# 程序代写代做代考 algorithm EE 5393 UMN

EE 5393 UMN

Circuits, Computation, and Biology Winter 2017

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Homework # 1

Due Thu. Feb 22, 2017

1. Analyzing Chemical Reaction Networks

The theory of reaction kinetics underpins our understanding of biological and

chemical systems. It is a simple and elegant formalism: chemical reactions

define rules according to which reactants form products; each rule fires at a

rate that is proportional to the quantities of the corresponding reactants that

are present. On the computational front, there has been a wealth of research

into efficient methods for simulating chemical reactions, ranging from ordinary

differential equations (ODEs) to stochastic simulation. On the mathematical

front, entirely new branches of theory have been developed to characterize the

dynamics of chemical reaction networks.

Most of this work is from the vantage point of analysis: a set of chemical

reaction exists, designed by nature and perhaps modified by human engineers;

the objective is to understand and characterize its behavior. Comparatively

little work has been done at a conceptual level in tackling the inverse problem

of synthesis: how can one design a set of chemical reactions that implement

specific behavior?

This homework will consider the computational power of chemical reactions

from both a deductive and an inductive point of view.

(a) A molecular system consists of a set of chemical reactions, each specifying

a rule for how types of molecules combine. For instance,

X1 + X2
k−→ X3,

specifies that one molecule of X1 combines with one molecule of X2 to

produce one molecule of X3. The rate at which the reaction fires is pro-

portional to (1) the concentrations of the participating molecular types;

and (2) a rate constant k. (This value is not constant at all; rather it

is dependent on factors such as temperature and volume; however, it is

independent of molecular quantities, and so called a “constant.”)

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Given a set of such reactions, we can model the behavior of the system in

two ways:

i. In a continuous sense, in terms of molecular concentrations, with

differential equations;

ii. In a discrete sense, in terms of molecular quantities, through proba-

bilistic discrete-event simulation.

Consider the reactions:

R1 : 2X1 + X2 → 4X3 k1 = 1
R2 : X1 + 2X3 → 3X2 k2 = 2
R3 : X2 + X3 → 2X1 k3 = 3

For a continuous model, let x1, x2 and x3 denote the concentrations of

X1, X2, and X3, respectively. (Recall that concentration is number of

molecules per unit volume.) The behavior of the system is described by

the following set of differential equations:

dx1
dt

= −x21x2 − 2x1x
2
3 + 6x2x3

dx2
dt

= −x21x2 + 6x1x
2
3 − 3x2x3

dx3
dt

= 4x21x2 − 2x1x
2
3 − 3x2x3

For the discrete model, let the state be S = [x1, x2, x3], where x1, x2 and

x3 denote the numbers of molecules of types X1, X2, and X3, respectively.

(Here we use actual integer quantities, not concentrations.) The firing

probabilities for R1, R2 and R3 are computed as follows:

p1(x1, x2, x3) =
1
2
x1(x1 − 1)x2

1
2
x1(x1 − 1)x2 + x1x3(x3 − 1) + 3x2x3

,

p2(x1, x2, x3) =
x1x3(x3 − 1)

1
2
x1(x1 − 1)x2 + x1x3(x3 − 1) + 3x2x3

,

p3(x1, x2, x3) =
3x2x3

1
2
x1(x1 − 1)x2 + x1x3(x3 − 1) + 3x2x3

.

Suppose that S = [3, 3, 3]. Then the firing probabilities for R1, R2 and R3

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are

p1(3, 3, 3) =
9

9 + 18 + 27
=

1

6
,

p2(3, 3, 3) =
18

9 + 18 + 27
=

1

3
,

p3(3, 3, 3) =
27

9 + 18 + 27
=

1

2
,

respectively.

N.B. In the continuous model, the rate of change of type is proportional

to xn where x is the concentration of a reaction and n is the coefficient. In

the discrete model, the probability is proportional to
(
x
n

)
. This is a subtle

difference. See the paper by Gillespie for an explanation.

Problem

Suppose that we define the following “outcomes”:

• C1: states S = [x1, x2, x3] with x1 > 7.
• C2: states S = [x1, x2, x3] with x2 ≥ 8.