Robustness and fragility of Boolean models for genetic regulatory networks
Introduction
Understanding how genetic information is translated into proteins to produce various cell types remains a major challenge in contemporary biology (Wolpert et al., 1998). Gene products often regulate the synthesis of mRNAs and proteins, forming complex networks of regulatory interactions. Concurrently with experimental progress in gene control networks (Davidson and et al., 2002), several alternative modeling frameworks have been proposed. In the continuous-state approach, the concentrations of cellular components are assumed to be continuous functions of time, governed by differential equations with mass-action (or more general) kinetics (Reinitz and Sharp, 1995, von Dassow et al., 2000, Gursky et al., 2001). Stochastic models address the deviations from population homogeneity by transforming reaction rates into probabilities and concentrations into numbers of molecules (Rao et al., 2002). Finally, in the discrete approach, each component is assumed to have a small number of qualitative states, and the regulatory interactions are described by logical functions (Mendoza et al., 1999, Sánchez and Thieffry, 2001, Yuh et al., 2001, Kauffman et al., 2003, Ghysen and Thomas, 2003, Bodnar, 1997, Albert and Othmer, 2003, Espinosa-Soto et al., 2004).
The kinetic details of protein–protein or protein–DNA interactions are rarely known, but there is increasing evidence that the input–output curves of regulatory relationships are strongly sigmoidal and can be well approximated by step functions (Yuh et al., 2001, Thomas, 1973). Moreover, both models and experiments suggest that regulatory networks are remarkably robust, that is, they maintain their function even when faced with fluctuations in components and reaction rates (von Dassow et al., 2000, Alon et al., 1999, Eldar et al., 2002, Carlson and Doyle, 2002, Conant and Wagner, 2004, Espinosa-Soto et al., 2004). These observations lend support to the assumption of discrete states for genetic network components and of combinatorial rules for the effects of transcription factors (Glass and Kauffman, 1973, de Jong et al., 2004). The extreme of discretization, Boolean models, consider only two states (expressed or not), closely mimicking the inference methods used in genetics (Kauffman et al., 2003, Thomas, 1973, Kauffman, 1993). It is straightforward to study the effect of knock-out mutations or changes in initial conditions in this framework, and the agreement between a real system and a Boolean model of it is a strong indication of the robustness of the system to changes in kinetic details (Albert and Othmer, 2003).
In discrete models the decision whether a network node (component) will be affected by a synthesis or decay process is determined by the state of effector nodes (nodes that interact with it). Typical time-dependent Boolean models use synchronous updating rules (Kauffman et al., 2003, Albert and Othmer, 2003, Bodnar, 1997, Kauffman, 1993), assuming that the time-scales of the processes taking place in the system are similar. In reality the time-scales of transcription, translation, and degradation can vary widely from gene to gene and can be anywhere from minutes to hours. Logical models following the formalism introduced by René Thomas (Thomas, 1973) allow asynchronism by associating two variables to each gene: a state variable describing the level of its protein, and an image variable that is the output of the logical rule whose inputs are the state variables of effector nodes. Whether the future state variable of a gene equals the image or current state variable depends on the update order and, in the absence of temporal information, the Thomas formalism focuses on determining the steady states, where the state and image variables coincide (Mendoza et al., 1999, Sánchez and Thieffry, 2001, Ghysen and Thomas, 2003, Bernot et al., 2004). The effect of asynchronous updates on the dynamics of the system, however, has not been explored yet.
In this paper, we present a methodology for testing the robustness of Boolean models with respect to stochasticity in the order of updates. Through this, we are also probing the system itself: will individual variations lead to unexpected gene expression patterns? In the asynchronous method, the synthesis/decay decision is made at different time-points for each node, allowing individual variability in each process’ duration, but more importantly, it allows for decision reversal if the dynamics of effector nodes changes. It becomes possible to reproduce, e.g., the overturning of mRNA decay when its transcriptional activator is synthesized, a process that synchronous update cannot capture. Thus, replacing synchronous with asynchronous updates is not merely a technical detail, but rather a fundamental paradigm shift from pointwise in time to potentially continuous communication between nodes. Indeed, the effective synthesis or decay times for a certain node are determined by the time interval between the latest update of its effector nodes and its current update time, and can be any positive fraction of the unit time interval. We propose three algorithms, with varying freedom in the relative duration of cellular processes, and find that very short transcription or decay times have the potential to derail the wild-type development process.
The steady states of a Boolean model will remain the same regardless of the mechanism of update, but its dynamical behavior can be drastically altered due to the stochastic nature of the updates; for instance, the same initial state may lead to different steady states or limit cycles. Since the duration of synthesis and decay processes is not known, we randomly explore the space of all possible time-scales and update orders, and derive the probability of different outcomes. Our methods offer a systematic way of exploring generic behavior of gene regulatory networks and comparing it to experimentally observed outcomes. To present a concrete example, we generalize a previously introduced Boolean model of the Drosophila segment polarity genes (Albert and Othmer, 2003). This model reproduces the wild-type steady state pattern of the segment polarity genes as well as the gene patterns of mutants, but its dynamic behavior is not directly comparable to that of the real system. Here we show that asynchronous update leads to a much more realistic model that gives further insights into the robustness of the gene regulatory network.
Section snippets
The segment polarity gene network in Drosophila
The Drosophila melanogaster segment polarity genes represent the last step in the hierarchical cascade of gene families initiating the segmented body of the fruit fly. While the preceding genes act transiently, the segment polarity genes are expressed throughout the life of the fly, and their periodic spatial pattern is maintained for at least 3 h of embryonic development (Wolpert et al., 1998). The regulatory roles of the previously expressed genes such as the pair-rule genes fushi tarazu, runt
Randomly perturbed time-scales
As in the context of parallel computation systems, the fundamental difference between synchronous and asynchronous updates is at the level of task coordination and data communication among nodes in a network (Bertsekas and Tsitsiklis, 1989). Synchronous algorithms are highly coordinated: at pre-determined instants, all the nodes “stop” and exchange the current information among themselves. For instance, suppose there are N nodes, where each node i “computes” the state of variable , according
Time-scale separation uncovers robustness of the model
In both of the previous algorithms we assumed no bias towards a preferred protein/mRNA updating sequence and, as a result, an unrealistic divergence from the wild-type pattern is observed, with high incidence of inviable states. Based on the fact that post-translational processes such as protein conformational changes or complex formation usually have shorter durations than transcription, translation or mRNA decay, we introduce a distinct time-scale separation by choosing to update proteins
A Markov chain process
As a Boolean model, there are only a finite set, say , of distinct states (in the total state space ) reachable by the system. Starting from any state , each permutation of takes the system to some other state . It is possible to theoretically identify all the distinct intermediate and final states of the system as well as all the possible transitions after one iteration. Thus the asynchronous algorithm consisting of the N node functions (2) together with assumptions
Identifying minimal pre-patterns
A necessary condition for convergence to the wild-type is that . Otherwise the trajectory immediately fails to enter the basin of attraction of the wild-type state: Fact 3 Assume . Then , for some . Proof Note that implies . Using Eqs. (6) and (7), (8), the equations for and simplify to where (respectively, ), if
Conclusions
In summary, we proposed an intuitive and practical way of introducing stochasticity in qualitative models of gene regulation. We explored three possible ways of incorporating the variability of transcription, translation, post-translational modification and decay processes (see Table 5 for a comparison between the synchronous and three asynchronous algorithms). Applying our methods on a previously introduced model of the Drosophila segment polarity genes gave us new insights into the dynamics
Acknowledgements
The work of M.C. was supported in part by NIH Grants P20 GM64375 and Sanofi-aventis. R.A. gratefully acknowledges an Alfred P. Sloan Research Fellowship. The work of E.D.S. was supported in part by NSF Grant CCR-0206789 and NIH Grants P20 GM64375 and R01 GM46383.
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