JHU 580.429 SB3
HW14: Diffusion and spatial patterning.
1. Diffusion.
(a) The diffusion constant of water in water is D = 105 cm2 /sec. We have interpreted D
as (1/2)x2 /t, where x is diffusive step and t is the typical time between steps.
For water, a ty

SB3 Exam 1 Review Problems
1. Basic integrals
for Laplace transforms are easily evaluated using contour integrals.
H
Evaluate dzf (z) for a closed counter-clockwise contour around the origin, z = r to
re2i , for the following choices of f (z).
(a) 1/z 2
(

Name:
SB3 Quiz 4
Please circle your final answers. Useful constants are e 2.7, ln 10 2.3, ln 5 1.6,
ln 2 0.70, 3.1. Morphogen M turns on cell fate 1 if M K and cell fate 2 if M < K.
The morphogen is produced by cells located at the origin that maintain a

Name:
SB3 Quiz 5
1. A graph has V vertices. Edges are added at random between pairs of vertices. What is the
critical value of J, the average number of neighbors, when the giant component begins to
form?
Answer: J =
2. A graph has 100 vertices and 200 und

Name: ANSWER KEY
SB3 Exam 1
SB3 Exam 1
Rules for all exams
1. You may not use any electronic devices, including calculators, computers, phones,
iPods, MP3 players, or neural implants.
2. This exam is also closed-book, closed-notes.
3. You may not obtain a

SB3 Exam 1 Review Problems
1. Basic integrals for Laplace transforms are easily evaluated using contour integrals.
Evaluate f dz f (z) for a closed counter-clockwise contour around the origin, 2: = r to
r627, for the following choices of f (z).
(a) l/z2 O

Motivation
In class we saw
i(tt )
d e i+ .
LT and inverse LT need complex integration.
Calculation
Pole singularity. For instance f (x) = 1/(1 + x2 ) has poles at x = i, i.
Cauchys THM C: contour/path, f (z) has no pole on and inside C (analytic). C is

Overview
Drug
S(t)
Cascade
TF
Transcrip4on
mRNA
Transla4on
Protein
Coopera4ve binding:
Simultaneous
Sequen4al
Independent
Overview
Linear Models for Cells signaling
Stochas4c Dynamics
Networks
Line

SB III 2011: HW #1
September 12, 2011
2. Laplace Transformation Provided: LT [f (t)] =
R
0
f (t)est dt
a. [1 point] f (t) = 1,
Z
LT [f (t)]
est dt =
=
0
1
.
s
b. [1 point] f (t) = I[t a]. Naturally let a 0; otherwise the answer is identical to (a).
Z
LT

Diffusion
Constant degradation rate and constant diffusion with impulse source at t=0
!
!
!
!
M x, t = D ! M x, t M x, t
!
At steady state:
M x, t = D
!
t ignore
M x, t M x, t = 0
!
!
!
M x = ! ! M x
Solving us

Name:
SB3 Quiz 3
Please circle your final answers.
1. Protein X is a transcription factor that activates the expression of protein Y , which
is itself an auto-activating transcription factor. The dynamics for Y are
Y (t) = X (X > K) + Y (Y > K) Y,
with X

Name: ANSWER KEY
SB3 Quiz 4
Please circle your final answers. Useful constants are e 2.7, ln 10 2.3, ln 5 1.6,
ln 2 0.70, 3.1. Morphogen M turns on cell fate 1 if M K and cell fate 2 if M < K.
The morphogen is produced by cells located at the origin that

JHU 580.429 SB3
HW2
The L operator is the
Laplace transform, L [ f (t)] = f(s) =
Rt 0
volution, f ? g(t) = 0 dt f (t t 0 )g(t 0 ).
R
0
dtest f (t). The ? operator is con-
1. Standard Laplace transform proofs.
(a) Prove that L [ f ? g(t)] = f(s)g(s).
(b) P

JHU 580.429 SB3
HW4
The Laplace transform operator is RL , defined as L [ f (t)] = f(s) = 0 dtest f (t). The convolution
operator is ?, defined as f ? g(t) = 0t dt 0 f (t t 0 )g(t 0 ). The real part of a complex variable z = x + iy
is denoted (z) = x.
R
1

JHU 580.429 SB3
HW12: Stochastic dynamics
1. Consider our standard model for the dynamics of a single mRNA species with copy number
n, with transitions
n n+1
n n1
and corresponding rates and n. Use parameter values = 1/min and = 0.1/min.
(a) What is avera

JHU 580.429 SB3
HW11: Combinatorial transcription factor binding
1. A gene exists in an unbound state, G, and also binds to two activating transcription factors,
X and Y . ODE models for binding are
G + nX
GXn
G + nY
GYn
with forward and reverse rate co

JHU 580.429 SB3
HW15: Networks.
1. Networks and degree distributions. Consider a network with N total vertices and E total
undirected, unweighted edges.
(a) What is the probability f that an edge connects two vertices connected at random?
(b) What is the

JHU 580.429 SB3
HW7: DNA information content
1. Assume that the length of the human genome is 3 109 base pairs, and that each of the 4
base pairs occurs with probability 1/4.
(a) How long in base pairs does a motif have to be to occur approximately once p

JHU 580.429 SB3
HW3
1. Evaluate dz 1z for the following closed contours expressed in polar coordinates, z = rei
with r > 0.
H
(a) A single counter-clockwise loop, = 0 to 2.
2i
(b) A single clockwise loop, = 2 to 0.
2i
(c) A double counter-clockwise loop,

JHU 580.429 SB3
HW8: Transcription factor binding
The following questions consider binding of a transcription factor protein, denoted X, to a gene
promoter with N binding sites. The number of transcription factors bound is denoted n, and in the
general ca

JHU 580.429 SB3HW13: Stochastic dynamics, fluctuation-dissipation, stability.
1. Stochastic protein dynamics. Consider a stochastic system in which particular protein in a
cell has copy number n. Starting in state n, the transition rate for n n + 1 is and

JHU 580.429 SB3
HW1
In this homework, L stands for the Laplace transform, f(s) = L [ f (t)] =
R
0
dtest f (t).
1. Simple Laplace transforms. Provide f(s) for the following f (t).
(a) f (t) = 1
(b) f (t) = 0 if t < a, f (t) = 1 if t a, and a 0.
(c) f (t) =

Laplace Transforms
!
f t = ! f t e!" dt = F(s)
! !
!
!
!
!" f t = ! !" f t e!" dt !" f t e!" = !" f t e!" sf t e!"
!
! !
!
!
! !" f t e!" + sf t e!" dt = ! !" f t e!" dt + s ! f t e!" dt
= 0 f 0 + sF s = sF s f(0)
!
!
f t

JHU 580.429 SB3
Study Questions for Part 3
1. Stochastic protein dynamics. Consider a stochastic system in which particular protein in a cell has copy number n. Starting in state n, the transition rate for n n + 1
is and the transition rate for n n 1 is n

JHU 580.429 SB3
HW6: Serial, Parallel, and Feedback Connections
We consider what happens when linear signal transduction cascades are connected in series and in
parallel, including positive feedback and negative feedback. In general, the signaling is init

JHU 580.429 SB3
HW4
The Laplace transform operator is RL , defined as L [ f (t)] = f(s) = 0 dtest f (t). The convolution
operator is ?, defined as f ? g(t) = 0t dt 0 f (t t 0 )g(t 0 ). The real part of a complex variable z = x + iy
is denoted (z) = x.
R
1

JHU 580.429 SB3
HW5
The Laplace transform operator is RL , defined as L [ f (t)] = f(s) = 0 dtest f (t). The convolution
operator is ?, defined as f ? g(t) = 0t dt 0 f (t t 0 )g(t 0 ). The real part of a complex variable z = x + iy
is denoted (z) = x.
R
1

JHU 580.429 SB3
HW7: Transcription factor binding
The following questions consider binding of a transcription factor protein, denoted X, to a gene promoter with N binding sites. The number of transcription factors bound is denoted n cfw_0, 1, 2, 3, . . .

JHU 580.429 SB3
1
HW15 (Review Questions)
Combinatorial regulation
1. Suppose a gene is combinatorially regulated by two transcription factors, X and Y , which
share the same binding site:
G + nX
GXn
(1)
G + mY
GYm .
(2)
For X, the forward and reverse r

JHU 580.429 SB3
1
HW15 (Review Questions)
Combinatorial regulation
1. Suppose a gene is combinatorially regulated by two transcription factors, X and Y , which
share the same binding site:
G + nX
GXn
(1)
G + mY
GYm .
(2)
For X, the forward and reverse r

JHU 580.429 SB3
Quiz 1
Name:
The Laplace transform operator is L , with L [ f (t)] = f(s) =
1. Provide f(s) for f (t) = 1.
Answer:
0
dtest f (t).
2. Provide f(s) for f (t) = t.
Answer:
3. Provide f(s) for f (t) = eat .
Answer:
4. Suppose that
and g(t) =

550.430, Fall 2016: Homework 1
Outline of Solutions
This homework consists almost entirely of problems that are review material from 550.420, Introduction to
Probability, although there is one simple R exercise as well. The homework is due at the beginnin