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Multiscale modeling &
Simulation methodology
Selected Ch 8 material
Time delays approx translocation dynamics
Transcription Factor Autoregulation Dynamics
K P 2 (t   )
dP (t )
 b  kdeg P (t )  2 1
dt
P (t   )  K 2
Constituitive
Production &
degradaton
Regulation:
NL + feedback
mRNA not modeled here.
Only the protein
1
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Modeling transcription factor autoregulation via homdimer
positive feedback
A Visual Tour from Cartoons To Differential Equations (& Back)
Doug Dang, BE Undergrad Student
Prof. Joe DiStefano III
CARTOON DIAGRAM
𝑘
τ
TF‐RE
𝑘
𝑘
TF‐Gene
2
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SIMPLE MODEL WITH TIME DELAY
τ1
𝑘
TF‐RE
TF‐Gene
𝑘
P
τ2
𝑘
τ1
τ2
τ
DIMERIZATION
3
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HOMODIMER TRANSLOCATION
MRNA
TRANSCRIPTION
AND TRANSLATION
4
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PROTEIN DIMERIZATION:
COMPLETE
𝑘
𝑘
P Pτ
TF‐RE
𝑘
P
TF‐Gene
DIRECTED GRAPH MODEL
𝑘
τ
𝑘
𝑘
5
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ODE MODEL
𝑏
𝑘
𝑃 𝑡
= constituent production ‐ degradation + positive f.b. regulation term
A MULTISCALE
MODEL
RIBOSOME
dP(t )
K1P 2 (t   )
 b  kdeg P(t )  2
dt
P (t   )  K 2
REGULATION
TERM
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Constituent production & degradation terms
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K P 2 (t   )
dP (t )
b 2 1
 kdeg P (t )
dt
P (t   )  K 2
SIMULATION CONDITIONS
P(0) = 200 molecules of TF in some unit volume of cytoplasm
over 0 < t < 180 min
basal synthesis rate b = 20 molecs/min
K1  2,000 min 1 , K 2 =400,000 molecules 2 and kdeg  1min 1
time delay τ = 0, 5,10 mins
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Simulink Implementation
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7
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kdeg
GAIN
–
1
s
+
+
×
INTEGRATOR
ADD
÷
DIVIDE
+
dP(t )
K P (t   )
b 2 1
 kdeg P(t )
dt
P (t   )  K 2
2
+
ADD
K2
CONSTANT
Pow
K1
Delay
GAIN
STEP‐
RESPONSES
Zero delay
Zero delay
mindelay
delay
55min
10 min delay
10 min delay
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subsystem
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Based on DSBMS JJ DiStefano III Academic Press 2015
16
8
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NUMERICAL ALGORITHMS
(“SOLVERS”) FOR SOLN OF ODEs
BASIC THEORY IN A NUTSHELL
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Initial Value Problems
Consider the NL vector ODE:
x  f (x, u, t )
x (t0 )  x0
Two basic categories of methods for numerically solving ODE's:
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•
One-step methods:
No iterations required. Self-starting.
•
Multi-step methods:
Not self-starting
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x  f ( x, u, t )  f ( x, t ),
x(t0 )  x0
Most algs use Taylor series expansion of f:
x (tm 1 )  x (tm )  x (tm ) t 
t  stepsize
x
m 1
x(tm ) 2
t  
2!
x (tm )  x m , shorthand for "mth step"
xm 2
 x  x t 
t  
2!
m
m
m = 0, 1, 2,….
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Simplest Solver – the Euler Method (Euler Formula) for
x  f
Given the solution x (t m )  x m at the discrete-time instant tm,
x(tm 1 )  x(tm )  f [ x(t m ), u(tm ), t ]t
t  t m 1  t m
abbreviated: x m 1  x m  f m t
x m 1  x m  f m t  ET2
O(t2)  truncation error  𝐸
10
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Euler's Method: First-order Runge-Kutta
Uses only first two terms of Taylor series expansion to predict soln at tm+1:
x = xm + (t – tm)f m
x m 1 ~ x m  f m t
xm+1
ET2
xm
TRUE x(t)
t
tm+1
tm
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Problem: Truncation error 0(∆t2)ET can be large.
Example:
dy
 y2  0
dt
or
dy
 y 2
dt
y(0) = 1
Exact solution is:
y = 1 / (1 + t)
2
Apply Euler algorithm: ym 1  y m  t ( ym )
Starting at t = 0 (m = 0), with t = 0.1, y at t = 0.1:
y1 = 1 + (0.1)(-12 ) = 0.9
Exact solution at this point is
yexact (0.1) 
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1
 0.9090909
1  0.1
Based on DSBMS JJ DiStefano III Academic Press 2015
22
11
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Next, at t = 0.2,
2
y2 = 0.9 + (0.1)(-(0.9) ) = 0.819
Exact solution is
1
 0.8333333
1  0.2
yexact (0.2 ) 
After 10 steps of t = 0.1,
y10 = 0.4627810
yexact (1.0) 
1
 0.5
1 1
ET is building up fast
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Example of Roundoff error buildup for small step-sizes
1.00
0.00
TOTAL ERROR vs. STEP-SIZE
𝑑𝑥
𝑑𝑡
Euler’s Method
𝑥
x  x
x(0) = 1
-1.00
16-bit word length precision
TOTAL
ERROR
-2.00
-3.00
-4.00
-5.00
h = e-15
-6.00
-30.00
-25.00
-20.00
-15.00
h =1
-10.00
-5.00
0.00
Ln h
Increasing step-size 
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12
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t  stepsize= tm  tm1
x (tm-1)
x (tm)
x (tm+1)
tm-1
tm
tm+1
t
x (tm )  x m , shorthand for "mth step"
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Based on DSBMS JJ DiStefano III Academic Press 2015
25
ERRORS IN DIGITAL COMP SOLNS of ODEs
Truncation Error: ET .. error in finite approx of  series (incl deriv approxs)
Roundoff Error: ER ..error in truncating numbers in calcs to finite
word size (e.g. 64 bits)
Total Error:
ET + ER
Relative Error: (E T+ ER) / Exact Soln x*(t)
Overall, most important
Numerical (computational) Stability (NS) … bounded relative errors
Stable algorithm: numerically stable
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Based on DSBMS JJ DiStefano III Academic Press 2015
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RUNGE-KUTTA SELF-STARTING METHODS
Properties
(1) One-step methods: to find xm+1, only need info avail at prior point, xm.
(2) agree with Taylor series thru terms in ∆tp, where p is called
the order of the method.
(3) do not require eval of any derivs of f, only f itself.
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Based on DSBMS JJ DiStefano III Academic Press 2015
27
HIGHER-ORDER R-K METHODS
f (x, t ) evaluated not only at (x m , tm ) but also at one or more
m
closely adjacent points BETWEEN ( x , tm )
Then soln at
tm+1, i.e., x m +1
and (x
.
m +1
, t m+1 )
, is obtained as a
.
weighted sum of these expressions for the fcn
f (x, t )
at different points btwn tm and tm+1, over t
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Based on DSBMS JJ DiStefano III Academic Press 2015
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Most Popular (ODE4): 4th-Order Formula (Evals half‐way btwn)
x m 1  x m 
where
k1  f ( x m , tm )
t
6
 k1  2k2  2k3  k4 
t
t 

k2  f  x m  k1 , tm    f  x1/2* , t1/2*  half‐way pt
2
2

t
t 

k 3  f  x m  k 2 , tm  
2nd half‐way pt
2
2

k4  f  x m  k3t , tm  t 
and
ET ~ O(h5)
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Based on DSBMS JJ DiStefano III Academic Press 2015
P(t)
2000
P R O T E IN C O N C E N T R A T IO N
29
dP(t )
K P 2 (t   )
b 2 1
 kdeg P(t )
dt
P (t   )  K 2
1500
RUNGA-KUTTA
4TH-ORDER
Varying Fixed
Step-Sizes
1000
 t = 0.05
 t = 0.1
 t = 0.5
t = 1
500
t = 2
0
0
20
40
60
80
100
MINUTES
120
140
160
180
4th–order Runge‐Kutta solutions for 5 different fixed stepsizes
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15
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EXTRAS
MULTISTEP PREDICTOR-CORRECTOR METHODS (Advanced)
ꞏ
Use vals of y determined at 2 or more preceding points.
ꞏ
Therefore, not self-starting.
ꞏ
Also, can be explicit or implicit formulas.
Example - EXPLICIT FORMULA: 2ND-order ADAMS-BASHFORD:
x m 1  x m 
t
( 3 f m  f m 1 )
2
past vals of f
Example IMPLICIT FORMULA: ADAMS-MOULTON 2nd-order
x m1  x m 
t
( f m1  f m )
2
current value
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Derivation of Multistep Algorithms: The Approach
NL Eq:
x  f ( x , t )
Taylor Series Solution:
t  m


x m 1  x m  t  f m 
f   ( t 3 ) 
2!


Method: Replace derivs f & fdot, etc by backward-difference approxs,
e.g. another 2nd‐order (EXPLICIT) ADAMS FORMULA:
:
f m  f m1
f m 
t


t  f m  f m1 
m 1
m
3
x  x  t  f m  
  O ( t ) 

2
t




1
3

x m1  x m  t  f m  f m 1   O ( t 3 )
2
2


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Based on DSBMS JJ DiStefano III Academic Press 2015
33
Or also substitute for the 2nd deriv (backward diff):
m
m 1
 f m2
f m  f  2 f
( t ) 2
23 m 16 m 1 5 m 2 
 O(t 4 )
x m 1  x m  t 
f 
f

f


12

12
12
A 3rd-order explicit ADAMS FORMULA
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Based on DSBMS JJ DiStefano III Academic Press 2015
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Another Example: 4TH-ORDER ADAMS Predictor Algorithm
x m 1  x m 
t
55 f m  59 f m 1  37 f m 2  9 f m 3 
24 
where
f m i  f ( x m  i , tm i )
i  1, 2,3
f m 1  f ( x m1 , tm1 )
• alg requires 4 prev vals of x, i.e., xm-3, xm-2, xm-1, xm to predict xm+1.
• 4th-order algorithm: errors are order of
t5,
similar to 4th-order Runge-Kutta algorithm
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Based on DSBMS JJ DiStefano III Academic Press 2015
35
Digression: common kinetic terminology
• Time constant T
• Half‐life T1/2
• Rate constant k
1
exp kt  exp t /T  0.37 for T  t
 0.5 for t  T1/2
k  ln2 / T1/2
0.5
0.37
18