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Immune response via interacting three dimensional network of cellular automata
R.B. Pandey, D. Stauffer
To cite this version:
R.B. Pandey, D. Stauffer. Immune response via interacting three dimensional network of cellular au- tomata. Journal de Physique, 1989, 50 (1), pp.1-10. �10.1051/jphys:019890050010100�. �jpa-00210897�
1
LE JOURNAL DE PHYSIQUE
Immune response via interacting three dimensional network of cellular automata
R. B. Pandey (*) and D. Stauffer
Institute for Theoretical Physics, Cologne University D-5000 Cologne 41, F.R.G.
(Reçu le 15 juin 1988, accepté sous forme définitive le 16 août 1988)
Résumé. 2014 Nous étudions numériquement la dynamique d’un système de type réseau de
neurones à trois sortes de cellules sur un réseau cubique. Notre but est de comprendre la réponse
immunitaire (a) dans les maladies autoimmunes et (b) dans le cas de faiblesse immunitaire. Nous utilisons dans chaque cas deux types d’interactions indépendantes. Dans le cas (a), le nombre de sites infectés croît avec la loi (temps )3,3. Le temps de saturation nécessaire pour infecter tous les sites varie approximativement comme p-0,3, où p est la concentration initiale de cellules, sauf dans le cas de cellules tueuses activées où, pour une interaction, il varie comme p-0,5. Dans le cas (b),
nous étudions l’évolution des cellules infectées pour des mélanges binaires d’interactions aléatoires d’intensité B. Quand nous augmentons l’intensité B, le nombre de cellules infectées viralement croît et le nombre de cellules T4 décroît de façon monotone. Dans un cas spécial
d’interactions aléatoires fluctuantes, nous observons une anomalie dans la variation des cellules T4 en fonction du temps dans un domaine restreint de B (près de B = 0,9).
Abstract. 2014 A computer simulation is used to study the dynamics of neural-network like system of three cell types on a cubic lattice in an attempt to understand the immune response (a) in
autoimmune disease and (b) in immune weakness each with two kind of independent interactions.
In case (a) the number of infected sites seems to grow with (time )3.3. The saturation time required
to infect all the sites varies roughly, with the initial concentration p of the cells, as
p-0.3 for all the cell types except for the activated killer cells where it varies as p-0.5 with one
particular interaction. In case (b), the evolution of infected cells is studied for binary mixtures of
random interactions of strength B. The number of viral infected cells grows and the number of T4- cells decreases monotonically on increasing the interaction intensity B. For a special case of
annealed random interaction, an anomaly is observed in the variation of the T4-cells as a function of time in a narrow regime of B (near B = 0.9).
Tome 50 N° 1 JANVIER 1989
J. Phys. France 50 (1989) 1-10 1er JANVIER 1989,
Classification
Physics Abstracts 05.50 - 87.10
1. Introduction.
This paper deals with biologically motivated [1] studies of complex cellular automata. For the
reader not interested in biological applications we can explain this and similar models [2-5]
(*) Alexander von Humboldt fellow ; address after July 1988 : Department of Physics and Astronomy, University of Southern Mississippi, Box 5046, Hattiesburg, MS 39406-5046, U.S.A.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019890050010100
even simpler : each site carries several (here three) spins si = 0 or 1, each of which is either up
or down. Each spin can interact with an (non-symmetric) exchange energy which is 0, 1, or - 1. The orientation of each spin at time t + 1 is determined by the sum of the interactions it is
feeling from the other spins at time t. The details of the interaction can be formulated by
thresholds for the sum of the interactions or, more easily, by logical operations, as detailed
later. In a mean field approach one considers only one such site, in a computer simulation one
studies sites on large lattices. Already in a mean field approach with only three spins, we have 39 = 19 683 possibilities to construct models ; only three of them are checked here for their time development and fixed points.
Biologically, the study of the dynamic evolution of interacting cellular automata in neural-
network like models for immune response has attracted a lot of interest. An immune system
responds in a specific but complex fashion when foreign antigens enter the body as a result of
viral attack, for example. Immune systems consist of a variety of cells like T4-cells, T8-celles, B-cells along with other elements such as myeloid. These elements participate randomly but cooperatively in chain reactions trigerred by the antigen-invasion to fight against infections. A
variety of mechanisms have been proposed in the literature for such complex response, but here we will concentrate on the network approach proposed by Jeme [1], and recently implemented by Weisbuch and Atlan (WA) [2] for the control of the immune response. They
consider five automata in which the first automaton represents killer cells in a resting state,
the second represents killers activated by the presence of the antigen, the third and fifth act as
suppressor cells while the fourth is a helper cell. The binary state of each automaton represents the concentration of the corresponding cell type : zero refers to small concentration while one to high concentration. These cells interact with each other with strength
Cik = 1, 0 or - 1 (see Ref. [2] for details). One starts with a random configuration (with binary states si = 0 or 1) of these cells and then, at the next time step, the binary state (i. e . the concentration) of one cell type is unity if the sum of interactions (with itself and with other
cells) is positive and zero for nonpositive sums. With this interacting network model of WA, if
one starts with any one of the 32 possible configurations, one ends up in one of two basins
(fixed points) in which one fixed point represents a state in which all five binary cells are zero (i.e. with their concentration zero) and the other fixed point describes the state in which all but the second cell, are in binary state one (with high concentration) ; the two configurations (S5, S4, S3, S2, Sl) may be represented as binary numbers (00000) and (11101) and correspond
to 0 and 29 respectively. The two attraction basins are interpreted as a virgin state (absence of
any cell type specific to the antigen, (0)) and a healthy carrier state (29).
Dayan et al. [3] have recently extended this model from its neural-network like description (of an infinite range interacting system or a mean field approximation) to a nearest neighbor interacting network model on a two dimensional lattice whereas Wiesner [4] has studied it in
three dimensions. We leave it open here whether the different lattice sites correspond to
different individua or to different parts of one single organism ; the answer to that question depends on the ratio of the time for the immune system to react to the time for the infection
spreading throughout one body. Starting from a random distribution, they [3, 4] end up with
only one attraction basin, a healthy carrier state 29, where at all sites all cell types except activated killers are present. This generalization of the simple WA model of MF-type to an interacting lattice model leads to a simpler result, from two attractor basins to one attractor basin. Instead of adding more complexities to the WA model, one of us [5] has recently
considered a more simplified version in which only three types of cells are involved ; interestingly this simpler version brings more informative results. Decreasing the number of the different types of cells from five to three increases the number of fixed points from two to
five. In the following section 2 we describe this simplified model, a more simplified version of
3
WA and two other three-cell models in an attempt to understand the immune response in AIDS ; their mean field results are briefly mentioned. In section 3 we present results from computer simulations when the first two interactions are placed on a simple cubic lattice
where cells at each site interact with their neighboring cells ; in section 4 we describe a
computer simulation of latter two interactions on this lattice. We summarize our results in section 5.
2. Models.
In the following we describe three-cell interactions and discuss their mean field results : 2.1 AUTOIMMUNE DISEASE. - (i) Stauffer’s simplified version, as we have mentioned
before, consists of three types of cells : killer cells of type 1, activated killer cells of type 2 and suppressor cells of type 3. All cells are self propagating and also have intercell interactions such that killer cells enhance and suppressor cells reduce the concentration of activated killers. If si represents the binary state for the concentration (0 for low and 1 for high) of cell
type i at time t, then interactions lead to a binary state,
where Cik are interaction strengths (Cii = 1, c21= c31=1, c23 = - 1 and all other Cik = 0), hi are thresholds which are taken to be 1/2 as in the original WA-model ; the step function
8 (x ) = 1 for x > 0 and 0 for x , 0. This equation involves only one site with 23 = 8 states (3
types of cells) and therefore is in accord with the mean field description of WA. It is easy to
verify that interaction (1 leads to five fixed points, denoted by (S3 S2 S1 ) : uninfected (000),
immunized (100), carrier (101), death by infection (111) and death by poisoning (010) with probability 1/8, 1/4, 1/8, 3/8 and 1/8 respectively, if we start randomly from one of the eight
initial possibilities. (The model was constructed to give such non-trivial results ; we do not
assert to have biological evidence for each value of Cik-)
(ii) A more simplified version which involves only two cells of types 1 and 2 with thresholds 1/2 and 3/2 respectively, gives four attraction basins, the same as in the above version except that death by poisoning is missing. Note that all the 4 possible configurations are 4 fixed points
with equal probabilities.
When we place these interacting three cell types on a lattice with a nearest neighbor interaction, as did Dayan et al. [3] and Wiesner [4] with the five cell model of WA, then we end up with only one configuration in which all sites reach the state (111) of death by
infection. In section 3 we study the temporal evolutions of these cells.
2.2 IMMUNE WEAKNESS. - There are various other three cell interactions which lead to
interesting fixed points. In the following we discuss two such interactions which may have
some application in understanding the immune response in AIDS where the immune system is weakened.
The chain reaction in the immune response to the AIDS virus involves a complex
interactions among, and a cooperative random evolution, of various cells such as B-cells, T- cells, macrophages and antigens produced by virus. At present, a complete understanding of
how these interacting components of the immune system evolve and fight against the deadly infection, is lacking. However, it is known that, in the case of the AIDS retrovirus, immunosuppression results from viral infection of T4 lymphocytes which act as inducer and
helper cells. As a helper, T4-cells play a very important role in fight against infection. Having recognized a specific antigen [6], the helper T4 cells enable cytotoxic T lymphocytes (a killer
T8 cell) to destroy cells bearing the antigen and B lymphocytes to secrete appropriate antibody. On a very crude level, we introduce here interactions which involve three cells i.e.
T4 cells, T8 cells and the antigens produced as a result of virus, we call them the viral (V)
cells.
We explicitly write the assumed interactions in Boolean expressions. We define the current status of a cell type i by IS (i ) = 0 (« false ») or = 1 (« true »). These binary states refer to low
and high concentration of cell type 1 and 2 respectively, while for the cell type 3, it is other way around i. e - IS (3 ) = 0 (« false ») refers to the high concentration. ISN(I) represents the
new binary state of the cell type i with the interactions considered in the following. (The
Boolean notation used here is not a deviation from the threshold principles used in WA. For
example, IS(1) .and. IS(3) corresponds to threshold 3/2, whereas IS(1) .or. IS(2) leads to
threshold 1/2. This Boolean notation merely helps to implement the model in bit-by-bit
Fortran computer statements ; using different thresholds for different cell types looks more complicated also to the reader even through it is equivalent).
(iii) Since, the cell type 1 (i.e. the T4 cell) is infected by virus (antigen), i.e. the cell type 3,
it is plausible that the concentration of T4 cells, after their proliferation induced by viral invasion, is high when the concentration of viral cells is low. The number of cell type 2 (T8 cells) grows as the immune system responds, but they do not act independently. T4 cells
prepare the viral infected cells, which are then attacked by T8 cells. This mechanism can be described by,
which leads to two fixed points : (010) in which T8 cells and viral cells are present with probability 3I4 and (000) in which only viral cells are present with probability 1/4 leading to
death.
(iv) We can consider another interaction which may also take into account the above mechanism in immune response in which the rules for updating the status of cells type 1 and 2 remains the same, but for the cell type 3, the interaction becomes,
This interaction gives three fixed points : (1) (111) where T4 cells and T8 cells are present with
probability 1/4, (2) (110) where only T8 cells are present with probability 1/2, and (3) (100)
where all the cell types are absent with probability 1/4. (There may be many other rules and combinations thereof which may yield similar or more realistic effects [6], also with more than only three cell types ; work along these lines is continuing.)
3. Interactions (i) and (ii) on a simple cubic lattice.
We consider a simple cubic lattice of size L * L * L with three types of cells at each site. The
multi-spin-coding trick [7] is used to perform simulation on large lattices. One cell type configuration is stored in one bit of a word with a « shift » operation on a scalar CDC Cyber
machine. Three words are used to address three cells, each word with sixty sites of one cell type.
5
At the beginning, a fraction p of each type of these cells is randomly assigned a binary state
1 of high concentration and the rest (fraction (1 - p)) a binary state 0. Biologically this might correspond to an early stage of the sickness, when an initial point-like infection has already spread. For very low concentrations p the whole lattice initially corresponds to one or several
well seperated pointlike infections. Each type of cell interacts with its neighboring cells of the
same type. The binary state of the cells depends upon the states of their six neighbors and
their own state, at the previous time step. Let us consider a cell k at site i. To update the state
of this cell, first we look into the states of the cell type k at the center site and that of the k- cells at the neighboring sites. If the sum of sk over these seven k-cells is positive then a temporary state of high concentration (sk =1 ) is assigned to the cell k. Similar updates are
made with the other sk with the nearest neighbor interactions for k = 1, 2 and 3 at this site. All the cells at this site then interact with the specified interactions i.e. simplified version (i) or (ii) depending upon the system. The whole process is repeated for each site for several
independent initial configurations. We have studied the temporal evolution of the number of each cell type separately for the two interactions (i and ii) for various small initial concentrations p with binary state 1. A typical variation of the number Nk of k-cells with time
t is shown in figure 1. We find that the number of infected cells grows with time with a power law exponent z,
with an effective z of about 3.3. From our data we observe a very small variation in the value of the exponent z on varying the concentration p, which may be due to statistical fluctuations.
In the limit of very small concentrations we expect z = 3 since then the infected cells form clusters around isolated infection centers, with cluster radii proportional to time.
Let us define the saturation time tk as the time required for the k-cells at all sites to be in binary state 1. There may be two different values for the saturation time : first the average value (tk) and second, the maximum value max tk. Figure 2 shows the variation of these saturation times with concentration p. We note that the growth of the infected cells (Fig. 1)
and therefore their saturation time tk does depend upon the concentration p - the lower the
Fig. 1. - Number Nk of infected sites versus time on a log-log plot. For interaction (i) circles and for interaction (ii) triangles are used with thickness increasing with increasing cell type 1-3. System size
60 x 60 x 60 with 500 independent samples is used. The initial concentration is p = 0.0002.
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concentration, the slower the growth. In model (i) the variation of the saturation times with concentration p for cell type 1 (killer) and type 2 (active killer) are essentially the same while
that of cell type 3 (suppressor) shows a slight shift (Fig. 2). On the other hand, for interaction
(ii), the variation of cell types 1 and 2 with concentration p shows a substantial shift, while the
variation of cell type 3 is very close to that of cell 1. To show the finite size effect we have
plotted data for two sample sizes 60 x 60 x 60 and 15 x 15 x 15. Apart from some statistical fluctuations, these data show a good linear fit for a power law dependence of the saturation
time tk with concentration p, i.e.
We use data for 60 x 60 x 60 lattices to evaluate the exponent x. For almost all cell types we find x - 0.3 except for the cell type 2 (the activated killer cells) in model (ii) (Fig. 2a) where x
is about 0.5. These exponents are quite different from the easily explained logarithmic variation, j - 0, observed by Dayan et al. [3] in the WA five-cell model.
4. Interaction (iii) and (iv) on a simple cubic lattice.
Now, let us consider the interacting cells with interactions (iii) and (iv) on a simple cubic
lattice with the same rules for the nearest neighbour interactions as with interactions (i) and (ii) in section 3. However, we study here the evolution of the number of cells for the random mixture of interactions (iii) and (iv) i.e. we use inhomogeneous cellular automata [8]. At each
lattice site, interaction (iii) is used with a probability B while interaction (iv) with probability (1- B ), where 0 -- B = 1 . For limiting values of B, 0 and 1, one recovers the deterministic cellular automata for interactions (iv) and (iü) respectively. A variety of random mixing is possible, but we consider here the following two kinds [10] :
1. Annealed interaction : at each time step, each lattice site is, randomly and indepen- dently, assigned interaction (iii) with probability B and interaction (iv) with probability
(1- B ).
2. Quenched interaction : interactions (iii) and (iv) are assigned with probabilities B and (1 - B) to cells at each lattice sites randomly in initialization and these binary random
interactions are then fixed the same throughout the evolution of the simulation.
We have studied in detail the evolution of the infected T4 cells, T8 cells and viral cells as a
function of B for their various initial concentrations p, both random interactions (1) and (2).
A typical variation of the number of cells with time t for interaction (2) at p = 0.0005 is shown in figure 3 for several values of B. At B = 0, the number of T4 and T8 cells (Ni and N2) increase with time and reach quickly their saturation (equilibrium) value, i. e. the size of the sample, while the number of viral cells N3 remains at the lowest level (N3 = 0 ). On increasing the value of B, the asymptotic value of the number of the viral cells (type 3)
increases while that of T4 cells decreases monotonically, and at the extreme value of B (= 1), N becomes 0 whereas N2 and N3 attain the maximum value, the size of the system. At this limit (B = 1 ), all the lattice sites are infected by AIDS virus and all the T4 cells are killed by
them. The qualitative nature for the evolution of the number of these cells as a function of B
and p remains the same for annealed random interaction (1). The mean field model also
makes sense in the annealed case ; we then end up with no T4 cells, either always or never a
T8 cell (depending on the initial configuration), and a fraction B of viral cells. If we update all sixty sites within one line by the same random number, i.e. the same random interaction for all sixty sites in the annealed case, then we observed an anomaly near B = 0.9 in which both the number of T4 cells and viral cells exhibit a maximum peak at a certain time (see Fig. 4).
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9
Fig. 3. - Number of T4 cells (0), T8 cells (D) and viral cells (0) versus time on a 60 x 60 x 60 lattice
with 200 independent samples at p = 0.0005 for B = 0.0 (a), 0.2 (b), 0.4 (c), 0.7 (d), 0.9 (e) and 1.0 (f)
with quenched interactions (iii) and (iv).
Fig. 4. - Number of T4 cells versus time at p = 0.0005 for B = 0.9 with a special annealed interaction in which sixty consecutive sites along one line of the cubic lattice are assigned the same random
interaction. Statistics are the same as that in figure 3.
