Analyzing the systemic
function of genes and proteins
Rui Alves
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?
How to predict behavior from
network?
•
Build mathematical models!!!!
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:
A
B
Function
B
A
Function
A
B
Function
A
B
Function
B
A
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
A
B
C
+
What is the form of the function?
A
B
C
+
A or C
Flux
Linear
Saturating
Sigmoid
What if the form of the function is
unknown?
A
B
C
+
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
+
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
+
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 system is non

linear?
•
Use a first order approximation in a
non

linear space.
Logarithmic space is non

linear
A
B
C
+
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
X
0
X
1
_
+
X
2
X
3
X
4
Constant
Protein
using X
3
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
How is homeostasis of the flux
affected by changes in X
0
?
Log[X
0
]
Log[V]
Low g
10
Medium g
10
Large g
10
Increases in X0 always increase the
homeostatic values of the flux through the
pathway
How is flux affected by changes in
demand for X
3
?
Log[X
4
]
Log[V]
Large g
13
Medium g
13
Low g
13
How is homeostasis affected by
changes in demand for X
3
?
Log[X
4
]
Log[X
3
]
Low g
13
Medium g
13
Large g
13
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
Solve numerically
Specific adaptive response
Get parameter values
Get concentration
values
Substitution
Solve
numerically
Time
[X
3
]
Change in X
4
General adaptive response
Normalize
Solve numerically
with
comprehensive
scan of parameter
values
Time
[X
3
]
Increase in X
4
Low g
13
Increasing g
13
Threshold g
13
High g
13
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?
X
0
X
1
_
+
X
2
X
3
X
4
X
0
X
1
_
+
X
2
X
3
X
4
_
_
Overall
feedback
Cascade
feedback
Overall feedback improves
functionality of the system
Time
Spurious
stimulation
[C]
Overall
Cascade
Proper
stimulus
Overall
Cascade
[C]
Stimulus
Overall
Cascade
Dual Modes of gene control
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
How to do it
•
Download programs/algorithms and do it
–
BST Lab, Mathematica, Maple
Summary
•
From networks to physiological behavior
•
Network representations
•
Mathematical formalism
•
Studying a mathematical model
Papers to present
•
Vasquez et al, Nature
•
Alves et al. Proteins
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
The Taylor theorem
C
f(C)
0 order
f(C)
1
st
order
2
nd
order
i
th
order
i
th
+ j
th
order
Are all terms needed?
A
B
C
+
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
+
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
How does the transformation
between spaces work?
X
Y
X
Y
How does the Taylor approximation
work in another space?
Variables:
A, B, C, …
f(A,B,…)
Variables:
A, B, C, …
f(A,B,…)
~
f(A,B,…)
Taylor
theorem
Transform to new space
Return to original space
~
f(A,B,…)
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