A formal model of an artificial immune system
Alexander Tarakanov
a, Dipankar Dasgupta
b,*
aSt Petersburg Institute for Informatics and Automation of the Russian Academy of Sciences,St Petersburg,Russia bMathematical Sciences Department,The Uni
6ersity of Memphis,Dun Hall,Room378,Memphis,TN38152-6429,USA
Abstract
The paper presents a mathematical model based on the features of antigen – antibody bindings in the immune system. In the natural immune system, local binding of immune cells and molecules to antigenic peptides is based generally on the behavior of surface proteins. In particular, immune cells contain proteins on their receptors, and apparently, these proteins play the key role both in immune response and recognition processes. In this work, we consider the immune cells in the form of formal B-cell and formal T-cell and develop a mathematical model of their interactions. We refer this model as the formal immune system (FIS). The paper provides an analysis of a network of bindings (or interactions) among the formal proteins of the FIS. © 2000 Elsevier Science Ireland Ltd. All rights reserved.
Keywords:Formal B-cell; Formal T-cell; Formal protein; Immune networks
www.elsevier.com/locate/biosystems
1. Introduction
The primary role of the immune system is to distinguish between the self (body cells and tis-sues) and the non-self (antigens). In general, the immune system recognizes foreign cells and molecules by producing antibody molecules that physically bind with antigens (or antigenic pep-tides). Specifically, antibody – antigen interactions take place among different peptides (paratopes and epitopes), and the discrimination between peptides are arising from self-proteins and those derived from foreign proteins (Perelson and
Weis-buch, 1995). Apparently, proteins play the key role both in immune response and recognition processes. However, for the antigen and antibody molecules to bind, three-dimensional shapes of proteins must match in a lock-and-key manner. For every antigen, the immune system must be able to produce a corresponding antibody molecule, so that the antigen can be recognized and defended against (Janeway et al., 1999).
From the computational viewpoint, the main functions of information processing in cells and tissues are realized by proteins. In fact, proteins recognize and execute programs (instructions) represented in the form of genetic code. Accord-ing to neurologists, proteins control the electrical activity of the brain as the neuro-mediators and the receptors of neurons. Similarly, proteins are also considered as the main components of the
* Correponding author. Tel.: +1-901-6784147; fax: + 1-901-6782480.
E-mail addresses:[email protected] (A. Tarakanov), [email protected] (D. Dasgupta)
immune system, i.e. the receptors of B-cells and T-cells, antibodies and messengers (Anderson 1995).
Based on the principles of the immune system, a new computational technique, called the artifi-cial immune system (AIS), is rapidly emerging, which appears to offer powerful and robust infor-mation processing capabilities for solving complex problems. Like artificial neural networks (ANNs), AISs can learn new information, recall previously learned information, and perform pattern recogni-tion in a highly decentralized fashion. Researchers have started using AISs in many applications including information security, vaccine design, fault detection, data mining, robotics, etc. (Das-gupta, 1999). However, compared with ANNs, the field of AISs has not yet developed its own mathematical basis. In fact, most of the AISs represent some hybrid and heuristic algorithms, using ideas from genetic algorithms, cellular au-tomata, ANNs, etc.
This paper introduces a notion of formal protein as a mathematical abstraction for key biophysical mechanisms of proteins’ behavior. In the frame of interactions between formal proteins, we determine networks of binding refer to as formal immune system. We show that even the simplest variants of such networks demonstrate important properties of immune response and im-mune memory.
2. Formal protein
Though the biological information processing in cells and tissues is very complex, abstract mod-els of their functionality can be represented by rather simple and general mechanisms (Matsuno and Paton, 1999). A good example is the repre-sentation of the double helix for chains of molecules that stores the genetic code. This spatial structure is formed by weak interactions between strictly adjusted molecular shapes, disposed in the same plane. No doubt that this is one of the most significant examples of geometrical correspon-dence at the biomolecular level.
There exist several immune system models (De-Boer et al., 1992) that highlight spatial
conforma-tions of proteins. However, most of them are complicated because of the consideration of three-dimensional (3D) geometry of the protein chain as well as many types of natural interactions that determine the native form of any protein. For example, Van der Vaals’ interactions, electrostatic forces, momentum of dipoles, hydrophobic forces, etc. (Cantor and Schimmel, 1980).
Nevertheless, there exist convincing evidence for the following principles of information pro-cessing by proteins:
1. function of any protein depends on its spatial conformation;
2. this conformation, in its own turn, is deter-mined by the sequence of amino-acids that encode a given protein.
Based on the above principles, Tarakanov (1998) has introduced a notion of formal protein, or formal peptide (FP). This notion represents a mathematical abstraction for the biophysical prin-ciple of free energy dependence over the space conformation of protein chain, rather than a com-putational model of natural proteins. In that sense, this notion may be analogous to the notion of artificial (or formal) neuron, which is highly simplified relative to the natural neuron, and in-tended to develop problem-solving techniques.
However, the proposed FP unites the two main ideas:
algebraic description of protein chain geometry by means of quaternions;
definition of free energy as a function over elements of quaternions.
In our mathematical formulation, the FP is defined as an ordered 5-tuple
P=n, U, Q, V,6
which comprises the following components: 1. number of links n\0;
2. set of angles U={8k,ck}, k=1,...,n, where
−p58k5p, −p5ck5p;
3. set of unit quaternions Q={Q0,Qk}, where
quaternions Qk=Qk{8k,ck}defined by the
Eqs. (2) and (3), and the resultant quaternion of FP, Q0 defined as their product:
4. set of coefficientsV={6
ij}, i=1, 2, 3, 4, j]i;
5. function6 (without index) is defined over the
elements of the resultant quaternionQ0by the
following quadratic:
6= −% j]i
6
ijqiqj (1)
In this definition, unit quaternions Q1,...Qn
have the following expression:
Q(8,c)=q1H0+q2H1+q3H2+q4H3 (2)
matrices of Pauli. It is to be noted that the angles of FP, u, h correspond to the fixed angles of valence between atoms of nitrogen and carbon within the molecule of a protein, and the angles
8k, ck correspond to the angles of rotation of
protein’s chain around the axes of valence, called torsion angles in biophysics.
Consider the values of torsion angles U as states of FP, and values of coefficients V as controls of the FP. Consider function 6 as free
energy of FP, and let us introduce a configuration vector of FP as column-vector [Q]=[q1q2q3q4]T,
where the elements are the constituents of resul-tant quaternion Q0. Consider also the symmetric
energy matrix M(6)=[mij], i,j=1, 2, 3, 4,
deter-mined by coefficients of quadratic Eq. (1) as
follows: mii=6ii, mij=mji=1/26ij, i"j. Hence,
the free energy of FP can be represented in vec-tor – matrix form as:
6= −[Q]TM(6)[Q] (4)
Therefore, FP described by four real numbers (quaternion), which can be computed as a product of elementary quaternions, corresponding to the links of a protein (amino-acids). Every elementary quaternion can be computed as a function over the two torsion angles of rotation. Besides, the FP’s free energy is computed as a sum of mutual products of quaternion’s elements with given weight coefficients.
As the result, we have torsion angles as inner parameters (states) of FP, and the weight coeffi-cient of the quadratic form — as outer parame-ters (controls). The values of inner parameparame-ters, which give local or global minimum to free energy calculation as a threshold, correspond to a stable state of the FP. The model demonstrates some important properties of proteins, such as self-or-ganization resulting in stable state (self-assembly), and its dependence from the number and the order (non-commutativity) of links. Besides, the transitions between the stable states of FP, outer influence correspond to storing and retrieving in-formation from memory.
3. Networks of bindings
The main function of a protein is its binding with another protein (or a molecule). Such a binding is highly specific (selective), because it depends on the existence of highly adjusted local shapes of interacting proteins. Fig. 1 shows that the protein P binds only with the molecule Q3.
As a result of binding, the protein can change its spatial shape (called allosteric effect). Further-more, by allosteric effect a protein can receive an ability to bind with such a molecule (antigen, antibody, messenger, transmitter, etc.), which it could not bind with before. Also new proteins can be involved in such processes in subsequent bind-ing, forming networks of binding. It is worth mentioning that similar networks under the name of molecular circuits have been postulated as a
Ì Ã
à Â
Å
Fig. 2. Diagram of bindings between FPs.
3. self-assembly for every bound FP determined by the energy of their complex.
Hence, the stationary and stable states of every bound FP are also determined by the energy of their complex. Therefore, we have shown that by binding and the subsequent breaking of bound FP can transfer to its other stable state. Such trans-formation of the FP’s state after binding is re-ferred to as (formal) allosteric effect.
Based on the above facts, we introduced the notion of (formal) network of binding, which implies any subsequence of binding with allosteric effects. It can be proved constructively that net-works of binding do exist.
For example, consider a special kind of FP, which depends only on one torsion angle 8, and assume a system with three such FPs: {P1, P2, P3}. Let 8*1, 8*2, 8*3 are their stable states as
given below (shown in Fig. 2):
8*i=
2p
3 (i−1) for i=1, 2, 3
In Fig. 2a, the bold lines represent the states of three FPs, which are drawn at different angles with respect to the positive direction of horizontal axe.
Let through binding, any of the FPs can change from one stable state to a new one:8**=i 8*i+p.
If we define the threshold of binding, wh=0
and the energy of interaction between Pi and Pj
as:
w(Pi,Pj)= −cos(8i−8j)
then no one pair of given FPs can bind to each other.
Let the fourth FP, denoted by A (an antigen), is added to the system. Let this FP is in its stable state 8*4=p/3 (shown by dotted lines in Fig. 2).
This FP binds with P1: P1lA (Fig. 2a), and P1 changes its stable state (Fig. 2b). Now P1 is able to bind with P2 or P3.
Let us consider that P1 binds with P2 (Fig. 2b) and P2 transfers to a new stable state (as in Fig. 2c). Now P2 is able to bind with P3 (Fig. 2c). If P3 change its stable state by this binding, then P3 changes to its new stable state and is able to bind with the initial antigen, A (shown in Fig. 2d). possible molecular basis of neural memory in the
human brain (Agnati, 1998).
For the mathematical abstraction of these facts, we consider the natural spread of quadratic Eq. (4) on the interactions between FPs (Tarakanov, 1999a). Let us define the binding energy between two FPs by the following bilinear form:
w(P,Q)= −[P]TW[Q] (5)
where [P], [Q] are configuration vectors of the first and the second FP, correspondingly,Wis the binding matrix, W={wij}, and wij are the given
coefficients,i,j=1, 2, 3, 4.
Let us consider that for any FP (P or Q), we have so called active center matrixAP,AQ, and we
also have a symmetric environmental matrix S, which is the same for all FPs. Let any binding matrix from Eq. (5) is equal to
W=AT PSAQ
Then the permutation of arguments in the left-hand side of Eq. (5) leads to
w(Q,P)= −[Q]TAT
QSAP[P]= −[Q]TWT[P]
and the direct representation of bilinear forms, determined by the matrices W and WT, gives
w(P,Q)=w(Q,P) for any configuration. In other words, the binding energy between two FPs is invariant to the permutation of configuration vec-tors as in Eq. (5).
Defining now the (formal) binding as satisfying condition w5wh for binding energy of FPs,
where wh is some threshold of binding. If this
condition is satisfied, then FPs considered as bound FPs and the following suppositions hold:
1. bound FPs form the unite complex;
The network of bindings between FPs: {A, P1, P2, P3} is shown schematically in Fig. 3, where dotted lines denote transitions between the stable states. In particular, the scheme depicts the fol-lowing events:
P1 binds with A and P1 changes state; P1 binds with P2 and P2 changes state; P3 binds with P2 and P3 changes state. Obviously, there exist other ways of bindings among the given FPs. For example, the initial activation of the system {P1, P2, P3} by antigen A could be realized by binding AlP2 and several variants of bindings may also exist subsequently. However, the condition of the system, corre-sponding to Fig. 2d, shows that there is only one way of such binding.
4. Formal B-cells
For an abstract modeling of the properties of immune networks, we need to supply networks of binding of FPs with the models of reproduction and death of cells. For this purpose, we have introduced a notion of formal B-cell as a set
B=P, Ir, G6
which include formal protein P, indicator of regime Ir, and map of controls G6 with the
fol-lowing restrictions:
1. B-cell can be in the regimes Ir={0, 1, 2}, whereIr=0 indicates the death of B-cell;
Ir=1-regime of receptor: B-cell possesses the abilities of its FP (as P);
Ir=2-regime of reproduction: another copy of B-cell is created so that for both B-cells,
Ir=1 and their new control parameters are determined by the map of controls G6;
2. transition from the regimeIr=1 to the regime
Ir=2 occurs only as a result of binding be-tween FPs.
We now define the B-FIS as a network of bindings, which includes B-cells:
B-FIS=FPs, B-cells
We have studied the simplest variant of such displacements within one-dimensional lattice. Ac-cordingly, a one-dimensional FIS is defined with the following elements and rules.
The elements include:
1. there arentypes of FPs {0, 1,...,n−1}, which are denoted byP0,P1,...,Pn−1; accordingly,Bi
denotes B-cell, which includes FP of the type
Pi;
2. a whole-number threshold of binding nh is
given;
3. energy of interaction between FPs defined by the formula
w(Pi,Pj)=min{(i–j)mod(n), (j–i)mod(n)}.
Rules of dispositions:
1. B-cells form one-dimensional sequence (popu-lation) without gaps from left to right; 2. if cell Bj reproduces then one of its copy
remains on the former place, and the other copy added at the end of the population; 3. if cellBjdies then other cells shift to the left to
fill the gap.
We have introduced and studied the two kinds of FISs, called the AB-network and the BB-net-works. The AB-network, AB(n,nh) is defined as a
one-dimensional FIS that possesses, apart from B-cells, also the free FPs (antigens) of n different types such as A0,A1,...,An−1 with the following
rule of displacements and interactions:
1. a sequence of antigens introduced in the popu-lation of B-cells such that every B-cell corre-spond to no more than one antigen;
2. interaction is allowed only with one B-cell and an antigen by the following rules:
a B-cell dies, if there is no antigen bind to
it, or if the energy of interaction between the B-cell and the antigen is higher than the threshold, w\nh;
if w=0, then the B-cell reproduces by
dif-ferentiating into two precise copies of itself (without mutations);
Fig. 4. Shows the death of the B-cell,Bj.
if the energy of interaction,wis 0Bw5nhthen
the second B-cell in the pair reproduces with mutations, where the first copy remains at the former place, and the second copy is delayed (Fig. 5):
after all pairs of population have interacted one time, B-cells shifted to the left for filling gaps of the died cells;
delayed copies are then added at the end of the population in proper order.
Previous studies (Tarakanov, 1999b) show that a system of B-cells with such rules of interactions can be determined, i.e. the initial population of B-cells can determine completely all the future populations. However, different variants of BB-networks have been studied and found that only one of the three regimes is possible for any initial population of B-cells:
death of all B-cells;
unbounded reproduction of B-cells;
cyclic reproduction of the initial population (formal immune memory).
In addition, it has also been shown that (Tarakanov, 1999b) the existence of such threshold nh for any n, a BB-network BB (n,nh)
can possess in a cyclic regime. Moreover, there exist variants of cyclic regimes with several peri-ods and dimensions of populations, including those, where the number of B-cells changes from population to population.
The results obtained so far (Tarakanov, 1999b) show that even the simplest variants of the FIS demonstrates such important effects as:
immune response in AB-networks under the control of antigens;
immune memory and generation of new im-mune repertoire in the absence of outer antigen by means of the cyclic regimes of BB-networks.
5. Formal T-cells
Consider a set of FPs, {S1,..., Sn}. Let two FPs
can bind, if and only if, they belong to the same type. Using such a set, we define a notion of formal T-cell T as a set of rules
Si0TSi1…Sir (6)
Fig. 5. Reproduction of a B-cell,Bj.
if 0Bw5nh, then the B-cell reproduce by
differentiating into two copies of its nearest type (with mutations);
the result of interaction has no influence on
the antigen;
interactions are realized consequently from
left to right; when the end of population is achieved, return to the beginning of the population.
The AB-networks studied so far considered that all antigens are of same type. It has been proved constructively (Tarakanov, 1999b) that even if only one B-cell is bind to an antigen, then after a finite number of steps, for every antigen there will be corresponding matching B-cells. This result affirms that even the simplest variant of FIS shows the mechanisms by which FPs (antigens) can control the reproduction and the death of B-cells.
The second variant, called the BB-network, BB (n,nh), where several types of B-cells are
gener-ated and stored through interactions among themselves, in the absence of any antigen. For this purpose, we defined a one-dimensional (whole number) FIS with the population of B-cells, where interactions are allowed only among the neighbor-ing B-cells with the numbers 2k−1, 2k, where
k=1, 2..., represents a number of pair of B-cells. These interactions are guided by the following rules:
if the last B-cell in population is odd (without pair) then it dies;
if the energy of interaction, w\nh then the
where i0, i1,...,ir are indexes of types and belong
to the set 1,...,n. The rules in the Eq. (6) mean that:
T-cell has receptors of types,Si1,…,Sir(on the
right side of the rule);
T-cell synthesizes the FP of the type, Si0 (left
side of the rule), when all the receptors
Si1,…,Sir are matched (T-cell activated).
According to Tarakanov (1999a), T-FIS is a set: Types, FPs, T-cells. Consider now a set of FPs:Pk1,…,Pkm, belong to the types,Sk1…,Skm,
respectively, and an antigen Ag, is of typeSj. We can denote these facts by the following rules:
Sk1Pk1,…,Sk1Pk1 (7)
AgSj (8)
Studies (Tarakanov, 1999b) showed that the T-FIS added with the rules in Eqs. (7) and (8), is equivalent to a special kind of attributive context-free grammar, where antigen Ag corresponds to the axiom of the grammar, typesS1,…,Snmap to
the non-terminals, FPs such as Pk1,…,Pkm
indi-cate the terminals, T-cells T as the synthesized attributes. It worth to note that this method can produce some kind of grammars for solving tasks as an inference engine (Tarakanov, 1999a).
6. Conclusion
The natural immune system is a highly dis-tributed adaptive learning system. Its learning takes place through interaction between peptides arising from self-proteins and those derived from foreign proteins. The immune cells contain formal proteins as their receptors those take part in binding and mutual recognition. We developed a mathematical model of these interactions and re-ferred to as an FIS.
The developed FIS theory has appeared to be useful in solving a number of important practical tasks, including computation of map of complex evaluation for ecological atlas of Kaliningrad city, detecting concrete dependence between quality of environment and morbidity of children for Tula districts, detecting similarity in dynamics of infec-tional morbidity in Russia, detecting dangerous
ballistic situations in near Earth space, synchro-nizing events in computer networks, etc. (Tarakanov, 1998; Kuznetsov et al., 1999; Tarakanov, 1999a,b).
It is worth to note that the theory of the FIS gives a mathematical basis for developing compu-tational algorithms for problem solving. The prin-cipal difference between the FIS and the ANN can be determined by functions of their basic elements. Unlike the ANN, whose elements con-sidered as fixed in space, elements of the FIS allowed to change their places. While an artificial neuron is considered as an aggregator with given threshold, the FP, according to its biological pro-totype, ensures self-organization (self-assembly) of its parameters, as well as free binding with any other FP, depending on their reciprocal states. However, it is to be noted that the ideas and the model proposed in this paper are still under devel-opment and further analysis and applications will be addressed in our future work.
Acknowledgements
Dipankar Dasgupta acknowledges the support of the Office of Naval Research (grant N00014-99-1-0721) and the University of Memphis. The authors would like to thank Ray Paton and re-viewers for their valuable comments and suggestions.
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