Simulation of Gene Regulatory Networks

Transcription

1 Simulation of Gene Regulatory Networks Overview I have been assisting Professor Jacques Cohen at Brandeis University to explore and compare the the many available representations and interpretations of genetic regulatory networks. The term regulatory network usually refers to the complex system of interactions that controls gene transcription and translation. A simplified description of the transcription-translation process is as follows: First, RNA polymerase attaches to a promoter site on a gene and produces an mrna mirror of the gene s DNA sequence. Next, a ribosome translates a protein product from the strand of mrna. At either step in this process proteins and other chemical and physical factors in the cell s environment can interfere to either promote or suppress the production of a protein from its source gene. Terminology In this work, G x represents a gene s rate of protein (or mrna) production and P x the concentration of the gene s product. The term product state refers to the vector, P, of product concentrations. A set of constraints describe the relationship between each gene s production rate, G x, and the vector of product concentrations, P. Constraints can be expressed as a set of differential or boolean equations, a directed graph, or as a table of product states. Given a set of constraints, a network may converge to either a cycle of states or a single state, usually called a stable state. G 0 G 1 (a) P curr P next [01] 01 [10] (b) G 0 = P 1 G 1 = P 0 (c) dp 0 dt dp 1 dt = = θ m0 0 γ P m0 1 + θ m0 0 P 0 0 θ m1 1 γ P m1 0 + θ m1 1 P 1 0 (d) Figure 1: Types of constraints: (a) Directed Graph where edges are marked + or - to indicate activation or suppression of the target gene, (b) Table where brackets indicate a stable state, and a superscript of plus or minus indicates that the product will increase or decrease at the given table entry, (c) Boolean Equation, (d) Differential Equation where θ is a measure of production rate and γ is a measure of decay rate. 1

2 Simulation Given a set of constraints it is useful to determine the sequence of product states in the network. This process is referred to as simulation. To efficiently simulate a regulatory network, we often attempt to make simplifying assumptions. The diagram below depicts a hierarchy of simplifications used in simulation. Simulation Discrete Continuous Asynchronous Synchronous Differential Equation Boolean Integer Boolean Integer Linear Non-Linear Figure 2: A rough hierarchy of simulation types Synchronous and asynchronous simulations Both the synchronous and asynchronous models are discrete simulations as shown in Figure 2. In discrete simulations, product concentrations are restricted to non-negative integer values 0 <= P x <= P xmax. The rates of product-concentration, G x, are also limited: G x = 1 denotes that the gene is producing its product at a rate greater than the product s natural rate of decay, G x = 1 denotes the opposite, and G x = 0 indicates that the rates of production and decay are equal. Constraints in discrete simulations can be specified as a set of boolean equations, as a directed graph, or in table form. Synchronous simulations possess two key features: first, product concentrations change at regular time intervals, t. Second, for all P x at time t, P x P x + G x at time t + t. The asynchronous model generalizes the synchronous model in a way that provides a more accurate descretization of the continuous model. Thomas and Kaufman [1] remark that in actual data, a set of non-zero product concentration rates rarely leads to the presence or absence of every increasing product as is the case in the synchronous model. The asynchronous model improves upon the synchronous model by associating a synthesis time, t x, and a decay time, t x, with each product concentration. These time delays correspond to the rates of product synthesis and decay in a continuous system. In asynchronous simulation, only one gene s product concentration changes with each iteration of time. Time iterations change by the product s t x values. Different t x s generally have different values, hence the term asynchronous. At a given product state, the P x whose t curr +t x (or t curr +t x if 2

3 gene x s product concentration is decreasing) has the smallest value will change to P x + G x. Cell Regulation Simulator (CRS): A Java application for simulating regulatory networks The Cell Regulation Simulator application consists of an engine for simulating regulatory networks according to Thomas asynchronous model and provides a convenient way of visualizing how gene product concentrations change over time. To illustrate the asynchronous model, we use CRS to simulate the simple network represented by the constraints in Figure 1. Figure 3: CRS provides a mechanism for the user to input the initial conditions of the simulation. The GUI allows the user to set initial and maximum values of product concentrations, t x s, and t x s. The program refers to t x s as Start Times and t x s as Stop Times. In the example above t 0 and t 0 are set to 200ms and t 1 and t 1 are set to 300ms. Figure 4: The screen-shot above shows CRS logical constraint interface using the constraints in Figure 1. 3

4 Figure 5: This screen-shot shows CRS table constraint interface with constraints equivalent to the boolean formulas in Figures 1 and 4. CRS calculates a set of table constraints given a boolean formula when the user switches from logical to table constraint modes. Figure 6: The run-mode simulation graphs product concentrations as they are updated according to the given time scale. The example above was initialized with the values from Figure 3. In the initial data, t 0 < t 1 so P 0 increases before P 1. If two values of t x are ever the same, CRS randomly selects a corresponding P x to change. At 200ms P 0 reaches a concentration of 1. CRS then detects that the concentration vector < 1, 0 > is a stable state and stops the simulation. 4

5 Figure 7: The diagram above shows a directed graph of all the possible product states in our example network. The states where CRS began its recursive calculation of the graph are indicated by red font on a gray background. Stable states are indicated by a black font on a red background. When the user holds the mouse cursor over of one of the nodes, the program displays the inequalities between t x s for each of the node s edges (shown on the node < 1, 0 >). We interpret the directed graph as one would interpret a decision tree: if the system begins at state P 0 = 0 P 1 = 0 then: if t 0 < t 1 then P 0 will increase to 1 at time t 0 ; if t 1 < t 0 then P 1 will increase to 1 at time t 1. Similar statements can be associated with paths starting at other nodes of the graph. 5

6 References [1] R. Thomas and M. Kaufman. Multistationarity, the basis of cell differentiation and memory. ii. logical analysis of regulatory networks in terms of feedback circuits. Chaos, 11(1): ,