The spatiotemporal expression of genes in an organism is determined by regulatory systems that involve a large number of genes connected through a complex network of interactions. As an intuitive understanding of the behavior of these systems is hard to obtain, computer tools for the modeling and simulation of genetic regulatory networks will be indispensable. This report reviews formalisms that have been employed in mathematical biology and bioinformatics to describe genetic regulatory systems, in particular directed graphs, Bayesian networks, ordinary and partial differential equations, stochastic equations, Boolean networks and their generalizations, qualitative differential equations, and rule-based formalisms. In addition, the report discusses how these formalisms have been used in the modeling and simulation of regulatory systems.

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Modeling and simulation of genetic regulatory systems: a literature review.

Engineered systems are often built of recurring circuit modules that carry out key functions. Transcription networks that regulate the responses of living cells were recently found to obey similar principles: they contain several biochemical wiring patterns, termed network motifs, which recur throughout the network. One of these motifs is the feed-forward loop (FFL). The FFL, a three-gene pattern, is composed of two input transcription factors, one of which regulates the other, both jointly regulating a target gene. The FFL has eight possible structural types, because each of the three interactions in the FFL can be activating or repressing. Here, we theoretically analyze the functions of these eight structural types. We find that four of the FFL types, termed incoherent FFLs, act as sign-sensitive accelerators: they speed up the response time of the target gene expression following stimulus steps in one direction (e.g., off to on) but not in the other direction (on to off). The other four types, coherent FFLs, act as sign-sensitive delays. We find that some FFL types appear in transcription network databases much more frequently than others. In some cases, the rare FFL types have reduced functionality (responding to only one of their two input stimuli), which may partially explain why they are selected against. Additional features, such as pulse generation and cooperativity, are discussed. This study defines the function of one of the most significant recurring circuit elements in transcription networks.

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Structure and function of the feed-forward loop network motif

http://mcb.berkeley.edu/courses/mcb137/exercises/Tyson%20(2003).pdf

Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell

Motivation: Our goal is to construct a model for genetic regulatory networks such that the model class: (i) incorporates rule-based dependencies between genes; (ii) allows the systematic study of global network dynamics; (iii) is able to cope with uncertainty, both in the data and the model selection; and (iv) permits the quantification of the relative influence and sensitivity of genes in their interactions with other genes. Results: We introduce Probabilistic Boolean Networks (PBN) that share the appealing rule-based properties of Boolean networks, but are robust in the face of uncertainty. We show how the dynamics of these networks can be studied in the probabilistic context of Markov chains, with standard Boolean networks being special cases. Then, we discuss the relationship between PBNs and Bayesian networks—a family of graphical models that explicitly represent probabilistic relationships between variables. We show how probabilistic dependencies between a gene and its parent genes, constituting the basic building blocks of Bayesian networks, can be obtained from PBNs. Finally, we present methods for quantifying the influence of genes on other genes, within the context of PBNs. Examples illustrating the above concepts are presented throughout the paper.

Probabilistic Boolean networks: a rule-based uncertainty model for gene regulatory networks

This review describes a body of work on computational immune systems that behave analogously to the natural immune system. These artificial immune systems (AIS) simulate the behavior of the natural immune system and in some cases have been used to solve practical engineering problems such as computer security. AIS have several strengths that can complement wet lab immunology. It is easier to conduct simulation experiments and to vary experimental conditions, for example, to rule out hypotheses; it is easier to isolate a single mechanism to test hypotheses about how it functions; agent-based models of the immune system can integrate data from several different experiments into a single in silico experimental system.

Computer immunology

Dynamical behaviour of biological regulatory networks--I. Biological role of feedback loops and practical use of the concept of the loop-characteristic state.

A biological introduction serves to remind us that differentiation is an epigenetic process, that multistationarity can account for epigenetic differences, including those involved in cell differentiation, and that positive feedback circuits are a necessary condition for multistationarity and, by inference, for differentiation. The core of the paper is comprised of a formal description of feedback circuits and unions of disjoint circuits. We introduce the concepts of full-circuit (a circuit or union of disjoint circuits which involves all the variables of the system), and of ambiguous circuit (a circuit whose sign depends on the location in phase space). We describe the partition of phase space (a) according to the signs of the ambiguous circuits, and (b) according to the signs of the eigenvalues or their real part. We introduce a normalization of the system versus one of the circuits; in two variables, this permits an entirely general description in terms of a common diagram in the “circuit space.” The paper ends with general statements concerning the requirements for multistationarity, stable periodicity, and deterministic chaos.

Multistationarity, the basis of cell differentiation and memory. I. Structural conditions of multistationarity and other nontrivial behavior.

Circuits and their involvement in complex dynamics are described in differential terms in Part I of this work. Here, we first explain why it may be appropriate to use a logical description, either by itself or in symbiosis with the differential description. The major problem of a logical description is to find an adequate way to involve time. The procedure we adopted differs radically from the classical one by its fully asynchronous character. In Sec. II we describe our “naive” logical approach, and use it to illustrate the major laws of circuitry (namely, the involvement of positive circuits in multistationarity and of negative circuits in periodicity) and in a biological example. Already in the naive description, the major steps of the logical description are to: (i) describe a model as a set of logical equations, (ii) derive the state table from the equations, (iii) derive the graph of the sequences of states from the state table, and (iv) determine which of the possible pathways will be actually followed in terms of time delays. In the following sections we consider multivalued variables where required, the introduction of logical parameters and of logical values ascribed to the thresholds, and the concept of characteristic state of a circuit. This generalized logical description provides an image whose qualitative fit with the differential description is quite remarkable. A major interest of the generalized logical description is that it implies a limited and often quite small number of possible combinations of values of the logical parameters. The space of the logical parameters is thus cut into a limited number of boxes, each of which is characterized by a defined qualitative behavior of the system. Our analysis tells which constraints on the logical parameters must be fulfilled in order for any circuit (or combination of circuits) to be functional. Functionality of a circuit will result in multistationarity (in the case of a positive circuit) or in a cycle (in the case of a negative circuit). The last sections deal with “more about time delays” and “reverse logic,” an approach that aims to proceed rationally from facts to models.

Marcelle Kaufman

Papers

Dynamical behaviour of biological regulatory networks--I. Biological role of feedback loops and practical use of the concept of the loop-characteristic state.

Multistationarity, the basis of cell differentiation and memory. I. Structural conditions of multistationarity and other nontrivial behavior.

Multistationarity, the basis of cell differentiation and memory. II. Logical analysis of regulatory networks in terms of feedback circuits

Towards a logical analysis of the immune response.

A new necessary condition on interaction graphs for multistationarity.