This paper presents an algorithm for approximating certain types of dynamical systems given by a system of ordinary delay differential equations by a Boolean network model. Often Boolean models are much simpler to understand than complex differential equations models. The motivation for this work comes from mathematical systems biology. While Boolean mechanisms do not provide information about exact concentration rates or time scales, they are often sufficient to capture steady states and other key dynamics. Due to their intuitive nature, such models are very appealing to researchers in the life sciences. This paper is focused on dynamical systems that exhibit bistability and are desc ribedby delay equations. It is shown that if a certain motif including a feedback loop is present in the wiring diagram of the system, the Boolean model captures the bistability of molecular switches. The method is appl ied to two examples from biology, the lac operon and the phage lambda lysis/lysogeny switch.
Since the discovery of the first gene regulatory network, the lactose metabolism network in the bacterium E. Coli by Jacob Monod, [8], such networks have been modeled extensively, traditionally with differential equations. But other modeling techniques like Boolean networks [1], stochastic models [15], Petri nets, or Bayesian networks [7] have also been used successfully. Work by Kauffman in 1969 [9] suggested that gene networks behave like Boolean switching nets and therefore Boolean networks are suitable to model them. Using a Boolean network model, it has been shown, for instance, that in some cases the network topology and the logic of the interactions among the different molecular species is sufficient to determine the qualitative dynamics of the network, without taking into account the detailed kinetics [1]. In cases where not enough information about the kinetic parameters is available to build a detailed continuous model, one often still has enough information to build a Boolean network model which can provide important information.
The purpose of this paper is to show for a particular family of continuous models that they can be approximated by a Boolean network model that retains the key information about model dynamics. Our intent is to demonstrate that Boolean models can be used to study a variety of structural and dynamic features of biological and other systems that have traditionally been modeled using continuous models. They have the added advantage that they are intuitive and do not require much mathematical background. This makes them particularly suitable for use in the life sciences. The focus of this paper is on two features: bistability and time delays. We give a general algorithm how to approximate a dynamical system given by a system of ordinary delay differential equations by a Boolean network model and validate it with two well-known examples from systems biology. The relation between discrete and continuous models has been examined before, e.g., [6] describes how a logical network can be used to create the analogous differential equations. Other methods allow to systematically create a continuous model from a discrete dynamical system [11], or to reduce the model space by using a finite state machine [4]. We briefly review the main concepts related to Boolean network models.
We use a time discrete deterministic Boolean network with synchronous update. In a Boolean model, a variable can only be in the “on” (1) or “off” (0) state. In systems biology applications, each variable typically corresponds to a molecular entity, e.g., the concentration of a gene product such as mRNA or proteins. A Boolean network of n variables consists of an update function
, also called a transition function. The dynamics of the system are generated deterministically by iteration of the transition function f . It can be visualized by its phase space, which shows all possible states and their transitions. A cycle in the phase space is called a limit cycle; if it has length one, a fixed point. Fixed points of a discrete system are the equivalent to steady states of a continuous system.
In the dependency graph, also referred to as wiring diagram, each variable is a vertex and an edge from variable x i to x j is drawn if x i shows up in the local transition function f j . A directed cycle in the dependency graph is called a feedback loop.
Consider the following Boolean network with three variables:
The phase space of this function is depicted in Fig 1 . The network has a limit cycle of length two because the state (1 0 1) transitions into (0 1 1), which turns back to (1 0 1). It also includes the fixed point (0 0 0). Discrete Visualizer of Figure 1:
Dynamics (DVD tool) [10] was used to create the phase space. It calculates the number and length of limit cycles and fixed points, as well as the trajectories from any initial state. It also generates a graph of the phase space along with the dependency graph.
We begin by describing several relevant features of biochemical networks that a model needs to capture and describe in each case how this is done in the Boolean network case. We use lower case letters for continuous variables and upper case letters for discrete ones.
It is common in biochemical networks that the concentration of a substance X decreases with time because of dilution and degradation. In a differential equations model this is usually accounted for with a negative degradation term:
In the discrete model, for simplicity it is assumed that the degradation rate for a substance is either vanishingly small and can therefore be ignored, or the substance is degraded below the threshold for discretization after two discrete timesteps.
To model the decrease of concentration of a substance by dilution and degradation, which is modeled in the continuous case by a negative degradation term for x, a new variable X old is introduced. If X old is “on,” then any quantity of X that is available has al
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