成功案例设置

Department of Bioengineering
BIOE97033 Modelling in Biology
(duration: 120 minutes)
The paper has 2 COMPULSORY questions.
Marks are shown next to each question.
The marks for questions (and parts thereof) are
indicative and may be slightly moderated at the
discretion of the Examiner.
PLEASE NOTE:
WE EXPECT HAND-WORKED SOLUTIONS TO ALL
QUESTIONS.
Question 1 Nonlinear dynamical systems.
a) A biochemical model. /30
The production and decay of a biochemical compound is represented by the
following Ordinary Differential Equation model:
x˙ = x3 − ax2 + x (1)
where we have used the notation x˙ = dx
dt
.
By definition, the variable x is non-negative, while the parameter a is
assumed to be strictly positive.
i) Find the analytical expression of the fixed points of the model in Eq. (1).
/8
ii) Considering a > 2, use a graphical approach (i.e. draw a graph of x˙ vs
x) to establish the position and stability of the fixed points of the model
in Eq. (1). Indicate the direction of motion (flow) on the x-axis of your
graph and consequently label each fixed point as stable or unstable. /10
iii) Considering a > 0 as a bifurcation parameter, draw the bifurcation
diagram for the system in Eq. (1). Indicate on your bifurcation diagram
the directions of motion. State and briefly justify the name of this
b) A model of civilisation dynamics. /70
The following set of Ordinary Differential Equations has been proposed to
capture the time evolution of civilisations:
x˙1 = x1(−a+ x2) (2)
x˙2 = x2
(
−x1 − x2
k
+ 1
)
(3)
where we have used the following notation: x˙1 = dx1dt and x˙2 =
dx2
dt
.
x2 represents the amount of resources available to the civilisation population
and x1 represents the members of the civilisation population consuming
these resources.
By definition, the variables x1 and x2 are non-negative, while the values of
the parameters a and k are such that k > a > 0.
i) State the type (linear or nonlinear) and order of the civilisation model in
(2)-(3). /6
ii) Find the analytical expression of the fixed points of the model in (2)-(3).
/15
iii) Perform a local stability analysis of the system in (2)-(3) around each
fixed point. For this:
* Provide the analytical expression of the Jacobian matrix of the
system in (2)-(3). /10
* Evaluate the Jacobian matrix at each fixed point, calculate the
corresponding eigenvalues and deduce the type and stability of
the associated fixed point. /18
iv) Confirm the fixed points values you obtained analytically above by
drawing the nullclines of the system in (2)-(3) in a x2 vs x1 phase plane
diagram. Using the nullclines, indicate the position and value of the
fixed points. /21
Question 2 Stochastic processes and networks
A protein is produced at time t = 0 at a certain point in a cell. The protein is
initially inactive and must undergo a maturation process before it is active.
Active proteins are then degraded, but inactive proteins are not.
a) Modelling the maturation and degradation of a single protein /50
We model the state of the protein as a discrete-state, continuous-time
Markov process Y (t) with three states: inactive (state 1), active (state 2) and
degraded (state 3). Maturation to the active state occurs with a transition rate
of km and degradation from the active state with a transition rate of kd.
i) Give a graphical representation of this stochastic process. /6
ii) Give the rate matrix K for this stochastic process. /6
iii) Write down and solve the differential equation describing the time
evolution of pinactive(t) = p1(t), given that the protein is inactive at t = 0.
/8
iv) Write down the differential equation describing the time evolution of
pactive(t) = p2(t). Hence verify that
p2(t) =
km
km − kd (exp(−kdt)− exp(−kmt)) . (4)
/15
v) Sketch p2(t) as shown in Eq. (4), labelling the times of any maxima or
minima. Assume km > kd. /15
b) Simultaneous diffusion and state dynamics /50
The cell can be modelled as one-dimensional. The protein is produced at
position x = 0, and diffuses with diffusion coefficient D prior to degradation.
Activation and degradation occur independently of diffusion.
i) q(x, t|y) is the conditional probability density to find the protein at
position x at time t, given that it is in the state y. The letter q is used
here to avoid confusion with the letter p that describes the activation
state. Since diffusion is independent of maturation and degradation,
q(x, t|2) = q(x, t|1) and q(x, t|2) obeys a Fokker-Planck equation:
∂q(x, t|2)
∂t
= − ∂
∂x
(A(x)q(x, t|2)) + 1
2
∂2
∂x2
(B(x)q(x, t|2)) . (5)
Describe the physical effect of the two terms on the right-hand side, and
thus explain qualitatively why A(x) = 0 and B(x) = 2D in this case. /8
ii) Verify that
q(x, t|2) =
√
1
4piDt
exp(−x2/4Dt) (6)
is the correct solution to the Fokker-Planck equation for this system.
When checking normalisation, you may assume that∫ +∞
−∞
exp(−αz2)dz =
√
pi
α
. (7)
/15
iii) Let r(x, t) be the probability density that the protein undergoes
degradation at time t and position x. Explain why
r(x, t) = kdp2(t)q(x, t|2), (8)
where p2(t) and q(x, t|2) are given by Eq. (4) and Eq. (6), respectively.
/8
iv) Without detailed calculation, give the mean position at which the protein
undergoes degradation. /4
v) Calculate the variance of the position at which the protein undergoes
degradation. You may find the following results helpful:∫ +∞
−∞ z
2 exp(−αz2)dz = 1

留学ICU™️ 留学生辅助指导品牌

在线客服 7*24 全天为您提供咨询服务

咨询电话(全球): +86 17530857517

客服QQ：2405269519

微信咨询：zz-x2580

服务项目

服务流程

客户保障

关于我们

微信订阅号