david cornepierluigi frisco school of mathematical and computer sciences heriot-watt university...

David Corne Pierluigi Frisco

School of Mathematical and Computer Sciences

Heriot-Watt University

Edinburgh

Workshop on Membrane Computing 25-28 June 2007, Thessaloniki (Greece)

Dynamics of HIV infection studied with cellular automata and conformon-P

systems

Conformon-P systems

X

Y

Z

Z

W

12

3

4

Image downloaded on the 24/7/2007 from

http://www.enchantedlearning.com/subjects/animals/cell/anatomy.GIF

http://www.enchantedlearning.com/subjects/animals/cell/anatomy.GIF

[ G, 9 ][ R, 9 ][ G, 5 ]

Conformon-P systems: conformons

[ G, 2 ]

r

[ R, 12 ]

interaction rule: r: G R3

[ G, 5 ]

[ R, 9 ]

Conformon-P systems: interaction

Conformon-P systems: example

Conformon-P systems: module

A group of membranes with conformons and interaction rules in a conformon-P system able to perform a specific task.

Module:

[R, 2]

[G, 3] [R, 0]

1

2

Conformon-P systems: modules

[A, ]only conformon [A, ], N can pass from membrane 1 to membrane 2.

1 2

A() B()

a conformon with name A can interact with B passing only if the value of A and B before the interaction is and respectively, , , N.

Conformon-P systems: modules

A(5) B(4)

a conformon with name A can interact with B passing 3 only if the value of A and B before the interaction is 5 and 4 respectively.

3

[A, 5] [B, 7]

[A, 3] [B, 4]

[A, 5] [B, 4]

Conformon-P systems: probabilities

When a simulation of a conformon-P system is performed, then probabilities are associated to

interaction and passage rules.

Grid of conformon-P systems




Edinburgh



systems

Cellular automata

Rule




Edinburgh



systems

Dynamics of HIV infection

1. the amount of virus in the host grows in exponential way;

2. the viral load drops to a “set point”;

3. the amount of virus in the host increases slowly, accelerating near the onset of AIDS.

first weeks later years1 2 3

HH

I

I

I

DD

Healthy

Infected

Dead

Healthy

Infected

Dead




Edinburgh



systems

Studied with

R. M. Z. Dos Santos and S. Coutinho. Dynamics of HIV infection: a cellular automata approach. Physical review letters, 87(16): 168102, 2001.

Healthy cell;

A-infected cell: infected cell free to spread the infection;

AA-infected cell: final stage of an infected cell before it dies due to action of the immune system;

Dead cell: killed by the immune response.

If an healthy cell has at least one A-infected neighbour, then it becomes infected.

If an healthy cell has no A-infected neighbours but at least 2 < R < 8 AA-infected neighbours, then it become A-infected.

An A-infected cell becomes AA-infected after time steps.

AA-infected cells become dead cells.

Dead cells can become healthy with probability prepl.

Each newly introduced healthy may be replaced by an A-infected cell with probability pinfec.

Studied with conformon-P systems

Healthy:

A-infected:

AA-infected:

Pre-dead:

Dead:

[R, 1] [V, 10] [E, 0] [W, 0]

copies

[H, 1] [A, 0] [AA, 0] [PD, 0] [D, 0]

[H, 0] [A, 1] [AA, 0] [PD, 0] [D, 0]

[H, 0] [A, 0] [AA, 1] [PD, 0] [D, 0]

[H, 0] [A, 0] [AA, 0] [PD, 1] [D, 0]

[H, 0] [A, 0] [AA, 0] [PD, 0] [D, 1]


[H, 0] [A, 1] [AA, 0] [PD, 0] [D, 0]

[R, 1] [V, 10] [E, 0] [W, 0]

copies

if a cell is A-infected, then it can generate [V, 11]

<if a cell is A-infected, then it can generate a virus>

R(1) A(1)

1A(2) V(10)

1

[A, 2] [R, 0] [A, 1] [V, 11]


[H, 1] [A, 0] [AA, 0] [PD, 0] [D, 0]

[R, 1] [V, 10] [E, 0] [W, 0]

copies

an healthy cell can become A-infected if it contains a virus

[V, 11]

V(11) H(1) H(12) A(0) A(12) W(0)

11 12 11

[V, 0] [H, 12] [H, 0] [A, 12] [A, 1] [W, 11]

Studied with conformon-P systems and cellular automata

If a cell is A-infected, then it can generate a virus.

An healthy cell can become A-infected if it contains a virus.

An AA-infected cell can generate [E, 1].

[E, 1] conformons can generate [E, 4].

An healthy cell can become A-infected if it contains [E, 4].

An A-infected cell can become AA-infected.

An AA-infected cell can become pre-dead.

A pre-dead cell removes viruses and E conformons present in it.

A pre-dead cell can become a dead cell.

If an healthy cell has at least one A-infected neighbour, then it becomes infected.

If an healthy cell has no A1-infected neighbours but at least 2 < R < 8 AA-infected neighbours, then it become A-infected.

An A-infected cell becomes AA-infected after time steps.

AA-infected cells become dead cells.

Dead cells can become healthy with probability prepl.

Each newly introduced healthy may be replaced by an A-infected cell with probability pinfec.

Studied with: rules

If a cell is A-infected, then it can generate a virus.

An healthy cell can become A-infected if it contains a virus.

An AA-infected cell can generate [E, 1].

[E, 1] conformons can generate [E, 4].

An healthy cell can become A-infected if it contains [E, 4].

An A-infected cell can become AA-infected.

An AA-infected cell can become pre-dead.

A pre-dead cell removes viruses and E conformons present in it.

A pre-dead cell can become a dead cell.

Studied with: neighbourhood

[V, 11] [E, 1] [E, 2] [E, 4]

Tests

cellular automata conformon-P systems

grid

neighbourhoods

pHIV

pinfec

400x400 torus 50x50 torus

3 kinds 3 kinds

0.05, 0.00005 0.05, 0.0004

0.00001, 0.00005 0.2, 1

Results: qualitative agreement

cellular automata

conformon-P systems

first weeks later years

Tests

cellular automata conformon-P systems

grid

neighbourhoods

pHIV

pinfec

400x400 torus 50x50 torus

3 kinds 3 kinds

0.05, 0.00005 0.05, 0.0004

0.00001, 0.00005 0.2, 1

M. C. Strain and H. Levine. Comment on “Dynamics of HIV infection: a cellular automata approach”. Physical review letters, 89(21):219805, 2002.

Results: overall

The conformon-P system model proved to be more robust to a wide range of conditions and parameters, with more reproducible

qualitative agreement to the overall dynamics and to the densities of healthy and infected cells observed in vivo.

The number of infected, healthy, and dead cells at the end of the third phase is not in accordance with the observed values.

About the rules

• rules are divided in two sets: part 1 and part 2;

• state-change rules and filling rules;

• the probabilities of the filling rules are equal in the two sets;

• the probabilities of the state-change rules are smaller in part 2

Future work

• obtain a better fit of the curve;

• study the simulation on bigger grids;

• simulate the best cure the infection;

• ...

Thank

you

david cornepierluigi frisco school of mathematical and computer sciences heriot-watt university...

Documents

grid of conformon

ba conformon

membrane computing

exampleconformonp systems

interactionconformonp

modulesa5 b4a conformon

interaction rule

g r3 g