Strategies to synchronize biological synthetic networks

Giovanni Russo Ph.D. Student

University of Naples FEDERICO II

Department of Systems and Computer Science

LAB Meeting 28/04/2008

Outline

• In the previous episode…

• A condition for synchronization and generalizations

• Future work

In the previous episode…

Giovanni Russo

The RepressilatorThe 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)

The mathematical modelThe mathematical model can also be rewritten in a form that underlines the structure of the biological system.

i i i i i i a a a ex = A x +B f x +B f x + ηS

i

s1 s0

-1 0 0 0 0 0 0

0 -1 0 0 0 0 0

0 0 -1 0 0 0 0

A = β 0 0 -β 0 0 0

0 β 0 0 -β 0 0

0 0 β 0 0 -β 0

0 0 0 K 0 0 -K - η

i

1 0 0

0 1 0

0 0 1

B = 0 0 0

0 0 0

0 0 0

0 0 0

2i

ii a2 2

i i

2i

α

1+C

kSαf = f =

1+ A 1+S

α

1+B

Te se ext e extS K N S x G

7 1

1 7,14,...7

0

N

i

G R

i NG

otherwise

0

0

1

0

0

0

0

aB

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)

Theorem:

Differentiation of the virtual system yields the dynamics for the virtual displacements and velocities:

Network of identical nodes (2)Thus, we have to prove that the matrix:

Is contracting.Theorem:Let A be a square matrix, and let

1

1n

i ijjj i

R a i n

Then all the eigenvalues of A are located in the union of n discs: : ii iz C z a R G A 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

1

1:

n

ii j ijj ij i

z C z a p ap

Application of the Gersghorin theorem (1)

Using the previous Corollary, the condition warranting the negativity of all the eigenvalues of the Jacobian matrix is:

6

221

21

1

p C

p C

422

21

2 1

p A

p A

5 7

2 223 3

21

111

i

ii

p p KSB K

p p SSB

1

4

p

p 2

5

p

p 3

6

p

p

40 1

7s s

pK K

p

Application of the Gersghorin theorem (2)

Since the derivative of the activation and inhibition functions are bounded, from the previous relations we have:

4 1

5 2

6 3

6

1 1

4

2 1

5 7

3 31 2

0 4

1 7

1

max

1

max

1 1

max max i

s

s

p p

p p

p p

p

p f C

p

p f A

p p

p pf B f S

K p

K p

This set of inequalities seems to have no solution for the set of parameters used in the literature!

...and now?

1 4

2 5

3 6

p p

p p

p p

If the first three inequalities were

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 modelThe set of inequalities becomes:

4 1

5 2

6 3

6

1 1

4

2 1

5 7

3 31 2

0 4

1 7

1

max

1

max

1 1

max max i

s

s

dp p

dp p

dp p

p

p f C

p

p f A

p p

p pf B f S

K p

K p

Then, if d>>β, the system can be solved.

The network can be better synchronized if max{f(*)} are decreased

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

1 1,2 1,

2,1 2 2,

,1 , 1

...

...

... ... ... ...

...

n

n

n n n n

f f

f f

f f

Consider a matrix of the form:

It’s possible to see that all its eigenvalues are negative if

,0

1... : max 1...ii ji n f j n

n 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 nearto 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

A w a A

The contracting property, warrant (stochastic) synchronization in presence of noise (since the stochastic contracting property is preserved in systems combinations).

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 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.

What happens if the nodes are not all identical?

Giovanni Russo

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

Simulation resultsIf noise is included into the differential equation of the coupling protein….

Simulation results

Simulation resultsIf, 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

Te e eS KS G x

2 2 2T T T

ij i j i j e e e ei j

V U x x PA x x S KS S G x

i i i i i i a a a ex = A x +B f x +B f x + ηS

How to study this closed loop system (even in the case of linear systems at the nodes)?

1. Lyapunov function:

2. Hyperstability….

T Tij i j i j e e

i j

V U x x P x x S S

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• …

Giovanni Russo

Thanks!