Organization of the talk
• From networks to physiological behavior
• Network representations
• Mathematical formalisms
• Studying a mathematical model
In silico networks are limited as predictors of physiological behavior
What happens?
Probably a very sick mutant?
Organization of the talk
• From networks to physiological behavior
• Network representations
• Mathematical formalisms
• Studying a mathematical model
Network representation is fundamental for clarity of analysis
A B
What does this mean?
Possibilities:
AB
Function
BA
Function
AB
Function
A B
Function
BA
Function
Defining network conventions
A B
C
Full arrow represents a flux between A and B
Dashed arrow represents modulation of a flux
+
Dashed arrow with a plus sign represents positive modulation of a flux
-
Dashed arrow with a minus sign represents negative modulation of a flux
Organization of the talk
• From networks to physiological behavior
• Network representations
• Mathematical formalism
• Studying a mathematical model
Representing the time behavior of your system
/dA dt
/ ,dA
dA dt A f A Cdt
/
dAdA dt
dt
A B
C
+
/dA
dA dt Adt
What is the form of the function?
A B
C
+
A or C
Flux1 2k A k CLinear 1 2
1 2 3 4
k A k C
K K A K C K AC
Saturating
4 41 2
44 41 2 3 4
k A k C
K K A K C K AC
Sigmoid
What if the form of the function is unknown?
A B
C
+
int intintint int
2 2
2int int
int
, ,, ,
, ,
operating operatingoperatingpo popo operating operating
po po
operating operatingpo pooperating
po
df A C df A CdAf A C f A C A A C C
dt dA dC
d f A C d f A CA A C C
dAdC d C
2
int
int
2 2
2int
int
,...
operatingpooperating
po
operatingpooperating
po
C C
d f A CA A
d A
Taylor Theorem:
f(A,C) can be written as a polynomial function of A and C using the function’s
mathematical derivatives with respect to the variables (A,C)
Are all terms needed?
A B
C
+
int
intint
intint
, ,
,
,
operatingpo
operatingpooperating
po
operatingpooperating
po
dAf A C f A C
dt
df A CA A
dA
df A CC C
dC
f(A,C) can be approximated by considering only a few of its mathematical derivatives
with respect to the variables (A,C)
Linear approximation
A B
C
+
1 20, ,
dAf A C f A C k A k C
dt
Taylor Theorem:
f(A,C) is approximated with a linear function by its first order derivatives with respect to
the variables (A,C)
Logarithmic space is non-linear
A B
C
+
1 2, g gdAf A C A C
dt
g<0 inhibits flux
g=0 no influence on flux
g>0 activates flux
Use Taylor theorem in Log space
Why log space?
• Intuitive parameters
• Simple, yet non-linear
• Linearizes exponential space
–Many biological processes are close to exponential → Linearizes mathematics
Why is formalism important?
• Reproduction of observed behavior
• Tayloring of numerical methods to specific forms of mathematical equations
Organization of the talk
• From networks to physiological behavior
• Network representations
• Mathematical formalism
• Studying a mathematical model
A model of a biosynthetic pathway
10 13 111 1 0 3 1 1/ g g hdX dt X X X
11 222 1 1 2 2/ h hdX dt X X
X0 X1
_
+
X2 X3
X4
22 33 343 2 2 3 3 4/ h h hdX dt X X X
Constant
Protein using X3
What can you learn?
• Steady state response
• Long term or homeostatic systemic behavior of the network
• Transient response
• Transient of adaptive systemic behavior of the network
What else can you learn?
• Sensitivity of the system to perturbations in parameters or conditions in the medium
• Stability of the homeostatic behavior of the system
• Understand design principles in the network as a consequence of evolution
Steady state response analysis
10 13 111 1 0 3 1 1/ 0g g hdX dt X X X
11 222 1 1 2 2/ 0h hdX dt X X
22 33 343 2 2 3 3 4/ 0h h hdX dt X X X
How is homeostasis of the flux affected by changes in X0?
0 3 0 10 33 13( , ) ( , ) /L V X L X X g h g
Log[X0]
Log[V]
Low g10
Medium g10
Large g10
Increases in X0 always increase the homeostatic values of the flux through the pathway
How is flux affected by changes in demand for X3?
4 13 34 13 33( , ) / 0L V X g h g h
Log[X4]
Log[V]Large g13
Medium g13
Low g13
How is homeostasis affected by changes in demand for X3?
3 4 4 13 34 13 33( , ) ( , ) / / 0L X X L V X g h g h
Log[X4]
Log[X3]
Low g13
Medium g13
Large g13
What to look for in the analysis?
• Steady state response
•Long term or homeostatic systemic behavior of the network
• Transient response
•Transient of adaptive systemic behavior of the network
Transient response analysis
10 13 111 1 0 3 1 1/ g g hdX dt X X X
11 222 1 1 2 2/ h hdX dt X X
22 33 343 2 2 3 3 4/ h h hdX dt X X X
Solve numerically
Specific adaptive response10 13 11
1 1 0 3 1 1/ g g hdX dt X X X 11 22
2 1 1 2 2/ h hdX dt X X 22 33 34
3 2 2 3 3 4/ h h hdX dt X X X
Get parameter valuesGet concentration
valuesSubstitution
Solve numerically
Time
[X3]
Change in X4
General adaptive response10 13 11
1 1 0 3 1 1/ g g hdX dt X X X 11 22
2 1 1 2 2/ h hdX dt X X 22 33 34
3 2 2 3 3 4/ h h hdX dt X X X Normalize
Solve numerically with comprehensive scan of parameter values
Time
[X3]
Increase in X4
Low g13
Increasing g13
Threshold g13
High g13
Unstable system, uncapable of homeostasis if feedback is strong!!
Sensitivity analysis
• Sensitivity of the system to changes in environment–Increase in demand for product causes increase in flux through pathway
–Increase in strength of feedback increases response of flux to demand
–Increase in strength of feedback decreases homeostasis margin of the system
Stability analysis
• Stability of the homeostatic behavior
–Increase in strength of feedback decreases homeostasis margin of the system
How to do it
• Download programs/algorithms and do it– PLAS, GEPASI, COPASI SBML suites,
MatLab, Mathematica, etc.
• Use an on-line server to build the model and do the simulation– V-Cell, Basis
Design principles
•Why is a given pathway design prefered over another?
•Overall feedback in biosynthetic pathways
•Why are there alternative designs of the same pathway?
•Dual modes of gene control
Why regulation by overall feedback?
X0 X1
_
+
X2 X3
X4
X0 X1
_
+
X2 X3
X4
__
Overall feedback
Cascade feedback
Overall feedback improves functionality of the system
TimeSpurious stimulation
[C]Overall
Cascade
Proper stimulus
Overall
Cascade
[C]
StimulusOverall
Cascade
Demand theory of gene control
Wall et al, 2004, Nature Genetics Reviews
• High demand for gene expression→ Positive Regulation
• Low demand for gene expression → Negative mode of regulation
Summary
• From networks to physiological behavior
• Network representations
• Mathematical formalism
• Studying a mathematical model
Computational tools in Molecular Biology
• Predictions & Analysis– Identification of components– Organization of components– Conectivity of components– Behavior of systems– Evolution & Design
• Prioritizing wet lab experiments– Most likely elements to test– Most likely processes to test
Are all terms needed?
A B
C
+
int
intint
intint
, ,
,
,
operatingpo
operatingpooperating
po
operatingpooperating
po
dAf A C f A C
dt
df A CA A
dA
df A CC C
dC
f(A,C) can be approximated by considering only a few of its mathematical derivatives
with respect to the variables (A,C)
Linear approximation
A B
C
+
1 20, ,
dAf A C f A C k A k C
dt
Taylor Theorem:
f(A,C) is approximated with a linear function by its first order derivatives with respect to
the variables (A,C)
What if flux is non linear?
A B
C
+
Use Taylor theorem in Non-Linear space!
Use Taylor theorem with large number of terms
or