Trajectory

From PBDN

This page contains a somewhat technical description of a trajectory. For an introduction to trajectories in the context of SSA, see the Stochastic Simulation Algorithm page.

Introduction

A trajectory in SSA is a time-ordered sequence of states of the chemical system in question. For example, in a 4-dimensional model (a model with 4 species) such as the metabolite-enzyme model, each state is a tuple (ordered set) of 4 integers where each element's value is the population of the corresponding species.

At any time, the state of the system after the next timestep depends only on the current state, and not on any previous states. That is to say that the system has no memory of earlier states. This is called the Markov property. Trajectories are Markov chains over the state-space of the chemical system, which consists of all possible combinations of populations of the species.

Although in principle, the populations of each species could grow to any number, in the metabolite-enzyme model, for example, the practical range of values for each species might be:

Metabolite A: $a \in S_A = \{0,1,\dots,100\}$
Metabolite B: $b \in S_B = \{0,1,\dots,100\}$
Enzyme A: $e_a \in S_{Ea} = \{0,1,\dots,20\}$
Enzyme B: $e_b \in S_{Eb} = \{0,1,\dots,20\}$

The state-space, $S$ , is the Cartesian product, $S=S_A \times S_B \times S_{Ea} \times S_{Eb}$ , of the sets of the possible values of each state. The state, $(a, b, e a, e b)$ , is always an element of $S$ . Each trajectory is a random walk through the state-space, $S$ .

Animation

The animation below (which also appears on other pages) shows 4 individual trajectories in the metabolite-enzyme model. Only the two metabolite species are shown, since including the two enzyme species at the same time would require a 4-dimensional video! The population of the A species (between zero and 100) is shown by horizontal position, while the population of the B species (between 0 and 100) is shown by the vertical position. All 4 trajectories start with 100 of each species (in the top right corner).

Get Flash Player to see this video

Back to Simulation of Biochemical Reaction Networks main page.

Retrieved from "http://polacksbacken.net/wiki/Trajectory"

Category: Simulation of Biochemical Reaction Networks