PhD oral defense_Yang

Epidemic Disease
Phyllotactic Patterns
Gene Regulatory Networks
Metabolic Networks
Optimization Models and Computational Methods
for Systems Biology
PhD candidate: Cong Yang
Supervisors: Dr. Ching Wai-Ki, Dr. Tsing Nam-Kiu
Jan 17th 2012
PhD candidate: Cong Yang Optimization Models and Computational Methods for Systems Bi

Epidemic Disease
Metabolic Networks
Systems biology
Systems biology treats a biological system as a network, and
focuses on all its components and their interactions.
Research topics
1 Dynamics and control of the spread of HIV in a system of
prisons
2 Dynamics and robustness of phyllotactic patterns
3 Long-run behavior and control of gene regulatory networks
4 Robustness of metabolic networks

Epidemic Disease
Metabolic Networks
Dynamics of HIV spread in a system of prisons
Dynamics of HIV spread under control policy
Optimal control policy
Section I: Epidemic Disease
1 Background and main reference
2 Dynamics of HIV spreading in a system of prisons
general model of HIV spreading
Newton’s method and equilibrium point
3 Dynamics of HIV spreading in a system of prisons under
screening and quarantine policy
general model of HIV spreading under screening and quarantine
existence and stability of equilibrium point
numerical examples
4 Optimal screening and quarantine policy

Epidemic Disease
Metabolic Networks
Background and main reference
The spread of HIV by both sexual contacts and needle sharing
is a serious problem in prisons .
[39] J. Gani, S. Yakowitz and M. Blount, The Spread and quarantine of HIV
infection in a prison system, SIAM J Appl Math., 57(6)(1997):1510-1530.
1 deterministic model and stochastic analogue for a single prison
and a two-prison system
focus on the change of HIV infectives along with time,
without considering the existence of equilibriums
2 screening and quarantine in a prison and cost-eﬀectiveness
analysis
quarantine all the HIV infectives by screening

Epidemic Disease
Metabolic Networks
For a single prison without control manner
N the number of prisoners
y(t) the number of infectives
β the infection rate
µ the mean proportion of infectives in the outside world
n the number of prisoners exchanged with the outside world
dy
dt
= βy(t) N − y(t)
new infectives
−
n
N
y(t)
outﬂow
+ nµ
inﬂow
[39] J. Gani, S. Yakowitz and M. Blount, The Spread and quarantine of HIV
infection in a prison system, SIAM J Appl Math., 57(6)(1997):1510-1530.

Epidemic Disease
Metabolic Networks
For a system of s prisons without control manner
Ni , the number of prisoners in Prison i
yi (t), the number of infectives in Prisont
βi , the infection rate in Prison i
µ, the proportion of infectives in the outside world
ni , the number of prisoners exchanged between the outside
world and Prison i
m, the number of prisoners exchanged between any two
prisons
dyi (t)
dt
= βi yi (t)(Ni − yi (t))
new infectives
−
m(s − 1)yi (t)
Ni
−
ni yi (t)
Ni
outﬂow
+
s
k=1,k=i
yk (t)m
Nk
+ ni µ
inﬂow

Epidemic Disease
Metabolic Networks
Equilibrium point
Let (y∗
1 , y∗
2 , . . . , y∗
s ) be the non-negative equilibrium point, if
N1 = N2 = . . . = Ns and βi <
ni
N2
i
then the equilibrium point is asymptotically stable.
Newton’s method
A suﬃcient condition for Newton’s method to be convergent with
the initial guess a = 1
2(N1, N2, . . . , Ns)t is
s
i=1
β2
i
s
i=1
ni (µ −
1
2
) +
βi N2
i
4
2 2 s
i=1
Ni
ni
4
≤
1
16
.

Epidemic Disease
Metabolic Networks
For a single prison with control manner
y(t) the number of infected but not detected
m(t) the number of detected but not quarantined
q(t) the number of quarantined
x(t) = N − y(t) − m(t) − q(t) the number of suspected
n the number of prisoners exchanging with the outside world
τ the proportion of prisoners to be screened (0 ≤ τ < 1)
κ the proportion of detected to be quarantined (0 ≤ κ ≤ 1)

Epidemic Disease
Metabolic Networks
Dynamics of HIV spreading in a single prison under screening and
quarantine



y(t + 1) = (1 − τ)(1 − n
N
) (y(t) + βx(t)(y(t) + m(t))) + (1 − τ)nµ
m(t + 1) = (1 − κ)(1 − n
N
)m(t) + (1 − κ)τnµ
+(1 − κ)τ(1 − n
N
)(y(t) + βx(t)(y(t) + m(t)))
q(t + 1) = (1 − n
N
)q(t) + κ(1 − n
N
)m(t)
+κτ(1 − n
N
)(y(t) + βx(t)(y(t) + m(t))) + τκµn
x(t + 1) = N − y(t + 1) − m(t + 1) − q(t + 1).
Existence and stability of equilibrium point
Given that µ < N
2N+n ,
The equilibrium point exists.
The equilibrium point is asymptotically stable.
µ is the mean proportion of infectives in the outside world

Epidemic Disease
Metabolic Networks
Eﬃciency of the control strategy
N = 500, β = 0.0005, n = 50, y(0) = 50, µ = 0.01, τ = 0.1, κ = 0.1.

Epidemic Disease
Metabolic Networks
For a system of s prisons with control manner
Ni , the number of prisoners in Prison i
yi (t), the number of infected but not detected in Prison i
mi (t), the number of detected but not quarantined in Prison i
qi (t), the number of quarantined in Prison i
xi (t) = Ni − yi (t) − mi (t) − qi (t), the number of suscepted in
Prison i
βi , the infection rate in Prison i
µ, the mean proportion of infectives in the outside world
ni , the number of prisoners in Prison i exchanging with the
outside world
h, the number of prisoners exchange between any two prisons

Epidemic Disease
Metabolic Networks
Dynamics of HIV spreading in a system of s prisons under
screening and quarantine



yi (t + 1) = (1 − τ)(1 −
ni +(s−1)h
Ni
)(yi (t) + βi xi (t)(yi (t) + mi (t)))
+(1 − τ)
s
j=1,j=i
h
Nj
(yj (t) + βj xj (t)(yj (t) + mj (t))) + (1 − τ)ni µ
mi (t + 1) = (1 − κ)(1 −
ni +(s−1)h
Ni
)mi (t) + (1 − κ)
s
j=1,j=i
h
Nj
mj (t)
+(1 − κ)τ(1 −
ni +(s−1)h
Ni
)(yi (t) + βi xi (t)(yi (t) + mi (t)))
+(1 − κ)τ s
j=1,j=i
h
Nj
(yj (t) + βj xj (t)(yj (t) + mj (t))) + (1 − κ)τni µ
qi (t + 1) = (1 −
ni +(s−1)h
Ni
)qi (t) +
s
j=1,j=i
h
Nj
qj (t) + κ(1 −
ni +(s−1)h
Ni
)mi (t)
+κ
s
j=1,j=i
h
Nj
mj (t)) + κτ(1 −
ni +(s−1)h
Ni
)(yi (t) + βi xi (t)(yi (t) + mi (t)))
+κτ
s
j=1,j=i
h
Nj
(yj (t) + βj xj (t)(yj (t) + mj (t))) + κτni µ
i = 1, 2, . . . , s

Epidemic Disease
Metabolic Networks
Existence and stability of equilibrium point
1 Without exchange of prisoners h = 0
If µ < Ni
2Ni +ni
, i = 1, 2, . . . , s holds, then
Equilibrium solution exists.
The equilibrium point is asymptotically stable.
2 With exchange of prisoners h = 0
Given that µ < Ni
2Ni +ni
, if h Ni for i = 1, 2, . . . , s holds, then
the equilibrium point is asymptotically stable.
µ is the mean proportion of infectives in the outside world

Epidemic Disease
Metabolic Networks
A two-prison system
Assumption: Screening cost Csi (τ, Ni ) = aτNi and
quarantining cost Cqi (κ, Ni ) = bκNi only depend on Ni .
Optimization Problem



min C(τ, κ, N) = (aτ + bκ)(N1 + N2)
s.t. qi ≤ Qi , i = 1, 2
2
i=1
yi + mi + qi ≤ I
0 ≤ τ, κ ≤ 1.
grid search (τ, κ) ∈ {(0.01i, 0.01j) : i, j = 0, 1, . . . , 100}.

Epidemic Disease
Metabolic Networks
Generation of a new primordium
Interaction among primordia
Measurement of pattern uniformity
Robustness
Section II: Phyllotactic Patterns
1 Background
2 Generation of a new primordium
Hofmeister’s hypothesis (MaxMin principle)
Atela’s Max-Min model
a simpliﬁed Max-Min model
3 Interaction among primordia during growing process
Ridley’s contact pressure model
repulsion pressure model
4 Measurement of pattern uniformity
5 Robustness

Epidemic Disease
Metabolic Networks
Robustness
Phyllotactic patterns
(a) sunﬂower pattern (b) microscope image (c) simulated pattern
1 Where the new primordium appears?
2 How the primordia inﬂuence each other during the growing
process?

Epidemic Disease
Metabolic Networks
Robustness
Mathematical interpretation
The apex is represented by a circle with radius R0.
Primordia are generated at the periphery of the apex with
periodicity T.
ti the birth time of the i-th primordium
θi the angle between the straight line joining the origin and
the i-th primordium and the positive x axis
The distance between the apex center and the primordium at time t
after birth is R0 + vts
, where v is the growing velocity and s = 0.5.
At the birth of the k-th primordium tk (k > i) (R0 cos θ, R0 sin θ)
The Euclidean distance between the i-th and k-th primordia is
˜di = (R0 + v
√
tk − ti ) cos θi − R0 cos θ
2
+ (R0 + v
√
tk − ti ) sin θi − R0 sin θ
2
.

Epidemic Disease
Metabolic Networks
Robustness
Hofmeister’s hypothesis (MaxMin principle)
The new primordium should be born at the position with the
largest minimum distance with all other pre-existing primordia.
[49] W.F.B. Hofmeister, Allgemeine Morphologie Der Gew˝achse, Kessinger Publishing, 1868.
Mathematical interpretation
Atela’s Max-Min model θk = max
0≤θ<2π
min
i∈{1,2,...,k−1}
˜di
[7] P. Atela, C. Gol´e and S. Hotton, A dynamical system for plant pattern formation: a rigorous
analysis, J Nonlinear Sci., 12(6)(2002): 641-676.
Simpliﬁed Max-Min model θk = max
0≤θ<2π
min
i=k−1,k−2
˜di
cos(θk − θk−1) =
(1+τ
√
T)− (1+τ
√
T)2+4(1+τ
√
2T)(2+τ(4
√
2T−2
√
T)+τ2T)
4(1+τ
√
2T)
, τ = v/R0.

Epidemic Disease
Metabolic Networks
Robustness
Ridley’s contact pressure model
Contact pressure exists when two primordia are close enough.
[93] J.N. Ridley, Computer simulation of contact pressure in capitula, J
Theor Biol., 95(1)(1982):1-11.
repulsion pressure model
Repulsion pressure
a
d2
ij
exists between any two primordia.
a is the repulsion weight
dij is the Euclidean distance
The position of the i-th primordium is inﬂuenced by
all other primordia (repulsion by all others)
the nearest left and nearest right neighbors (repulsion by two)

Epidemic Disease
Metabolic Networks
Robustness
Comparison among mechanisms

Epidemic Disease
Metabolic Networks
Robustness
Measure of pattern uniformity
1 map the packing pattern into a disk with radius 1
˜Ri = Ri /R1, i = 1, 2, . . . , n
2 uniformity value is deﬁned as
rmax
dmin
rmax the radius of the largest circle that can be placed into
the pattern without covering any primordium
dmin the minimum distances between any two primordia
dmin = mini=j d(i, j)

Epidemic Disease
Metabolic Networks
Robustness
Repulsion by all others

Epidemic Disease
Metabolic Networks
Robustness
Repulsion by nearest left and right neighbors

Epidemic Disease
Metabolic Networks
Robustness
Robustness

Epidemic Disease
Metabolic Networks
Distribution and enumeration of attractors in PBN
Finite-horizon control with multiple hard-constraints
Section III: Gene Regulatory Networks
1 Distribution and enumeration of attractors in PBN
Background of the problem
Estimation of the number of singleton attractors
Algorithms for finding attractors
Finding singleton attractors
Finding cyclic attractors with a fixed period
Computational experiments
2 Finite-horizon control with multiple hard-constraints
Problem formation
Algorithms for finding optimal control strategies
Reserving Place Algorithm
Genetic Algorithm

Epidemic Disease
Metabolic Networks
Consider a PBN with n genes v1, v2, . . . , vn.
For gene vi , there are l(i) Boolean functions f
(i)
ji
(1 ≤ ji ≤ l(i))
to be randomly assigned.
There are N =
n
i=1
l(i) possible realization fj = (f
(1)
j1
, f
(2)
j2
, . . . , f
(n)
jn
)
for the PBN.
Attrators
In a PBN, a set of Gene Activity Proﬁles (GAPs)
{v1, v2, . . . , vp} is called an attractor of period p, if
Prob(v(t + 1) = vi+1 | v(t) = vi ) = 0
holds for all i = 1, . . . , p.

Epidemic Disease
Metabolic Networks
If the numbers of possible Boolean functions for gene vi are
identical (l(i) = L, i = 1, . . . , n), then the number of singleton
attractors of the PBN is 2 − 1
2
L−1 n
.
Initialize m = 1
Procedure PBNAttractor( v(t), m)
if m = n + 1 then output v1(t), v2(t), . . . , vn(t) return
for b = 0 to 1 do vm(t) := b
if vi (t) = f
(i)
j (v(t))(= vi (t + 1)) holds for any j for some
i ≤ m then continue
else PBNAttractor(v(t), m + 1)
return

Epidemic Disease
Metabolic Networks
Assumptions and notations
The numbers of possible Boolean functions for gene vi are identical, i.e.
l(i) = L holds for i = 1, 2, . . . , n.
For gene vi , the probabilities for choosing any Boolean function f
(i)
ji
are
identical c
(i)
1 = c
(i)
2 = · · · = c
(i)
l(i).
The maximum indegree of gene vi is bounded by constant K.
Computational complexity
For gene vi , IN(f
(i)
j ) denotes the set of input genes for corresponding Boolean
function f
(i)
j .
General Case: If IN(f
(i)
j1
) = IN(f
(i)
j2
) (1 ≤ j1, j2 ≤ l(i)) stands for some
gene vi , then the number of recursive calls executed for the ﬁrst m genes
is bounded by {(2 − 0.5L−1
sKL
)s
}n
, where s = m/n.
Special Case: If IN(f
(i)
j1
) = IN(f
(i)
j2
) (1 ≤ j1, j2 ≤ l(i)) holds for any gene
vi , then the number of recursive calls executed for the ﬁrst m genes is
bounded by {(2 − 0.5L−1
sK
)s
}n
, where s = m/n.

Epidemic Disease
Metabolic Networks
Finding attractors with period p
p-ancestor(vi , j1, . . . , jp): a set of nodes along paths reaching vi with length p when
the realization of the PBN at time t + q (1 ≤ q ≤ p) is given by fjq .

Epidemic Disease
Metabolic Networks
1 The numbers of possible Boolean functions for gene vi are identical, i.e.
l(i) = L holds for i = 1, 2, . . . , n.
2 For gene vi , the probabilities for choosing any Boolean function f
(i)
ji
are identical
c
(i)
1 = c
(i)
2 = · · · = c
(i)
l(i)
.
3 The maximum indegree of gene vi is bounded by constant K.
Computational complexity
For gene vi , IN(f
(i)
j ) denotes the set of input genes for corresponding Boolean
function f
(i)
j .
General Case: If IN(f
(i)
j1
) = IN(f
(i)
j2
) (1 ≤ j1, j2 ≤ l(i)) stands for some gene vi ,
then the number of recursive calls executed for the ﬁrst m genes is bounded by
2 − 0.5L
1−Kp
1−K −1s
L
1−Kp
1−K ×
K(1−Kp)
1−K
s n
, where s = m/n.
Special Case: If IN(f
(i)
j1
) = IN(f
(i)
j2
) (1 ≤ j1, j2 ≤ l(i)) holds for any gene vi ,
then the number of recursive calls executed for the ﬁrst m genes is bounded by
2 − 0.5L−1s
K(1−Kp)
1−K
s n
, where s = m/n.

Epidemic Disease
Metabolic Networks
Number of singleton attractors
The number of singleton attractors of the PBN is
2 − 1
2
L−1 n
.
based on 100 randomly generated PBN based on 1000 randomly generated PBN for p = 1, L = 2

Epidemic Disease
Metabolic Networks
For WNT5A network
For WNT5A network, 10,000 PBNs with 2 Boolean functions and 3 inputs
for each node are generated based on the structure of WNT5A network.
For comparison supers, we randomly generated 10,000 PBNs with 10
nodes and 2 Boolean functions with 3 inputs for each node.
In WNT5A network, IN(f
(i)
j1
) = IN(f
(i)
j2
) stands.

Epidemic Disease
Metabolic Networks
Goal manipulate external control approaches to alter the dynamics of
the genetic regulatory network with minimum cost
Constraints the number of times each control manner can be applied
of the network
[20] W.K. Ching, S.Q. Zhang, Y. Jiao, T. Akutsu, N.K. Tsing and A.S. Wong, Optimal control policy for
Probabilistic Boolean Networks with hard constraints, IET Syst Biol., 3(2)(2009):90-99.
Notations
The initial probability distribution is x0 = (v1(0), . . . , vn(0))t
.
At time j, Control ij ∈ {0, 1, 2} is applied to the system.
The control sequence from time 1 to T is σ = i1 i2 . . . iT .
Vector xT = (v1(T), . . . , vn(T))t
= PiT . . . Pi1 x0 is the state distribution
at time T by control strategy σ.
Set S contains all the possible states of the PBN.
Target states S ⊆ S

Epidemic Disease
Metabolic Networks
Determine a sequence of actions σ over a ﬁnite time horizon T
leads the system fall into a set of target states S with probability
higher than ¯p, i.e. i∈S [xT ]i ≥ ¯p,
minimizes the total control cost
T
i=1 C(σi ).
Problem formation



min
σ
T
i=1 C(σi )
s.t. i∈S [xT ]i ≥ ¯p,
0 ≤ s1 ≤ K1,
0 ≤ s2 ≤ K2.
Here si is the number of times that Control i is conducted, and Ki is the
maximum number of times that Control i can be applied, i = 1, 2.

Epidemic Disease
Metabolic Networks
1 generate set U = {σ = i1 i2 . . . iT , ij ∈ {0, 1, 2}, and 0 ≤ si ≤ Ki , i = 1, 2}
1 ﬁx 0 ≤ k ≤ K2 place for Control 2 (represented by 2) in the
control string of length T
2 place Control 0 and Control 1 in binary string of length T − k
string 11 . . . 1
K1
00 . . . 0
T−K1−k
is the largest one after decimalization
translate decimal digits from 0 to 2T−k
− 1 to binary digits of
length T − k while checking the number of times that Control
1 is applied
3 increasing k from 0 to K2
2 ﬁnd the feasible solution set V = {σ ∈ U : i∈S [xT ]i ≥ ¯p} ⊆ U
Computational cost
The computation cost of the Reserving Place Algorithm is bounded
above by MT22n
, where M =
K2
j=0
K1
i=0
T!
i!j!(T − i − j)!
.

Epidemic Disease
Metabolic Networks
Computational experiment



min
σ
10
i=1 C(σi )
s.t. [x]3 + [x]4 ≥ 0.8,
0 ≤ s1 ≤ 5,
0 ≤ s2 ≤ 2.
x0 = (0.1, 0.4, 0.3, 0.2)t
. The costs for conducting Control 1,
Control 2 and Control 3 are 2.5, 4 and 0 respectively.

Epidemic Disease
Metabolic Networks
Comparison among two algorithms

Epidemic Disease
Metabolic Networks
Impact degrees in metabolic network with cycles
Approximation for impact degree distribution
Section IV: Metabolic Networks
1 Impact degrees in metabolic network with cycles
Deﬁnition of impact degree
Algorithm design
Impact of single deletion
Impact of multiple deletion
2 Approximation for impact degree distribution
Branching Process Models
The Poisson model
Power-law model
Parameter extraction
Evaluation of approximation

Epidemic Disease
Metabolic Networks
Background
A metabolic network is composed of its compounds, linking
each other by reactions.
One can disable (delete) a reaction by disrupting a gene
corresponding to the enzyme which catalyzes the reaction.
Impact degree
Assume all the reactions and compounds are activated, the
impact degree of reaction Ri is the number of reactions
(including Ri ) that is inactivated due to the deletion of Ri .
[62] D. Jiang, S. Zhou and Y. Chen, Compensatory ability to null mutation in
metabolic networks, Biotechnol Bioeng., 103(2)(2009):361-369.

Epidemic Disease
Metabolic Networks
Reactions and compounds
For each reaction, there are consumed, produced and directly unrelated
compounds.
Reaction should be inactivated if any consumed compound or
produced compound is inactivated.
For each compound, there are consuming, producing and directly
unrelated reaction.
Compound should be inactivated if all its consuming reactions
or all its producing reactions are inactivated.

Epidemic Disease
Metabolic Networks
Simple Algorithm

Epidemic Disease
Metabolic Networks
Consider the impact of deleting reaction Ri only
related reactions all reactions inactivated
inactivated compounds all compounds inactivated
related compounds all substrates and products of all related reactions
remained compounds the related compounds but not inactivated
Deletion of reaction pair (Rh, Rg )
Overlapped compounds compounds that are remained compounds for both reaction
Rg and reaction Rh.

Epidemic Disease
Metabolic Networks
Improved algorithm

Epidemic Disease
Metabolic Networks
Medium-scale E. coli network from KEGG
253 reactions (150 are reversible) and 261 compounds
average impact degree of single deletion: 1.9331.
average impact degree of double deletion: 3.8461.
99.73% simpliﬁed case
Simple Algorithm: 3427.7 sec, Improved Algorithm: 88.9 sec

Epidemic Disease
Metabolic Networks
Whole E. coli network
single deletion double deletion triple deletion

Epidemic Disease
Metabolic Networks
Background
Estimation of impact degree distribution of single reaction
deletion.
The impact spread out in a cascade manner, it spreads
through downstream reactions whose substrates are
synthesized via unique metabolic reactions.
Branching process: each progenitor generates oﬀsprings
independently but according to the same probability
distribution.

Epidemic Disease
Metabolic Networks
metabolic network corresponding reaction network
Oﬀsprings
Assume the network is in tree structure.
The number of oﬀsprings of reaction Ri
di =
kout
i (if kin
i = 1)
0 (otherwise)
,
where kout
i and kin
i are the outdegree and indegree of reaction
Ri respectively.

Epidemic Disease
Metabolic Networks
The Poisson Model
1 The number of offsprings d follows Poisson distribution,
P(d) = µd
e−µ
/d!,
where µ is the mean of the offspring numbers.
2 The total number of offsprings (i.e., impact degree) r follows
Borel distribution:
P(r) = (µr)r−1 e−µr
r!
.
By Stirling formula, P(r) ∝ r−3/2
e−r(ln µ−µ+1)
.

Epidemic Disease
Metabolic Networks
Power-law Model
1 The number of offsprings d follows Power-law distribution,
P(d) = C /dγ+1
,
where C = 1/ζ(γ + 1).
2 If 1 < γ < 2 holds, the total number of offsprings (i.e., impact
degree) r follows
P(r) =
µ
νr1+1/γ
ϕ
(1 − µ)r − µ − 1
νr1/γ
,
where ϕ(x) =
∞
0 exp uγ cos πγ
2 cos uγ sin πγ
2 + ux du,
µ is the mean of the number of offsprings, ν = µ[γΓ(−γ)]1/γ,
here Γ(x) is the Gamma function.
If µ = 1, P(r) ∝ 1/r1+1/γ
.
If γ > 2, P(r) is similar to that in Poisson model.

Epidemic Disease
Metabolic Networks
Parameter extraction
Poisson Model The mean of the number of oﬀsprings for each reaction node
µ can be estimated by
µ =
1
N
N
i=1
di ,
where N is the total number of reaction nodes.
Power-law Model The exponent γ can be estimated by
γ = |N
∗
|


i∈N∗
ln
di
dmin


−1
,
where N∗ is the set of reaction nodes with di > 0, and |N∗| is the total number
of such reaction nodes. dmin is the minimum of di in the set of N∗.

Epidemic Disease
Metabolic Networks
Oﬀspring distribution
(A) Escherichia coli, (B) Bacillus subtilis, (C) Saccharomyces cerevisiae and (D) Homo sapiens

Epidemic Disease
Metabolic Networks
Impact degree distribution
(A) Escherichia coli, (B) Bacillus subtilis, (C) Saccharomyces cerevisiae and (D) Homo sapiens

Epidemic Disease
Metabolic Networks
Acknowledgment
1 Department of Mathematics, The University of Hong Kong
Dr. Tuen-Wai Ng
Ho-Yin Leung
Dr. Allen H. Tai
2 Bioinformatics Center, Kyoto University
Dr. Tatsuya Akutsu
Dr. Takeyuki Tamura
Dr. Morihiro Hayashida
3 Japan Science and Technology Agency
Dr. Kazuhiro Takemoto
4 Centre for Computational Biology, Mines ParisTech
Dr. Jean-Philippe Ver
5 Faculty of Mathematical Sciences, Fudan University
Dr. Shu-Qin Zhang
6 School of Mathematical Sciences, Xiamen University
Dr. Zheng-Jian Bai

Epidemic Disease
Metabolic Networks
Dynamics of HIV spreading in a single prison under screening and quarantine
y(t) the number of infected but not detected
m(t) the number of detected but not quarantined
q(t) the number of quarantined
x(t) = N − y(t) − m(t) − q(t) the number of suspected
n the number of prisoners exchanging with the outside world



y(t + 1) = (1 − τ)(1 −
n
N
) (y(t) + βx(t)(y(t) + m(t))) + (1 − τ)nµ
m(t + 1) = (1 − κ)(1 −
n
N
)m(t) + (1 − κ)τnµ
+(1 − κ)τ(1 − n
N
)(y(t) + βx(t)(y(t) + m(t)))
q(t + 1) = (1 −
n
N
)q(t) + κ(1 −
n
N
)m(t)
+κτ(1 − n
N
)(y(t) + βx(t)(y(t) + m(t))) + τκµn
x(t + 1) = N − y(t + 1) − m(t + 1) − q(t + 1).

Epidemic Disease
Metabolic Networks
After exchange with the outside world
infected but not detected prisoners (1 − n
N
)(y(t) + βx(t)(y(t) + m(t))) + nµ
detected but not quarantined prisoners (1 − n
N
)m(t)
quarantined prisoners (1 − n
N
)q(t)
After screening of τ of prisoners undetected and unquarantined
infected but not detected (1 − τ)(1 − n
N
)(y(t) + βx(t)(y(t) + m(t))) + (1 − τ)nµ
detected but not quarantined
(1 − n
N
)m(t) + τ((1 − n
N
)(y(t) + βx(t)(y(t) + m(t))) + nµ)
quarantined prisoners (1 − n
N
)q(t)
After quarantine of κ of detected prisoners
infected but not detected y(t + 1)
(1 − τ)(1 − n
N
)(y(t) + βx(t)(y(t) + m(t))) + (1 − τ)nµ
detected but not quarantined m(t + 1)
(1 − κ)(1 − n
N
)m(t) + (1 − κ)τ(1 − n
N
)(y(t) + βx(t)(y(t) + m(t))) + (1 − κ)τnµ
quarantined prisoners q(t + 1)
(1 − n
N
)q(t) + κ(1 − n
N
)m(t) + κτ(1 − n
N
)(y(t) + βx(t)(y(t) + m(t))) + τκµn
suspected prisoners x(t) N − y(t) − m(t) − q(t)

Epidemic Disease
Metabolic Networks
Newton’s Method
For solving F(x) = 0, Xn+1 = Xn − [F (xn)]−1
F(xn).
Convergence of Newton’s Method
Kantorovich Theorem Let a be a point in RK
, U be an open neighborhood
of a in RK
and F : U → RK
be a differential mapping with its derivative
[DF(a)] being invertible. Define
h = −[DF(a)]−1
F(a), a1 = a + h and U0 = B|h|(a1).
If U0 ⊂ U and the derivative [DF(x)] satisfies the Lipschitz condition
||DF(u1) − DF(u2)|| ≤ M|u1 − u2|
for all points u1 and u2 and if the inequality
|F(a)||DF(a)|−2
M ≤
1
2
is satisfied, then the equation F(x) = 0 has a unique solution in U0 and
Newton’s method converges with an initial guess a.

Epidemic Disease
Metabolic Networks
Stablity
Given that xe is the equilibrium point of equation system x = Ax,
Lyapunov Stable For ∀ > 0, ∃δ = δ( ) > 0 s.t. if x(0) − xe < δ, then
x(t) − xe < for ∀t ≥ 0.
Asymptotically Stable Lyapunov stable and if ∃δ > 0 s.t. if
x(0) − xe < δ, then lim
t→∞
x(t) − xe = 0.
Exponentially Stable Asymptotically stable and if ∃α, β, δ > 0 s.t. if
x(0) − xe < δ, then x(t) − xe ≤ α x(0) − xe e−βt
, t ≥ 0.
Asymptotically stable
The solution for X = AX is asymptotically stable as t → ∞, if and only if for
all the eigenvalues λ of A, Re(λ) < 0.
Gershgorin disc Theorem
For matrix A = (aij )n×n, closed disc D(aii , Ri ) centered at aii with radius Ri is
called Gershgorin disc, where Ri = j=i | aij | is the sum of the absolute values
of the non-diagonal entries in the ith row. Every eigenvalue of matrix A lies
within at least one of the Gershgorin discs D(aii , Ri ).

Epidemic Disease
Metabolic Networks
Phyllotactic patterns under diﬀerent s

Epidemic Disease
Metabolic Networks
Repulsion by all others under various repulsion weight a

Epidemic Disease
Metabolic Networks
Repulsion by nearest left and right under various repulsion weight a

Epidemic Disease
Metabolic Networks
Assume that the numbers of possible Boolean functions for gene vi
are identical, i.e., l(i) = L holds for all i. The expected number of
singleton attractors for a PBN with n genes is
2n
× 1 −
1
2
L n
= 2 −
1
2
L−1 n
.
If fj (v) = v holds for some realization fj , then GAP v is a
singleton attractor.
For each gene vi , for any PBN realization fj , the probability
that f
(i)
j (v(t)) = a, a ∈ {0, 1} is (1
2)L.
Thus Prob(v(t + 1) = u | v(t) = u) = {1 − (1
2)L}n, and the
expected number of singleton attractors is
2n × 1 − 1
2
L n
= 2 − 1
2
L−1 n
.

Epidemic Disease
Metabolic Networks
Branch and bound algorithm
It consists of a systematic enumeration of all candidate solutions,
where large subsets of fruitless candidates are discarded, by using
upper and lower estimated bounds of the quantity being optimized.
Initialize m = 1
Procedure PBNAttractor( v(t), m)
if m = n + 1 then output v1(t), v2(t), . . . , vn(t) return
for b = 0 to 1 do vm(t) := b
if vi (t) = f
(i)
j (v(t))(= vi (t + 1)) holds for any j for some
i ≤ m then continue
else PBNAttractor(v(t), m + 1)
return

Epidemic Disease
Metabolic Networks
The numbers of possible Boolean functions for gene vi are identical, i.e. l(i) = L holds for i = 1, 2, . . . , n.
(i)
ji
are identical c
(i)
1 = c
(i)
2 = · · · = c
(i)
l(i)
.
For gene vi , IN(f
(i)
j
) denotes the set of input genes for corresponding Boolean function f
(i)
j
.
Singleton attractor IN(f
(i)
j1
) = IN(f
(i)
j2
)
The number of recursive calls executed for the ﬁrst m genes is bounded by {2 − 0.5L−1 m
n
KL
}m
.
The probability that for some gene some vi , i ≤ m, vi (t) = f
(i)
j
(v(t))(= vi (t + 1)) holds for any Boolean function
f
(i)
j
, 1 ≤ j ≤ l(i) = L is
L
j=1



0.5 ×
C
|IN(f
(i)
j
)|
m
C
|IN(f
(i)
j
)|
n



≈
L
j=1
0.5 ×
m
n
|IN(f
(i)
j
)|
≥ 0.5
L m
n
KL
.
The probability that the algorithm examines the (m + 1)-th gene is smaller or equal to 1 − 0.5L m
n
KL m
.
Thus the number of recursive calls executed for the ﬁrst m genes is bounded by
2
m
× 1 − 0.5
L m
n
KL m
= 2 − 0.5
L−1 m
n
KL m
.

Epidemic Disease
Metabolic Networks
(i)
ji
are identical c
(i)
1 = c
(i)
2 = · · · = c
(i)
l(i)
.
For gene vi , IN(f
(i)
j
(i)
j
.
Singleton attractor IN(f
(i)
j1
) = IN(f
(i)
j2
)
The number of recursive calls executed for the ﬁrst m genes is bounded by {2 − 0.5L−1 m
n
K
}m
.
The probability that vi (t) = vi (t + 1) holds for some i is
0.5
L
×
C
|IN(f
(i)
1
)|
m
C
|IN(f
(i)
1
)|
n
≈ 0.5
L m
n
|IN(f
(i)
1
)|
≥ 0.5
L m
n
K
.
The probability that the algorithm examines the (m + 1)-th gene is no more than 1 − 0.5L m
n
K m
. The
number of recursive calls executed for the ﬁrst m genes is thus bounded by
2
m
× 1 − 0.5
L m
n
K m
= 2 − 0.5
L−1 m
n
K m
.

Epidemic Disease
Metabolic Networks
(i)
ji
are identical c
(i)
1 = c
(i)
2 = · · · = c
(i)
l(i)
.
For gene vi , IN(f
(i)
j
(i)
j
.
attractor of period p IN(f
(i)
j1
) = IN(f
(i)
j2
)
The number of recursive calls executed for the ﬁrst m genes is bounded by





2 − 0.5
L
1−Kp
1−K −1
s
L
1−Kp
1−K ×
K(1−Kp)
1−K



s 


n
, where s = m/n.
At the corresponding time, one out of L Boolean functions is chosen for each node of p-ancestor. There are at
most Kp
input nodes.
There are at most L
p−1
q=0
Kq
= L(1−Kp)/(1−K)
combinations of p realizations for each node vi .
The probability that vi (t) = vi (t + p) holds for some i is approximately at least



0.5 ×
C
p
q=1
Kq
m
C
p
q=1
Kq
n



L(1−Kp)/(1−K)
≈ 0.5 ×
m
n
p
q=1
Kq L(1−Kp)/(1−K)
.

Epidemic Disease
Metabolic Networks
(i)
ji
are identical c
(i)
1 = c
(i)
2 = · · · = c
(i)
l(i)
.
For gene vi , IN(f
(i)
j
(i)
j
.
attractor of period p IN(f
(i)
j1
) = IN(f
(i)
j2
)
The number of recursive calls executed for the ﬁrst m genes is bounded by
2 − 0.5
L−1
s
K(1−Kp)
1−K
s n
, where s = m/n.
The probability that vi (t) = vi (t + p) holds equals to
0.5
L
×
C
p
q=1
Kq
m
C
p
q=1
Kq
n
≈ 0.5
L m
n
p
q=1
Kq
.

Epidemic Disease
Metabolic Networks
The computation cost of the Reserving Place Algorithm is bounded above by
MT22n, where
M =
K2
j=0
K1
i=0
T!
i!j!(T − i − j)!
.
The matrix-vector multiplication PiT
. . . Pi1
x0 occupies the main computational time.
If one searches optimal solutions in the set
W = {σ = i1 i2 . . . iT : ij ∈ {0, 1, 2}},
then one needs to conduct T3T 22n basic operations, where n is the number of genes
in the PBN, where T is the time length.
For Reserving Place algorithm, one only needs to ﬁnd the optimal control strategies in
the set
V = {σ ∈ U :
i∈S
[xT ]i ≥ ¯p}.
The number of basic operation to be conducted equals to T22nn(V ), where n(V ) is
the size of set V .
Note that V ⊆ U, the computational cost is bounded above by T22nn(U). For the
Reserving Place algorithm, the number of basic operations is upper bounded by
MT22n, where
M = n(U) =
K2
j=0

C
j
T
K1
i=0
C
i
T−j

 =
K2
j=0
K1
i=0
T!
i!j!(T − i − j)!
.

Epidemic Disease
Metabolic Networks
Metabolic network
Mechanisms
reaction R = (C1 ∧ C2) ∧ (C3 ∧ C4)
compound
C = (R12 ∨ R2) ∧ (R11 ∨ R3 ∨ R4) = (R1 ∨ R2) ∧ (R1 ∨ R3 ∨ R4)

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