Transcript Lab_meeting_28_04_20..

Strategies to synchronize
biological synthetic networks
LAB Meeting 28/04/2008
Giovanni Russo Ph.D. Student
University of Naples FEDERICO II
Department of Systems and Computer Science
Outline
• In the previous episode…
• A condition for synchronization and
generalizations
• Future work
In the previous episode…
The Repressilator
The Repressilator is a network of three genes, the products of which inhibit the transcription of
each other in a cyclic way. Here is represented a modular addition with the aim of coupling a
population of cells.
Ref: J. Garcia-Ojalvo, M. B. Elowitz, S. H. Strogatz (2004)
Giovanni Russo
The mathematical model
The mathematical model can also be rewritten in a form that underlines the structure
of the biological system.
xi = Ai xi + Bifi  xi  + Bafa  xa  + ηSe
0
-1 0 0
 0 -1 0
0

 0 0 -1 0

A i =  β 0 0 -β
0 β 0
0

0
0 0 β
0 0 0 K
s1

0 0
0 
1 0 0 

0 1 0 
0 0
0 


0 0
0 
0 0 1 



0 0
0  B i = 0 0 0 
0 0 0 
-β 0
0 



0
0
0
0 -β
0 


0 0 0 
0 0 -K s0 - η


Se   Kse ext N  Se ext x G
T
Ba
0 
0 
 
1 
 
 0 
0 
 
0 
0 
 
 α 

2 
1
+
C
i 

 α 
kS i
fi = 
f
=
a
2
2
1
+
A
1
+
S
i
i


 α 

2 
1
+
B
i 

G  R 7 N 1
1 i  7,14,...7 N
Gi  
 0 otherwise
Studying synchronization…
In order to study synchronization Contraction theory is used.
Suppose now that all the nodes are identical
The following virtual system can be used:
Network of identical nodes (1)
Differentiation of the virtual system yields the dynamics for the virtual displacements and
velocities:
Theorem:
Network of identical nodes (2)
Thus, we have to prove that the matrix:
Is contracting.
Theorem:
n
Let A be a square matrix, and let Ri   aij
1 i  n
j 1
j i

z  C : z  aii 
Then all the eigenvalues of A are located in the union of n discs:
Corollary:
Let A be a square matrix, and let p1, p2,...pn be positive real numbers. Then all the
eigenvalues of A lie in the region


n
1


aij 
 z  C : z  aii  p j 
j 1 pi


j i


Ri   G  A 
Application of the Gersghorin
theorem (1)
Using the previous Corollary, the condition warranting the negativity of all the
eigenvalues of the Jacobian matrix is:

p6  2 C
1
p1  1  C 2



p5  2 B
1
p3  1  B 2


p1
 
p4




2



p4  2 A
1
p 2  1  A2



2







  p7   KSi  K 
2
 p3  1  S 2 1  Si  
i



p2
 
p5
Ks0  
p4
K s1
p7
p3
 
p6
Application of the Gersghorin
theorem (2)
Since the derivative of the activation and inhibition functions are bounded, from the
previous relations we have:
p4  p1
This set of inequalities seems to have no solution
for the set of parameters used in the literature!
p5  p2
p6  p3
p6
1

p1
max  f1  C 
p4
1

p2
max  f1  A 
p5
1

p3
max  f1  B 
p7
1

p3
max  f 2  Si 
Ks0 
p
 4
p7
K s1
...and now?
If the first three inequalities were
p1  p4
p2  p5
p3  p6
The whole system would be consistent
On the other hand if we slightly modify our system in the following manner:
A   a  dA
B   b  dB
C   c  dC
A modified mathematical model
The set of inequalities becomes:
dp4   p1
dp5   p2
dp6   p3
Then, if d>>β, the system can be solved.
p6
1

p1
max  f1  C 
p4
1

p2
max  f1  A 
p5
1

p3
max  f1  B 
The network can be better synchronized if
max{f(*)} are decreased
p7
1

p3
max  f 2  Si 
Ks0 
p
 4
K s1
p7
A condition for synchronization and
generalizations
A condition for synchronization in a
network of Repressilators
In the above case study we have seen that using contraction theory and the disks
theorem it is possible to obtain a condition on the set of parameters that warrants the
existence of a stable synchronous state
In particular the following procedure was used:
1. Definition of a virtual system and differentiation,
2. Derivation of a set of inequalities from the disks’ theorem,
3. Check if exists a set of parameters for which the inequalities hold!
Generalization of the results
Consider a matrix of the form:
  1
f
 2,1
 ...

 f n ,1
f1,2
2
...
...
...
...
...
f n ,n 1
f1,n 
f 2,n 
... 

n 
It’s possible to see that all its eigenvalues are negative if
i  1...n : max  f i , j  
i
n  n0
j  1...n
From a biological point of view this means that the maximum production/inhibition rate
must be less than the self degradation for at least one specie in the system.
Remarks (1)
• The procedure is made on the worst-case: indeed, the maxima of the Hill functions
are considered.
• The above matrix does not take into account the case in which the terms on the
diagonal elements are not constant: this is the case in which the degradation of a
protein (or mRNA) is determined by other proteins (or mRNAs)
For example, consider:
Remarks (2)
It is easy to check that differentiation yields a Jacobian matrix with diagonal terms
depending on the state variables
In this case one could give a condition on the minimum of those elements: however,
this criterion could be very restrictive in the cases in which the minimum is near zero
(as in the considered case, in which the minimum of the periodic trajectory is near
to zero). Another way to proceed is to satisfy the inequalities given by the application
of the circle criterion for the functions present in the Jacobian matrix. If the object of
the study is to design a synthetic circuit, then the set of inequalities could be
easily satisfied.
Network of nonidentical nodes
The contracting property, warrant (stochastic) synchronization in presence of noise (since
the stochastic contracting property is preserved in systems combinations).
What happens if the nodes are not all identical?
We can take into account the mismatch of parameters between the cells using white noise:
for our purposes it’s possible to modify each protein equation in the following manner:
A     w   a  A 
A    a  A  w  a  A
Since a and A are bounded and wβ is a white noise with mean equal to zero, we are in the
hypotheses in which the stochastic contraction theory holds.
A control strategy: centralized
controller
•
The controller is implemented in a different cell (or cell population) from that of the
population.
Advantages: the use of an external controller could be easily implemented, the
control action will be moderate, we foresee a robust control
Drawbacks: the main drawback is the lack of informations for the controller. It can
use only informations about the extra-cellular auto-inducer; this will be
the control input too.
Controllers population
Giovanni Russo
Simulation results
If noise is included into the differential equation of the coupling
protein….
Simulation results
Simulation results
If, on the other hand, time delay is included into the differential
equation of the coupling protein….
Simulation results
Future work
Quorum sensing as a protocol for
synchronization
Bacteria lives different environments: however, in each of them continually chemical signals
run. In other words, small molecule, called autoinducer link the population of bacteria and
carry informations.
The autoinducer molecules accumulate themselves near the bacteria: when the
concentration near a bacterium exceeds a certain threshold (quorum), some intracellular
reactions are activated.
Thanks to this mechanism, bacteria can coordinate their actions.
It’s interesting to note here that the global behavior of the population is driven by the local
measurement made by each bacterium.
Modeling quorum sensing
xi = Ai xi + Bifi  xi  + Bafa  xa  + ηSe
Se   KSe  eGT x
How to study this closed loop system (even in the case of linear systems at the
nodes)?
1. Lyapunov function:
V    U ij  xi  x j  P  xi  x j   SeT Se
T
i j
V  2  U ij  xi  x j  PA  xi  x j   2SeT KSe  2SeeGT x
T
i j
2. Hyperstability….
A biologically inspired consensus
protocol (1)
It is proven that bacteria move along gradients of specific chemicals: this process is
called bacterial chemotaxis.
Bacterial chemotaxis achieves remarkable performance considering the physical
limitations faced by bacteria. They can detect concentration gradients as small as a
change of one molecule per cell volume per micron and function in background
concentrations spanning over five orders of magnitude. All this is done under strong
white noise, such that if the cell tries to swim straight for 10 s, its orientation is
randomized by 90° on average.
How E. coli manage to move up gradients of attractants despite these physical
limitations?
The key stands in the fact that E. coli uses temporal gradients to drive its motion: in
particular a biased-random-walk strategy is used to sample space and convert
spatial gradients to temporal ones.
A biologically inspired consensus
protocol (2)
To sense gradients, E. coli compares the current attractant concentration to the
concentration in the past! If a positive net change of attractant concentration is
sensed than the movement will be in the corresponding direction.
Idea:
Is it possible to use this high-performance strategy in other
kinds of networks?
Applications would be, for example:
• Sensor networks
• Mobile agents
•…
Thanks!
Giovanni Russo