Modeling of cell’s functional regulatory
networks in cancer using variable order
fractional systems
Supervisor I : Dr. Keivan Maghooli
Supervisor II : Dr. Masoud Asadi Khiavi
Advisor: Dr. Nader Jafainia Dabanloo
Presentation: Hanif Yaghoobi
Regulatory Network
• a collection of DNA segments (genes or their products ) in a
cell which interact with each other and with other
substances in the cell, thereby governing the rates at which
genes in the network are transcribed into mRNA.
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Translation
Rate
Translation
Rate
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Gene
Expressi
on Level
Protein
Level
GENE A
Gene
Expressi
on Level
Protein
Level
GENE B
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Transcription
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Rate
Transcription
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Rate
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Data Required: Gene Expression Matrix
t1
t2
t3
t4
g1
0
1
2
1
0
g2
1
2
1
0
1
1.
g3
0
1
1
1.
1
0
g4
1
2
1
0
a1
a2
a3
a4
g1
0
3
1
1
g2
1
2
1
g3
0
1
g4
1
2
Snap Shot
Time serious
5
Model formalisms
• Many formalisms to model genetic regulatory networks
Graphs
Boolean equations
precision
abstraction
Differential equations
Stochastic master equations
The first step in modeling is to determine the structure of the network.
Then, a complete analysis of the dynamics of its components requires
solving the Master Equations. By writing the master equations, this
network is described as a dynamic system with nonlinear differential
equations in state space.
6
Dynamic models
• In systems theory, GRNs are complex nonlinear dynamic systems
with time delays, and complex nonlinear dynamics can be found
even within the simplified and constructive infrastructure of
these systems.
• Intrinsic properties of GRNs:
A) Time Delay
B) Non-Linear Dynamics
C) Stochastic Dynamics
D) Long Range Dependence (Epigenetic Long Term Memory )
E) Slow and Complex Dynamic
Simplified form of GRN in the cell cycle based on the VOF system model
Case Study
. Gene network in the mammalian
cancer cell during G1/s transition
stage of cell cycle
DataSet
Gene expression time series data of cell cycle process are suitable for
analysing GRNs. We used two sets of data. First is related to Human
Primary Skin Fibroblasts throughout the Cell Cycle, which is a normal
cell.
This dataset is available under the access number E-TABM-263 at
https://www.omicsdi.org/dataset.
The second is the Human Cell Cycle in the Hela cell line, which is a
cancer cell. This dataset is available at
http://genome-www.stanford.edu/Human-CellCycle/HeLa/.
Summary of Methodology
Basic definitions
Grünwald–Letnikov definition for the fractional variable
order derivative
Deterministic Solution
Stochastic Solution
dy(t )
dW
 f ( y (t ), t )  g ( y(t ), t )
dt
dt
General form of Stochastic Differential Equations
W is the Wiener Process (Standard Brownian Process)
 (t )
Dt
y (t ) 
d  (t )
dt
 (t )
dW
y (t )  f ( y (t ), t )  g ( y (t ), t )
dt
Generalize to Variable Order Fractional EQ
d  (t ) y(t )  f ( y(t ), t )dt  (t )  g ( y(t ), t )dWdt (t ) 1
Stochastic Solution
Final Solution
 (tk )
y (t k )  f ( y (t k ), t k )h
dW ~
dt  N (0,1)
g ( y(t ), t )  y(t )
 (t ) 1
 g ( y (t k ), t k )dWh
k
j   (t j ) 
  (1) 

j 1
j
 y (t k  j )

GRN modeling by Delayed VOFS
GRN modeling by Delayed VOFS
GRN modeling by Delayed VOFS
GRN modeling by Delayed VOFS
Gi (t k 1 )dWt h (t )1 to the above equation, a
It is easy to see that by adding
system of stochastic equations is obtained for each gene node.
Function Approximation for Variable Order
In real systems,  (t ) is a time-varying variable (parameter)
that affects the system. In the GRN system, for example, this
is most likely a time-varying temperature function. For
phenomena such as biological systems, it is almost
impossible to determine  (t ) analytically. So in the
optimistic case,  (t ) is obtained through measurements as a
time series of measurements.
Function Approximation for Variable Order
Expression Approximations for Data-less Genes
Unfortunately, time series data similar to those for coding
genes are not available for miRNAs and LNC-RNAs.
Parameters Identification and Optimization
The purpose of the parameter estimation step is to
determine and optimize all system parameters so that the
system's solved response (here predicted values of gene
expression) has the least error with real data (here gene
expression time series data). This is done by the
Imperialist Competitive Algorithm (ICA).
Results
Results of Deterministic System
Decision Tree Algorithm for Chaos Test
0-1 test
0-1 test uses a time series of data obtained from a dynamic
system

 8110
10
50
Chaos test
Gene Name
MYC
E2F1
RB1
CDK4
CDC25A
CDK2
CDKN1B
miR-17-92
Without Change in parameters
k
-0.33
0.25
-0.11
0.26
0.38
-0.00
0.32
-0.21
Permutation Entropy
4.17
3.94
3.64
3.66
3.84
3.85
3.90
2.61
Result
Periodic
Periodic
Periodic
Periodic
Periodic
Periodic
Periodic
Periodic
Change the Delay parameter   10
k
0.32
0.59
0.17
0.22
0.93
-0.18
0.03
0.38
Permutation Entropy
4.02
3.46
3.4
3.75
3.05
4.52
3.6
2.61
Result
Periodic
Chaotic
Periodic
Periodic
Chaotic
Periodic
Periodic
Periodic
Change the MYC Self-Synthesis Rate parameter   50
1
k
0.87
0.29
-0.1
0.36
0.46
0.08
0.32
-0.21
Permutation Entropy
3.94
4.08
3.63
3.66
3.87
3.85
3.9
2.61
Result
Chaotic
Periodic
Periodic
Periodic
Periodic
Periodic
Periodic
Periodic
Change the miR-17-92 Degradation parameter   10
8
k
-0.32
0.11
-0.06
0.32
0.41
0.03
0.34
-0.09
Permutation Entropy
4.17
3.71
3.63
3.66
3.84
3.85
3.9
0.00
Result
Periodic
Periodic
Periodic
Periodic
Periodic
Periodic
Periodic
Periodic
Change the Variable Order parameter K=2
k
-0.3
0.72
0.98
-0.01
0
-0.05
-0.47
0.95
Permutation Entropy
3.76
2.77
3.94
3.91
0.55
4.65
3.77
2.92
Result
Periodic
Chaotic
Chaotic
Periodic
Periodic
Periodic
Periodic
Chaotic
Phase Plane Plots for Chaotic States
P-Q Plots
Hurst Exponent test for Long Term Memory
Hurst Exponent
MYC E2F1 RB1 CDK4 CDC25A CDK2 CDKN1B miR-17-92
Normal Cell 0.92 0.92 0.93 0.92
0.89
0.85
0.91
0.94
Cancer Cell 0.94 0.99 0.97
1
0.95
0.99
0.95
0.63
Stochastic System Results
Chaos Test
Gene Name
MYC
E2F1
RB1
CDK4
CDC25A
CDK2
CDKN1B
miR-17-92
Without Change in parameters
k
-
0.26
-0.45
-0.18
0.04
0.32
0.11
-0.06
Permutation Entropy
-
3.96
4.35
4.51
5.22
5.46
4.39
4.14
Result
Stochastic
Periodic
Periodic
Periodic
Periodic
Periodic
Periodic
Periodic
Change the Delay parameter   10
k
0.38
0.12
0.01
0.12
0.06
-0.01
0.01
-0.24
Permutation Entropy
4.38
3.77
4.7
4.33
4.34
4.4
4.31
4.54
Result
Periodic
Periodic
Periodic
Periodic
Periodic
Periodic
Periodic
Periodic
Change the MYC Self-Synthesis Rate parameter   50
1
k
0.93
-0.14
-0.54
-0.21
0.08
0
0.2
-0.02
Permutation Entropy
4.22
5.22
4.35
4.51
5.42
5.41
4.39
4.14
Result
Chaotic
Periodic
Periodic
Periodic
Periodic
Periodic
Periodic
Periodic
Change the miR-17-92 Degradation parameter   10
8
k
-
-0.51
-0.14
0.13
0.27
0.28
-0.05
-0.31
Permutation Entropy
-
3.99
4.35
4.51
5.11
5.48
4.39
0.21
Result
Stochastic
Periodic
Periodic
Periodic
Periodic
Periodic
Periodic
Periodic
Change the Variable Order parameter K=2
k
-0.31
0.15
-0.08
0.32
0.34
0.14
0.98
0.06
Permutation Entropy
4.55
3.73
4.39
3.54
5.64
4.6
4.42
4.53
Result
Periodic
Periodic
Periodic
Periodic
Periodic
Periodic
Chaotic
Periodic
Comparison with previous works
. Modeled gene expression signals for PSK cell with
 (t )  0.8 .
Variable order effects
Variable order effects
Variable order effects
Average Value of  (t )
MYC
E2F1
RB1
Normal Cell 0.15 0.32 0.30
Cancer Cell 0.14 0.19 0.16
CDK4 CDC25A CDK2 CDKN1B
0.26
0.09
0.38
0.17
0.17
0.11
0.17
0.15
miR-1792
0.14
0.31
What we found
• the nonlinear variable order fractional systems have very good
flexibility in adapting to real data.
• Regulatory networks in cancer cells actually have a larger delay
parameter than in normal cells.
• It is also possible to create chaos, periodic and quasi-periodic
oscillations by changing the delay, degradation and synthesis rates.
• Profound effect of time-varying order on GRN, which may be related
to a type of cellular memory due to epigenetic and environmental
factors.
What we found
• By changing the delay parameter and the variable order function
for a normal cell system, its behavior changes and becomes quite
similar to the behavior of a cancer cell.
• This work also confirms the effective role of the miR-17-92 cluster
in the cancer cell cycle.
Thank You