On a discrete approach to cell dynamics in context to immune response against HIV

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On a discrete approach to cell dynamics in context to immune response against HIV

R. Pandey

To cite this version:

R. Pandey. On a discrete approach to cell dynamics in context to immune response against HIV. Jour-

nal de Physique I, EDP Sciences, 1991, 1 (12), pp.1709-1713. �10.1051/jp1:1991228�. �jpa-00246448�

Classification

Physics

Abstracts

05.50 87.10

On

_a

discrete approach to cell dynamics in context to immune response against HIV

R. B.

Pandey

Department

of

Physics

and Astronomy,

University

of Southem

Mississippi, Hattiesburg,

MS 39406-5046, U-S-A-

(Received

23 August 1991,

accepted

in

final form

2

September 1991)

Abstract. A network model of

interacting

cellular elements, such _as

macrophages,

viruses, T4 cells, and T8 cells, is introduced _to

study

the evolution of cell

population

in _an

early

HIV infection in _a cell mediated _response. The mean field treatment leads to two fixed

points (describing

«

immunocompetence

» and _«

immunodeficiency

») and _a limit

cycle

in infected states of

period

two.

Using

_a cellular _automata rule for the _nearest

neighbour

interaction, the

growth

of the cellular

population

is studied _{as a} function of their initial concentration p; on

increasing

p, a crossover from _an inununodeficient _{to an}

immunocompetent

state occurs at a characteristic

value _p~.

A considerable interest has been

recently

directed towards

studying

the immune response via theoretical models

[1-13].

In _a

typical

immune _response, the

antigen presenting

cells

(APC)

such _as

macrophages,

dendritic

cells,

and B-cells

engulf

the invader

(the antigen),

process

it,

and return it to the cell membrane. As _{soon as} the _«

helper (T4) lymphocytes recognize

the

antigens

_on the membranes' surface

they (T4 cells) begin

_to

proliferate

with activated T4 cells

releasing

chemical

signals,

the

lymphokines (especially IL2).

With the presence of

IL2,

the T4

lymphocytes

not

only multiply themselves,

but also induce

specific cytotoxic (T8) lymphocytes

to

proliferate.

As

such,

isolated T4 cells _are blind to free

antigens.

Suppressor

T8 cells inhibit the

production

of

lymphokines by

T4 cells in order to switch off the response. In AIDS

[14],

HIV viruses have an

ability

to escape such cell mediated attack

(as

well _as humoral

aspect involving specific antibodies).

Conforrnational

complementarity

of the viral

protein (gp120)

and the surface receptor

(CD4)

of T4 cells

help

in fusion of the viral membrane with the T4 cell

leading

to release of viral _cores into the interior of the T4 cell.

Production of _new virus from infected cells results in death of the infected T4 cells. A discrete

approach

is used here to model the

growth

of the cellular

population

in this _context. We would like to

point

out

that,

there have been _numerous theoretical studies

by

_a

variety

of models in _{recent years}

[2, 9, 12, 15-17] although

with very little _success in

understanding

the clinical

findings thoroughly

and

therefore,

_apart from _{a new} theoretical

study (see below),

one should _not expect a

breakthrough

in this article either.

To describe the state of _a

cell,

_we

employ

_a

simplified binary representation

that has been used

frequently

in recent years

[6-13]

in

modeling

the immune _response where

logical

1710 JOURNAL DE

PHYSIQUE

I M 12

expressions [18]

are used to construct interactions _among cellular elements. The

binary values,

0 _{» («} false

»)

and _« I _{» («} true

»)

^of a cell

type represent

the states in which the cell is in its low and

high

concentrations. Here, _we consider

only

four cell types

macrophages (M),

virus _or viral infected cells

(V),

T4

cells,

and T8 cells.

First,

_we consider the « mean field _» like interaction where the concentration of each cell

type

^is described

by

one value for the whole

system.

^In a cell mediated _{response, we}

attempt

to

incorporate

some of the main functions of cellular elements in which

macrophages (M) display antigenic _component

of virus to the T4 cell _; cells

bearing

CD4

proteins (which

includes both T4 cells _as well _as

macrophages)

attract HIVS. T4 cells _are killed

by

viruses whereas _most

macrophages

_are not, rather,

they provide

shelter to these viruses and _{act as a} reservoir in _a HIV infection. In coordination with T4

cells,

the T8 cells

proliferate

_as

they

receive

lymphokine signals

from T4 cells and kill the viral infected cells. In _our earlier

study [9],

the role of APCS _was

completely ignored

inclusion of _some of their functions not

only provides

a more realistic model but also

brings interesting

results. We propose a

following

set of intercell interactions _:

M(t

₊ I

=

M(t)

_or V

(t)

V(t

₊

Ii

=

(V(t)

_or

(M(t)

_or T4

(t))j

and

(not (T8 (iii j T4(t

+

Ii

₌

(M(tj

or

T4(tjj

and

(not (V(tjjj

T8(t

+

Ii

=

T4(tj

and

(M(tj

and V

(iii

where the

growth

states of the four cell

types

at time

step

t + I _are evolved from their

growth

states at

previous

time step t. The first

equation

describes the

growth

of activated

macrophage population

which will be in _a

high

concentration _{state at} time

step

t + I

(I.e. M(t

₊ I

=

Ii

if

it _was

already

in this state

(M(t)

=

I)

at time

step

t

(I.e.

the

macrophages

_are self-

propagating)

_or if _a viral infected cell _was present

(V (ii

=

I

),

_or both _are present ^{in their}

high

concentration state the presence of a virus activates

macrophages.

The second

equation represents

^{the virus}

multiplication

in which the viruses _grow

only

if the

cytotoxic

T8 cells _are absent

(T8(t)

=

0) and,

in

addition,

at least _one of the

remaining

cells

including

the virus itself is present at its

previous

time step. ^{Note that}

high

concentrations of T4 cells and

macrophages

enhance the viral

growth.

Proliferation of T4 cells is

accomplished

via the third

equation,

in which the virus must be absent

(V (11

=

0) and,

in

addition,

either

macrophage (along

with MHC

III

or T4 cells _or both be present in their

high

concentration state at time

step

t, in order _to

produce

T4 cells at time

step

t + I.

Finally,

the fourth

equation gives

the

growth

of

cytotoxic

T8 cells which _are activated

by

T4 cells in order to kill viral infected cells

presented by macrophages.

Thus _a T8 cell will

multiply (I.e. T8(t

+ I

)

=

Ii

if all other cell

types

(I.e.

T4

cells, macrophages,

and

viruses)

_are

_present

_at

previous

time

step

t.

The overall immune-status is described

by

the

configuration

of the four

binary

cells. There

are sixteen

possible

^such

configurations

but not all of them _are stable. Let _us

represent

the

configuration

of these cells in _a

binary representation (M, V, T4, T8).

If _we start with any of these sixteen

configurations randomly

_we either end _up in two fixed

points, (o000)

and

(1100),

or

get trapped

in _a triad

cycle

of

configurations, (1000) (1l10) (l101).

In the first stable

immunological configuration,

0

=

(oooo),

since there is _no

stimulatory

cell

present,

we call this _a

«fully immunocompetent»

state which is also referred _as the

« native _{» or «}

unchallenged

state. This does _{not mean} that there _{are no} cells in the

body,

it

simply implies

that cells that _were stimulated to grow

during

the immune response are no

longer

activated. On the other

hand,

in the second stable

configuration

12

=

(l100), only

macrophages

and viruses _are present, where the

macrophages (in

the absence of other

stimulatory cells)

act as reservoir for HIV.

Therefore,

_we call this fixed

point

the

fully

immunodeficient _» state. We

interpret

the triad

cycle configurations

as follows _: in

configur-

ation 14

=

II10)

all cells _are present except

cytotoxic

T8 cells and therefore _we call it _an

« infected _» state because the presence of

macrophages

and T4 cells may stimulate other cell types

including

T8 cells. The

configuration

13

=

(1101)

_represents the state in which all cells

are present except ^T4 cells which orchestrate the functions of other cells in immune response, and therefore _we call this

configuration

a «

severely

^infected » state. On the other

hand,

in

configuration

8

=

(1000)

all cells _are absent

except macrophages

and _we call it _{a «} suscept,

ible _» state. Note that the

competition

between the viral attack and immune defense cells

leads to this limit

cycle

in which the immune system ^{oscillates in} a triad

cycle

of _« infected _»,

«

severely

infected _», and _«

susceptible

» states

periodically.

Opposite

to the infinite range interaction

just discussed,

_{now we} consider the nearest

neighbour

intracellular interaction

[9]

^with a

spatial

distribution

[19]

^{of cell}

types.

^For

simplicity,

_we restrict ourselves here to _a

simple

cubic lattice _{as a} host medium.

First,

_we

generate

a

simple

cubic lattice of size L*L*L and then _we

place

four cell types ^{at each lattice} site.

However, only

_a small fraction

p(I)

of these cell

types I(=

I

(Ml,

2

(V),

3

(T4),

4

(T8 ii

is

assigned

the

high

concentration state « I _» while the rest, a fraction I

p(I)

of

cells,

is

assigned

the low concentration state «0». The

binary

states «0»

(empty)

and «I

(occupied)

of _a cell type ^I at _a site _may also later

specify

the _occupancy status of the site

by

cell

type

^I.

Thus,

_a lattice site _can be either

empty

_or

occupied by

each of the four cell

types

^and the number of sites

initially occupied by

_a cell type ^{I is}

N,

= p

(I)*

N where N is the total

number of lattice sites. Here _we

study

how

N~

grows with time and how it

depends

_on

interaction and other parameters relevant to immune

system.

In addition to the above basic intercell

interactions,

_we consider _an intersite intracell interaction _as follows. A cell

type

I is selected _{at a}

site,

_say

j.

The

binary

state of this

cell,

_say

c~(t),

at current time step t is then added to the

binary

states of the _same cell type I at the

neighboring

six sites. If this _sum of _seven

binary

states is

positive,

then _a temporary

binary

state

c;(tl'

of

high

concentration (« I

»)

is

assigned

to the cell type ^I at the site

j,

otherwise

(I.e.

if the _sum is

zero)

the temporary

binary

state of cell type I at site

j

remains in its low concentration state (« 0

»),

In other

words, c,(tl'

is the result of

logical

« or »

operation ii

_5] of the _seven lattice sites involved. The same

procedure

is used _to

assign temporary binary

states to all other cell

types

at site

j.

With their

temporary binary

states, c,

(ii',

the four cells at this site

j

then interact

according

to intercell interaction discussed

above the

resulting

states _are then

assigned

_as states at time step t ₊ I. This _process of

selecting

_a

site, evaluating

the intermediate cellular _states

by taking

into account the intersite

intracell interactions for all cell types ^at this

site,

and then

assigning

final

binary

states to each cell

emerging

from the intercell interaction at the next time step ^is

repeated again

and

again

for each cell

type

at all lattice sites for _a number of time

steps

^{in which the}

growth

and

decay

of each cell type ^{reaches their}

steady

state value, To obtain _a reliable estimate of the

steady

state

populations

of each cell types, we use several

independent

_runs.

The computer ^{simulation is}

performed

_{on a}

Cray

XMP and YMP machine where _a

multispin

code

[15]

is used for

storing

the lattice sites. The status of _a lattice site where _a cell

type

^{is in its}

high

_(« I

»)

_or low _(« 0

»)

concentration _state is described

by

a

single

^bit of _a 64- bit word

using

_{« yes »} and _{no »}

logic description. Samples

of two different

sizes,

64*64*64

and

192*192*192,

_are used _to generate ^{data for the}

population

of each cell type, ^{in which 5}

independent samples

_are used to find their average values. The maximum

speed

_was found to be about 152

steps

per site _per microsecond _per YMP processor and _was about 33

percent

slower _on the XMP.

On _a 64*64*64

lattice,

the lowest

possible

initial concentration of _a cell

type

^{I is}

1712 JOURNAL DE

PHYSIQUE

I M 12

p(I )

= 0.000005 in which

only

one site is

occupied by

_a cell type I. With the nearest

neighbor

interaction

(of

the cellular

automata)

rule described

above,

intercell interaction leads to a

steady

state in which the number of

macrophages

and T8 cells

approach

a maximum value

(typically

the size of the

sample,

I-e- all sites _are

occupied by

at least these two cell

types)

the number of viral infected cells

(virions)

and T4 cells oscillate about their _mean values say

N~

and

NT4.

A

typical

variation of the

population growth

of

virus,

T4 cells and T8 cells with

equal

initial concentration for all cells

except

that of virus which started at _one site

only,

is

shown in

figure

I. We observe

that,

the

steady

state

population

of virus is

larger

than that of the T4

cells,

if the initial concentration of defensive host cells

(T4 cells,

T8 cells and

macrophages)

is lower than _a characteristic value

P~(I)

above which the

steady

state

population

of T4 cells is

larger

than that of the virus. This exhibits _{a crossover} from _an

«immunodeficient» state to _an

«imrnunocompetent»

state at

p~(I).

The

population

of

macrophages

remains constant at its maximum value

(I.e.

the size of the

sample)

and that of

the T8 cells also remains constant but at _a lower value

depending

_upon its initial

concentration. The relaxation time _to

approach

the

steady

state

population depends

_upon the initial concentration of host cells in which the lower the

concentration,

the

longer

the time it takes to reach the

steady

state. Note the difference in results from the _mean field treatment of the

preceding

section. Instead of two stable fixed

points

and _an

oscillatory

state with the infinite range

interaction,

here in the _n-n-

interacting model,

_we observe

only

two states

(I.e.

« imrnunodeficient _» and «

immunocompetent »)

which

depend

_upon the initial concentration

O.75 t/l

d

m CJ

>

~ O.5

d

z o CJ

O.25

O

O-05 O.06 O.09 O-12 O-15

P

Fig.

I- Concentration of viruses

((+)

maximum,

(*) minimum),

and T4 cells

((O)

maximum,

(x) minimurn)

with respect to the

population

of T8 cells

(*,

on the

top) (I.e.

the concentration of virus

=

number of viral infected

cells/number

of T8

cells)

_versus initial concentration of host cells

(T4

cells, T8 cells and

macrophages).

A 64*64*64

sample

with 5

independent

runs is used to evaluate the

steady

state

populations ^{of the four cell} type ; number of macrophages reach their maximum value (I.e. ^the size of the

sample).

of the cell

types suggesting

that this

spatial approach

is _more

appropriate

than the _mean field

approach. Thus,

the

simple

model

presented

here does capture some of the essential features of the HIV

infection, especially

the decline of T4 cell

populations.

We

hope

to

incorporate

more realistic factors in _our continued efforts to understand

complex

issues in immune response in HIV infection.

The author would like to thank Dietrich Stauffer for discussion and valuable

suggestions.

The financial

support

^and

computer

^time

provided by

HLRZ

during

_{a summer} visit to HLRZ Juelich where this work _was done

partial support

^from a Research

Corporation

_grant

(#

C-

2903)

is also

acknowledged.

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