Special Topic Ppt

A GENERAL METHOD FOR THE COMPUTATION OF PROBABILITIES IN SYSTEMS OF FIRST ORDER CHEMICAL REACTIONSCHEM 112.1 Special TopicRhiza Lyne E. Villones

INTRODUCTION

Use of a stochastic way to model chemical reactions, rather than with the traditional differential equations, in small biological systems.

Due to: Small number of molecules involved in the

system Molecules are not homogenously distributed Change of molecular population level is a

discrete integer Population change is not a deterministic process

INTRODUCTION

Two approaches in the stochastic study of the number of molecules in biochemical reactions: Based on the analysis of the master equation Monte Carlo simulation methods

INTRODUCTION

McQuarrie in 1967 summarized the stochastic study using the master equation (ME) approach. For a reaction, it is assumed that in an

infinitesimal time interval, the probability of having one reaction per unit reactant molecule combination is proportional to the length of the time interval.

A probability difference function is first obtained based on the assumption, which leads to a differential-difference equation called the master equation of the reaction.

INTRODUCTION

Laurenzi in 2000 introduced a way to solve the master equation using the Laplace transform. In this way, solving the partial differential

equations is avoided, which is important for more complicated reactions.

Instead, one needs to solve a set of linear equations.

INTRODUCTION

Study of systems of first-order reactions Molecules which do not chemically interact

Independence of the molecules in a first-order reaction system is exploited to derive the population distributions of the molecules in the system.

ONE MOLECULE

Master equation for one molecule in a system of first-order reactions

Assumption: M molecule species, S1, S2, … SM

Probability rate constant of the reaction Si → Sj = cij

ME describes the evolution of the molecules’ population distribution as a function of time

ONE MOLECULE

When there is only one molecule in the system, ME gives the time evolution of the probability that this molecule has become a certain chemical species.

Let p(i)(t) = probability that the molecule is an Si molecule at time t.

For a simple system of reactions,

ONE MOLECULE

Source probability density function where f(i)(t) is the source’s probability density

Representation using the Markov chain with a transition matrix

ONE MOLECULE

The set of first-order linear differential equations has the following form:

ONE MOLECULE

The equations can be solved by applying the Laplace transform, followed by solving the resulting algebraic equation, and finally using the inverse Laplace transform.

MORE MOLECULES

Several molecules are injected in the system Can come from one source (start as the same

chemical species) More/multiple sources

EXAMPLE 1: FIRST-ORDER IRREVERSIBLE REACTION

Consider the first-order reaction: where only S1 molecules are injected into the

system with probability density function

Rewrite the equation (see slide 8) with


The equations can be solved by applying the Laplace transform, followed by solving the resulting algebraic equation, and finally using the inverse Laplace transform.

The final solution will be:


When is some other density function,

Substituting these probability values, one can get the distribution of molecules.

EXAMPLE 2: FIRST-ORDER REVERSIBLE REACTION

Consider which starts with x S1 molecules and y S2 molecules.

Each x S1 molecule gives rise to the following probabilities, which are obtained by solving the probability equation (slide8) with

For the effect of each y S2 molecule, we take the ff parameters and obtain the ff probabilities


Substituting these probabilities to

The distribution of S1 molecules are given by:


(More sources)

And similarly for the S2 molecules.

Consider the reaction system:

Assumption: x S1 molecules and y S2 molecules at t=0.

Solve prob’y eqn. with

Probabilities due to one S1 molecule are:

EXAMPLE 3: A COMBINATION OF THE F-O IRREVERSIBLE AND F-O REVERSIBLE REACTIONS

On the other hand, the probabilities due to one S2 molecule are:


The population distribution for the S1, S2 and S3 molecules can be obtained by substituting their probabilities in the right equation (see slide 18).


EXAMPLE 4: AN ENZYME-SUBSTRATE REACTION

Consider an enzyme reaction where a soluble substrate, with an AB structure (floats freely) reacts with immobilized enzyme molecules located on the surface of cells.

Formulation of the involved chemical reaction is given by:

where AB is the substrate molecule, R is the enzyme molecule, and ARB is the enzyme-substrate complex (intermediate product).


Several simplifications can be made. AR can be eliminated because the number of AR

molecules has the same probability distribution as the number of B molecules. This has no effect on the accuracy of the obtained

results. If the number of AB molecules is XAB, the product

of XAB and the reaction parameter c1(2) represents

the probability rate for an Rmolecule to tranfer into the ARB state. Therefore, c1

= c1(2) .

The reaction is now simplified into:



Mean populations of molecule species R (full line), ARB (dot-dashed line) and B (dashed line).


Zoomed plots for the mean populations of molecule species R (full line), ARB (dot-dashed line) and B (dashed line).

A property for many enzyme reactions is that the number of substrate molecules is sufficiently large. That is, the number of substrate molecules

consumed during the course of reaction is negligible in comparison to the total number of substrate molecules.

XAB is approx. equal to its initial value, and c1 can be seen as constant.

Thus, after simplification and approximations, same calculations as in Example 3 can be done.


Thank You.

REFERENCE

Zhang, X., De Cock, K., Bugallo, M.F., and Djuric, P.M. A General Method for the

Computation of Probabilities in Systems of First-Order Chemical Reactions. J. Chem. Phys. 122, 104101 (2005).

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