Session W29: Biological Networks and System Biology

Self-Organization of Networks Via Synchrony-Dependent Plasticity

We employ an adaptive parameter control technique based on a previously developed measure that detects phase/lag synchrony in the system to dynamically modify the structure of a network of non-identical, weakly coupled R\"{o}ssler oscillators. Two processes are simulated: adaptation, under which the initially different properties (such as frequency) of the units converge, and aggregation, in which coupling between units is altered and clusters of interconnected elements are formed based on the temporal correlations. We show that adaptation speed depends on connectivity and topology, with more global connections resulting in greater temporal order and faster convergence of adaptation. We find that aggregation leads to unidirectional clusters, and that asymmetric aggregation (with differing rates for increasing or decreasing coupling strength) has an optimum ratio of rates to make denser clusters that maintain their selectivity. Combining adaptation and aggregation results in clusters of identical oscillators with bi-directional coupling. An optimum ratio of process rates results in stable coupling between the units. Change from this ratio may result in annihilation of the network for slow aggregation, or more numerous, denser, and more transient clusters for faster aggregation.   

Phase reduction analysis of coupled neural oscillators: application to epileptic seizure dynamics

Epileptic seizures are generally held to be result from excess and synchronized neural activity. To investigate how seizures initiate, we develop a model of a neocortical network based on a model suggested by Wilson [1]. We simulate the effect of the potassium channel blocker 4-aminopyridine, which is often used in experiments to induce epileptic seizures, by decreasing the conductance of the potassium channels (g$_{K})$ in neurons in our model. We applied phase reduction to the Wilson model to study how g$_{K}$ in the model affects the stability of the phase difference. At a normal value of g$_{K}$, the stable phase difference is small, but the neurons are not exactly in phase. At low g$_{K}$, in-phase and out-of-phase firing patterns become simultaneously stable. We constructed a network of 20 by 20 neurons. By decreasing g$_{K }$to zero, a dramatic increase in the amplitude of mean field was observed. This is due to the fact that in-phase firing becomes stable at low g$_{K}$. The pattern was similar to local field potential in 4-aminopyridine induced seizures. Therefore, it was concluded that the neural activity in drug-induced seizure may be caused by a bifurcation in stable phase differences between neurons. [1] Wilson H.R., J. Theor. Biol. (1999) 200, 375-388 [2] Ermentrout, G.B. and Kopell, N., SIAM J. Math. Anal. (1984), 215-237   

Effect of Delays and Network Topology in Spatiotemporal Pattern Formation

Synchronization between connected neurons is believed to play a role in the processing of information within the brain. This implies a temporal ordering in the discharge of their electrical signals but since the axons have a finite length over which a signal must traverse, information relating to a particular process and emanating from different neurons reaches a target neuron after a time delay. We investigate the effects of delays on the formation of temporally ordered states in a model network with SWN topology. We show that incorporation of two different types of delay, length independent and length dependent, lead to dramatically different properties of the network. In the first case, the formation of global random connections leads to an increase in temporal ordering, while in the second case locally ordered clusters are annihilated and form a disordered state.   

Inhibitory Synaptic Coupling and Spatiotemporal Synchrony in a Neural Model

We study the behavior of an array of neurons, connected by excitatory and inhibitory synapses, when the relative proportion of such connections is varied. The cells, described by the Huber-Braun model [1], exhibit different bursting states as parameters such as temperature and coupling strength are tuned. In a recent paper [2], stochastic phase synchronization was studied in this model, using gap-junction type connections. Here, we extend this work to more realistic synaptic connectivities, to investigate the connection between bursting and synchronization, which has been implicated in the triggering of pathological processes such as epilepsy, since synchronous firing in neuronal populations is viewed as a hallmark of seizures. We also present evidence suggesting that noise could be responsible for transitions between various types of field potential oscillations, reminiscent of the transitions between spike-and-wave firing and low voltage fast activity observed in the epileptic cortex. [1] H. A. Braun, M. T. Huber, M. Dewald, K. Sch\"{a}fer, and K. Voigt. Computer simulations of neuronal signal transduction: the role of nonlinear dynamics and noise. Intl. J. Bifurcation and Chaos 8(5): 881-889, 1998. [2] S. Bahar. Burst-enhanced synchronization in an array of noisy coupled neurons. Fluctuation and Noise Letters 4(1):L87-L96, 2004.   

Attentional modulation of stimulus competition in a large scale model of the visual pathway

Neurons in cortical area V4 are sensitive to shape and have large receptive fields. In a typical visual scene there are multiple objects in the V4 cell's receptive field, only a few of which may be behaviorally relevant. The visual system is capable of selecting relevant objects by increasing the neural response to them and reducing the response to non-relevant objects. Neuronal synchrony may play an important role in this process. Using a large-scale network model of the visual pathway, we study the emergence of shape selectivity in V4, the competition between different objects for control of the firing rate of individual V4 neurons, the attentional modulation of this stimulus competition and the role of synchrony.   

Specificity, promiscuity, and the structure of complex information processing networks

Both the top-down designs of engineered systems and the bottom-up serendipities of biological evolution must negotiate tradeoffs between specificity and control: overly specific interactions between components can make systems brittle and unevolvable, while more generic interactions can require elaborate control in order to aggregate specificity from distributed pieces. Complex information processing systems reveal network organizations that navigate this landscape of constraints: regulatory and signaling networks in cells involve the coordination of molecular interactions that are surprisingly promiscuous, and object-oriented design in software systems emphasizes the polymorphic composition of objects of minimal necessary specificity [C.R. Myers, Phys Rev E 68, 046116 (2003)]. Models of information processing arising both in systems biology and engineered computation are explored to better understand how particular network organizations can coordinate the activity of promiscuous components to achieve robust and evolvable function.   

Modeling of signal transduction in bacterial quorum-sensing

Several species of bacteria are able to coordinate gene regulation in response to population density, a process known as ``quorum-sensing''. Quorum-sensing bacteria produce, secrete, and detect signal molecules called autoinducers. For several species of bacteria in the {\it Vibrio} genus, recent results have shown that the external autoinducer concentrations control the expression of regulatory small RNA(s) which are critical to the process of quorum-sensing. We present a theoretical analysis of the network which relates the rate of small RNA expression to the external autoinducer concentrations. We relate the results from our modeling to previous experimental observations and suggest new experiments based on testable predictions of the model.   

Sloppiness is universal in systems biology: making predictions nonetheless

Quantitative models of complex biological systems often possess dozens of unknown parameters. We argue that such systems are universally ``sloppy''; their behaviors are orders of magnitude more sensitive to moves in some directions in parameter space than others. To establish this, we survey models from the literature and show that their ``complete and perfect data'' Fisher Information Matrices possess eigenvalues typically spanning a range of more than $10^6$. Sloppiness implies that collectively fitting model parameters to even the best experimental data will tightly constrain only a few parameter combinations, perhaps suggesting the necessity of a difficult experimental program to measure each individual parameter. An example demonstrates, however, that a collective fit to a modest amount of real data may tightly constrain model behavior even though it only poorly constrains many parameter combinations. Low uncertainty predictions can thus be made without knowledge of precise values for individual parameters.   

Inference and analysis of gene-regulatory networks in the bacterium B.subtilis

We present the methods and results of a two-stage modeling process that generates candidate gene-regulatory networks of the bacterium B.subtilis from experimentally obtained, yet mathematically underdetermined microchip array data. By employing a computational, linear correlative procedure to generate these networks, and by analyzing the networks from a graph theoretical perspective, we are able to verify the biological viability of our simulated networks, and we demonstrate that our networks' graph theoretical properties are remarkably similar to those of other, more well-studied biological systems. We test the robustness of the inference process first by introducing noise into the experimental data, and then by comparing the graph theoretical properties of the resulting perturbed networks to those of the original networks.   

Sensitivity-based approach to optimal experimental design in a receptor trafficking and down regulation model

We apply the ideas of optimal experimental design to systems biology models: minimizing a design criterion based on the average variance of predictions, we suggest new experiments that need to be performed to optimally test a given biological hypothesis. The estimated variance in predictions is derived from the sensitivities of protein and chemical species in the model to changes in reaction rates. The sensitivities also allow us to determine which interactions in the biological network dominate the system behavior. To test the design principles, we have developed a differential equation model incorporating the processes of endocytosis, recycling and degradation of activated epidermal growth factor (EGF) receptor in a mammalian cell line. Recent experimental work has discovered mutant proteins that cause receptor accumulation and a prolonged growth signal. Our model is optimized to fit this mutant experimental data and wild type data for a variety of experimental conditions. Of biological interest is the effect on surface and internalized receptor levels after the overexpression or inactivation of regulator proteins in the network: the optimal design method allows us to fine tune the conditions to best predict the behavior of these unknown components of the system.   

Stochasticity in the Expression of LamB and its Affect on $\lambda $ phage Infection

$\lambda $ phage binds to \textit{E. Coli's} lamB protein and injects its DNA into the cell. The phage quickly replicates and after a latent period the bacteria bursts, emitting mature phages. We developed a mathematical model based on the known physical events that occur when a $\lambda $ phage infects an \textit{E.Coli} cell. The results of these models predict that the bacteria and phage populations become extinct unless the parameters of the model are very finely tuned, which is untrue in the nature. The lamB protein is part of the maltose regulon and can be repressed to minimal levels when grown in the absence of inducer. Therefore, a cell that is not expressing any lamB protein at that moment is resistant against phage infection. We studied the dynamic relationship between $\lambda $ phage and \textit{E. Coli} when the concentration of phage greatly outnumbers the concentration of bacteria. We study how the stochasticity of the expression of lamB affects the percentage of cells that the $\lambda $ phage infects. We show that even in the case when the maltose regulon is fully induced a percentage of cells continue to persist against phage infection.   

Control of lineage stability and its role in resolving cell fates

We synthesize experimental data from recent studies to construct a computational model for the gene regulatory network that governs the development of immune cells and use it to explain several surprising results. At the heart of the model is a cross-antagonism between the macrophage-promoting factor Egr and the neutrophil-promoting factor Gfi. This module is capable of giving rise to both graded and bistable responses. Increasing the concentrations of these factors forces the system into the bistable regime in which cells can decide stochastically between fates. This bistable switch can be used to explain cell reprogramming experiments in which a gene associated with one cell fate is induced in progenitors of another. In one such experiment, C/EBP$\alpha $, a neutrophil promoting factor, was induced in B cell progenitors which then differentiated to macrophages. Our model shows that if C/EBP$\alpha $ is induced early, it can induce differentiation to a neutrophil. In B cell progenitors, however, the bistable switch is already in a macrophage promoting state. Thus, expression of C/EBP$\alpha $ cannot activate the neutrophil pathway, but it can repress the B cell pathway and promote macrophage differentiation.   

Role of finite-size fragments in analysis of DNA replication

In higher organisms, DNA replicates simultaneously from many origins. Recent in- vitro experiments have yielded large amounts of data on the state of replication of DNA fragments. From measurements of the time dependence of the average size of replicated and non-replicated domains, one can estimate the rate of initiation of DNA replication origins, as well as the average rate at which DNA bases are copied. One problem in making such estimates is that, in the experiments, the DNA is broken up into small fragments, whose finite size can bias downwards the measured averages. Here, we present a systematic way of accounting for this bias by deriving theoretical relationships between the original domain-length distributions and fragment-domain length distributions. We also derive unbiased average-domain-length estimators that yield accurate results even in cases where the replicated (or non-replicated) domains are larger than the average DNA fragment. Then we apply these estimators to previously obtained experimental data to extract improved estimates of replication kinetics parameters.   

Stochastic dynamics of macromolecular-assembly networks.

The formation and regulation of macromolecular complexes provides the backbone of most cellular processes, including gene regulation and signal transduction. The inherent complexity of assembling macromolecular structures makes current computational methods strongly limited for understanding how the physical interactions between cellular components give rise to systemic properties of cells. Here we present a stochastic approach to study the dynamics of networks formed by macromolecular complexes in terms of the molecular interactions of their components [1]. Exploiting key thermodynamic concepts, this approach makes it possible to both estimate reaction rates and incorporate the resulting assembly dynamics into the stochastic kinetics of cellular networks. As prototype systems, we consider the \textit{lac} operon and phage $\lambda $ induction switches, which rely on the formation of DNA loops by proteins [2] and on the integration of these protein-DNA complexes into intracellular networks. This cross-scale approach offers an effective starting point to move forward from network diagrams, such as those of protein-protein and DNA-protein interaction networks, to the actual dynamics of cellular processes. [1] L. Saiz and J.M.G. Vilar, submitted (2005). [2] J.M.G. Vilar and L. Saiz, Current Opinion in Genetics {\&} Development, 15, 136-144 (2005).   

Selective advantage for sexual reproduction

We develop a simplified model for sexual replication within the quasispecies formalism. We assume that the genomes of the replicating organisms are two-chromosomed and diploid, and that the fitness is determined by the number of chromosomes that are identical to a given master sequence. We also assume that there is a cost to sexual replication, given by a characteristic time $ \tau_{seek} $ during which haploid cells seek out a mate with which to recombine. If the mating strategy is such that only viable haploids can mate, then when $ \tau_{seek} = 0 $, it is possible to show that sexual replication will always outcompete asexual replication. However, as $ \tau_{seek} $ increases, sexual replication only becomes advantageous at progressively higher mutation rates. Once the time cost for sex reaches a critical threshold, the selective advantage for sexual replication disappears entirely. The results of this talk suggest that sexual replication is not advantageous in small populations per se, but rather in populations with low replication rates. In this regime, the cost for sex is sufficiently low that the selective advantage obtained through recombination leads to the dominance of the strategy. In fact, at a given replication rate and for a fixed environment volume, sexual replication is selected for in high populations because of the reduced time spent finding a reproductive partner.