Chapter 11 Homework For any fixed 100-element background set S

Combinatorial Properties of Reaction Systems

September 4, 2008

1. Preliminaries

a. Define an (r-i)-reaction.

2. Probability That a Reaction Is Enabled

This section develops formulae for the probability that a random reaction is enabled for a

random state. In particular, we develop closed formulae for various forms of the

following definition:

Definition 1. Let r, i, n and m be integers with n ≥ r + i ≥ 2 and n ≥ m ≥ r.

Example 1. Let r = 3, i = 1, n = 100 and m = 50.

For any fixed 100-element background set S, there are 15,684,900 combinations of a

We next develop a closed formula for the special case of probenabled (3, 1, n, m). Later,

the formula is generalized for any r and i, and a limit version of the formula is shown for

any fraction s ∈ [0..1] to be:

2.1. Probability That a (3,1)-Reaction Is Enabled

Let a be some (3,1)-reaction over a background set S of n elements (with n ≥ 4). Recall

that our definition requires a’s one inhibitor is not also in its three-element reactant set

(since otherwise, it has no chance of ever being enabled).

We also require that the reactant set of

a

is a subset of

T

; given that the inhibitor is not in

T

, this probability (that

R

a

⊆

T

) is:

Multiplying all the terms together and simplifying gives:

Theorem 1.

Let n and m be integers with n ≥ 4 and n ≥ m ≥ 3. Then:

Theorem 2.

Let t ∈ [0..1] be a constant. Then:

2.2. Probability That an (r,i)-Reaction Is Enabled

This section generalizes the results of the previous section to the case of an (r,i)-reaction.

For this, we consider a to be an (r,i)-reaction over a background set of n elements with

n ≥ r + i. As always, the reactant set of a is disjoint from its inhibitor state.

Using factorials, this simplifies to:

Given that none of the inhibitors are in T, we can express the probability that all of the r

reactants are in T as:

Once again, this simplifies with factorials:

Multiplying the two parts of the probability together and canceling terms gives the first

result of this section:

Theorem 3.

Let n, m, r and i be natural numbers with n ≥ r+i and n ≥ m ≥ r. Then:

In the special case of (3,1)-reactions, this simplifies to Theorem 1 from the previous

section.

Theorem 4.

Let t ∈ [0..1] be a real number. Also let r and i be natural numbers.

Then:

3. The Size of a Result State

Throughout this section, let r, i, and n be integers, let t ∈ [0..1], and let b ∈ [0..∞). We

assume that n ≥ r + i ≥ 0. Also:

We will examine a particular case where b and t are related in a way that makes the

expected size of U close to that of T.

We want to determine the expected size of U. For this, consider any reactant u ∈ S.

What is the probability that u is not in U? If so, then it must not be the result of the first

enabled reaction in en

A

(T), which occurs with a probability of

Continuing this argument, the probability that u is not the result of any of the enabled

reactions is obtained by the multiplicative product:

For a large n, each of these factors approaches

n

n1

−

, so the probability that u is not the

Theorem 5.

Let r, i, and n be integers, let t ∈ [0..1], and let b ∈ [0..∞). We assume that

n ≥ r + i ≥ 0. Also:

•

Let S be a background set of n reactants.

•

Let B be a set of (r,i,1)-reactions over S; the number of reactions in B is

4. Simulations and Cycles

Consider a reaction system (S, B) and a state T ⊆ S. This section parameterizes these