signaling and verify the predictions from the minimal model.   

Sloppy systems biology: tight predictions with loose parameters

Directly measuring the parameters involved in dynamical models of cellular processes is typically very difficult, and collectively fitting such parameters to other data often yields large parameter uncertainties. Nonetheless, a collective fit which only weakly constrains model parameters may strongly constrain model \emph{predictions}, if the model is ill-conditioned: much more sensitive to some directions in parameter space than others. In the quadratic approximation, the model sensitivities are proportional to the inverse square roots of the hessian matrix eigenvalues. Using a collection of 14 models from the systems biology literature, we show that for large systems the eigenvalue spectra are universally \emph{sloppy}; they span huge ranges ($> 10^6$) and have approximately constant logarithmic spacing. Thus the models are ill-conditioned and have no well-defined cutoff between important and unimportant parameter combinations. This universal sloppiness suggests that collective fits will often poorly constrain parameters but usefully constrain many predictions.   

A Physical Theory of the Competition that Allows HIV to Escape from the Immune System

Competition within the immune system may degrade immune control of viral infections. We formalize the evolution that occurs in both HIV-1 and the immune system quasispecies [1]. Inclusion of competition in the immune system leads to a novel balance between the immune response and HIV-1, in which the eventual outcome is HIV-1 escape rather than control. The analytical model reproduces the three stages of HIV-1 infection. We propose a vaccine regimen that may be able to reduce competition between T cells, potentially eliminating the third stage of HIV-1. 1) G. Wang and M. W. Deem, Phys. Rev. Lett. 97 (2006) 188106.