Towards Immunocomputer
Alexander O. Tarakanov

Abstract — The paper shows how the principles of information
processing by proteins and immune networks could lead to a
new kind of computers. We propose to call such computers
'immunocomputers' by analogy to the widely spread
neurocomputers, which are based on the models of neurons
and neural networks. We consider a rigorous mathematical
basis and possible applications of the immunocomputer.
Index Terms — formal protein, formal immune network,
immunocomputer
I. INTRODUCTION
From the computational viewpoint we can consider that
proteins realize the main functions of information
processing in living nature. In fact, namely the proteins
recognize and execute programs (instructions) represented
in the form of genetic code. Being the neuro-mediators and
the receptors of neurons, proteins control the electrical
activity of the brain. Proteins are also the main components
of the immune system, which thus involves free proteins
(antibodies, messengers, etc.), and proteins as receptors of
immune cells (B-cells and T-cells). Apparently, proteins
should play the key role both for immune and intellectual
processes.
For example, specialists call the immune system “the
second brain of vertebrates'' [6]. Indeed, the immune system
possesses all the main features of Artificial Intelligence
(AI): memory, abilities to learn, to recognize, and to make
decisions how to treat any non-self protein (antigen), even
if the latter had never existed before on the Earth. Of the
especial interest for computer science is the widespread
theory of immune networks, formed on the interactions
between proteins of the immune system. Nowadays there is
no doubt that such networks exist. Their fragments and
interactions have been detected experimentally. It is worth
to note that similar networks under the name of molecular
circuits have been proposed as a possible molecular basis of
neuronal memory in the human brain [2].
Based on the biological principles of the immune system,
there arises a new and rapidly growing field of the Artificial
Russian Academy of Sciences
St.Petersburg Institute for Informatics and Automation
14 line, 39, VO,
St.Petersburg, 199178, Russia
tarakanov@togetherlab.nw.ru
Immune Systems (AIS), offering powerful and robust
information processing capabilities for solving complex
problems [3]. Like Artificial Neural Networks (ANN), the
AIS can learn new information, recall previously learned
information, and perform pattern recognition in a highly
decentralized mode. The AIS have already been applied in
several specific problems, such as information security,
faults detection in manufacturing, vaccine design, data
mining, robotics, etc.
However, comparing with the ANN, the field of AIS has
yet neither a clear and sound mathematical basis, nor a
hardware implementation analogous to the existing
neurocomputers that are based on ANN. Nowadays most of
the AIS represent some hybrid and heuristic algorithms,
using ideas from genetic algorithms, cellular automata,
ANN, etc. [3]. On the other hand, the role of proteins, as the
basic natural elements of the information processing, has
not yet been fully exploited by computer science, including
AI, ANN and AIS.
Therefore, this paper makes an attempt to bridging the
existing gaps.
We propose a concept of immunocomputer (IC) that is
based on the principles of information processing executed
by proteins and immune networks. We develop a proper
mathematical basis of the IC by introducing notions formal
protein (FP) and formal immune network (FIN) [15-17].
The need of such novel notions is caused by very specific
objects and interactions of immune networks, which differ
remarkably from genetic algorithm, cellular automata,
ANN, or intelligent agent.
We hope that such mathematical basis could raise the AIS
up to the level of the wide spread ANN, and even allow to
speak about hardware implementation of FIN in a special
electronic scheme (immune chip). Such chip could be
treated as a core of the future IC.
We demonstrate also potential applications of the IC in
terms of numerical algorithms, which are able to solve
specific complex problems. Among them we should
consider briefly complex evaluation of ecological and
medical indicators, information security, and some of the
other ones.
II. BIOPHYSICAL BACKGROUND
According to biological prototypes and their mathematical
models [19], the principal difference between the IC and the
neurocomputer should be determined by functions of their
basic elements. If artificial neuron is considered as a
summation with a threshold, connected with fixed neurons
[23], then protein as a basic element of the IC ensures quite
other conditions. Consider them more detailed.
From the computational viewpoint the living nature has the
uniformed information basis It consists of the universal
genetic code and the universal alphabet, where words are
molecules of proteins. Using analogy with computer, it can
be said that the genetic code is like “software”, i.e.
instructions of program that the cell receives from its parent
cell, while proteins are “hardware”, i.e. the biophysical
mechanisms that execute the program. No wonder that
proteins are the most complex of the known molecules and
the most universal in their properties and functions.
Although genes and proteins are exceptionally complex,
some of their features can be explained by rather simple and
general mechanisms. These biophysical mechanisms,
though, are not easy to uncover. A striking example is the
discovery in 1953 of double helix structure of the chain
molecules that store the genetic code. This spatial structure
is formed by the so-called weak interactions between very
strictly determined molecular shapes situated in the same
plane. This is one of the most significant examples of the
geometrical correspondence between biomolecules.
But for proteins the mechanisms with similar simplicity and
explanatory power have not been found yet. Nevertheless,
the following principles are evident [5]:
1.
2.
Spatial conformation of protein is determined by
the linear sequence (word) of its amino acid’s
code;
This conformation determines functions of any
protein.
The first correspondence between the code and the stable
conformation of protein (the so-called native form) is
realized by mechanisms of self-assembly (or folding). The
second correspondence between spatial conformation and
the function of protein is realized by mechanisms of
molecular recognition. Just as for the double helix, these
mechanisms are based essentially on the week interactions
between different parts of the protein molecule and between
different molecules of proteins.
As a result of such interactions, protein can bind with
another protein or molecule. As a result of binding, protein
can change its spatial shape (the so-called allosteric effect
[5]). Furthermore, from this effect the protein can receive
an ability to bind with another protein (antigen, antibody,
receptor, etc.), with which it couldn't bind before. And new
proteins can be involved in such process of subsequent
binding, forming molecular circuits or immune networks.
Therefore, binding could be seen as the main information
processing function of proteins. Like “key and lock”, such
binding is highly specific, because it depends on the
existence of highly adjusted local shapes of interacting
proteins. It could be said also, that the protein is able to
select or “recognize” the appropriate “pattern”, as well as
reject all the other ones.
The main biophysical characteristic of interaction between
proteins is a free energy [5]. The lower is the energy, the
stronger is the binding, and vice versa. Thus, the negative
energy, lower than the energy of the Brownian motion,
corresponds to the proper binding, while the positive energy
corresponds to repulsion between proteins.
According to [19], consider the free energy of interactions
between proteins as a binding energy, to distinguish it from
a free energy of protein’s folding.
III. BASIC COMPONENTS
In spite of the proteins, cells can be considered as the
second basic component of information processing by
immune networks. Two main sorts of immune cells can be
distinguished: B-cells and T-cells. Cells produce and
secrete proteins, as well as expose proteins as their
receptors.
Accordingly, let us distinguish two kinds of proteins: “free
proteins” independent from cells, and proteins anchored in
the membranes as cell’s receptors. The examples of free
proteins may be any peptides (small proteins), antigens,
antibodies produced by B-cells, and numerical peptides
(limphokines) produced by T-cells. The examples of
receptors may be the so-called proteins MHC I and MHC II
(Major Histocompatibility Complex class I and II). These
proteins are used by the immune system as universal
markers of any own (self) cell of the body to distinguish it
from non-self antigens.
On the other hand, the architecture of any computer
includes at least two basic components: memory and
processor. They can be gathered in the separate modules,
like RAM (Random-Access Memory) and CPU (Central
Processing Unit) of the traditional PC, or distributed among
other structural elements, like neuron of neurocomputer, or
cell of cellular automata [1]. Nevertheless, memory and
processing units are the intrinsic components of any
computer.
Thus, consider architecture of memory of the IC as shown
on Fig.1.
DNA (deoxyribonucleic acid) or antibodies (corresponding
to the array 3 of the IC). These probes are immobilized to a
solid surface, such as nylon, glass, or silicon substrates
(array 4), and exposed to a set of testing samples (array 2).
The results of binding between samples and probes are
determined by fluorescence or electric signals (array 1).
On the other hand, if any memory array of the IC is able to
store only few discrete states, and all units of the array
change states simultaneously in discrete time, then the wellknown cellular automata machines [22] or excitable media
[1] could realize horizontal interactions within the array.
Fig.1. Architecture of memory of the IC
Each memory unit is depicted in Fig.1 as a square. Units are
gathered in the four main arrays, depicted in the Fig.1 as
horizontal layers (top-down):
1.
2.
3.
4.
The output array corresponding to the binding
energies between free proteins (array 2) and
receptors (array 3);
The input array corresponding to the free proteins;
The array of the receptors corresponding to the
stored patterns;
The array of the cells corresponding to patterns’
control.
Consider, that the every memory unit has only the strictly
determined neighboring units. Namely, every memory unit
has:
1.
2.
Four vertical neighbors arranged strictly above
and/or below the unit in any other array;
Four horizontal neighbors arranged in a cross
manner in the same array, as shown in Fig.2.
However, such special kinds of the IC are obviously
insufficient to simulate features of immune networks.
Hence, consider that any memory unit of the IC is able to
store a set of real numbers, and processing units of IC are
able to compute this set using the set of any horizontal or
vertical neighbor.
IV. MATHEMATICAL BASIS
Designate states of the memory units as wij , Pij , Rij , Cij ,
where i and j are row and column numbers (address) of the
unit within the array, wij is real value of binding energy, Pij ,
Rij ,and Cij are vectors with real components of the
dimensions nP , nR,, and nC, correspondingly. Consider
vectors Pij , Rij ,and Cij as column vectors which code the
states of proteins, receptors, and cells, correspondingly.
Let binding energy be defined by a bilinear form
wij   PijT MRij ,
(1)
where M is a matrix nP nR of real values, and upper case
‘T‘ is a symbol of transposing.
Consider several special cases that implement important
mathematical constructions.
A. Singular Value Decomposition
Fig.2. Four horizontal neighbors of a memory unit
Consider the content of any memory unit as its state. Then
general function of the IC is to determine the states of the
output array by the states of the input array in accordance
with stored patterns that can change dynamically. For this
purpose processing units of the IC determine the
interactions only between states of the neighboring units.
It is worth to note, that vertical interactions in such IC could
be realized by already existing biochips, also called
microarrays [9, 13]. Actually, microarray of the biochip is
an orderly arrangement of probes, such as short strands of
Let matrix M be given. Consider a set of unit vectors Pij and
Rij of the dimensions nP and nR , correspondingly:
PijT Pij  RijT Rij  1, i , j .
Compute wij for all pairs of such vectors by Eq. (1). Select
the minimal wij=w* and the corresponding pair of the
vectors P*, R*:
w*  min{ wij }, w*  [ P*]T MR * .
(2)
i, j
Let value w*=w(P*, R*) satisfy to the following condition:
w*  w( P, R ), P, R : PT P  RT R  1 .
1 
2
,  k  k 1 , k  0 ,1,..., n  1 .
n
Then, according to [16], s1=w* is the maximal singular
value of the matrix M, while X1=P* and Y1=R* are the left
and the right singular vectors of this matrix.
Then we have exactly n types of the vectors. Designate
them as S(k), k=0,...,n1.
Compute the matrices Mk by the following recurrent rule:
Define matrix M as a unit matrix:
M k  M k 1  sk 1 X k 1YkT1 , k  2,...,r , M 1  M ,
1 0 
M 
.
0 1
where r is the rank of the matrix M.
Analogously to the matrix M, determine the maximal value
sk=w* and the corresponding vectors Xk=P* and Yk=R*
for the matrix Mk.
Finally, we obtain the so-called Singular Value
Decomposition (SVD) of the initial matrix M:
M  s1 X1Y1T  ... sr X rYrT .
Note, that IC allows minimizing the value wij at least in two
different ways.
Firstly, the IC can use a process of random “mutations” of
vectors’ coordinates so, that Pij and Rij still remain unit
vectors. For example, if the IC had received a value of wij ,
which satisfies to Eq. (2), and this value had not been
reduced after a big number of mutations, then this value
could be considered as a minimum.
Secondly, the IC is able to use more strictly determined
procedure. For example, let wij , Pij , Rij ,and Cij be
computed by the following recurrent scheme (we omit the
lower indexes for convenience):
[ R( k ) ] T  [ P( k 1 ) ] T M , C ( k )  MR ( k ) , P( k )  C ( k ) ,
w( k )  [ P( k ) ] T MR ( k ) , k  2 ,...,
(k )
while w
( k 1 )
w
 .
According to [16], such scheme converges to the maximal
singular value and singular vectors in general case of the
matrix M.
B. Formal Immune Networks
Then binding energy between vectors S k1  and S k2  is


w   cos  k1   k2 .
(3)
Define binding as such event, when wwh , where wh is a
given threshold of binding. Let an integer nh define the
threshold as follows:
wh   cosnh1  .
Hence, binding condition can be reduced to the following
inequality:
mink1  k2 , n  k1  k2  nh .
Let memory of the IC is one-dimensional. Then states can
be marked by an index j and represented in a form of a
matrix:
 w1
P
 1
 R1

C1
... w j
...
Pj
... R j
... C j
...
...
...

...
Designate an empty memory unit (gap) by the symbol .
Let the initial sequence (population) {R} of the length m
without gaps be given:
R : R j   j  m, R j   j  m.
(4)
Let the population {P} be an arbitrary, and the initial
population {C} be empty. Consider processing of the
population {R} by the following algorithm.
Algorithm 1.
Consider only unit vectors of the dimension 2. Then any
vector can be represented as depending on one angle, for
example:
S(  )  [cos , sin ] T .
Let this angle accept only one of n discrete values:
1.
2.
3.
4.
Compute wj between Pj and Rj by Eq. (3);
Change Rj and Cj according to wj and wh ;
Merge the sequences {R} and {C};
Repeat the steps 1-3 until {R} becomes empty or
overflows a memory limit.
Step 2 is performed simultaneously for all j by the
following rules:
If wj>wh or Pj= then Rj= .
If wj= 1 then Cj=Rj .
If 1<wjwh then
Cj=[Rot(1)]Rj , Rj=[Rot(1)]Rj ,
cos  1
Rot(  1 )  
 sin  1
 sin  1 
.
cos  1 
Simply say, if Pj doesn’t bind with Rj , then Rj dyes, else Rj
reproduces. If strength of the binding is the highest
possible, then Rj creates a copy, else Rj reproduces in the
two nearest types (mutates).
Step 3 is performed simultaneously for all j by the
following rules:
If wj>wh or Pj= then Rj= .
If wjwh then
Cj=[Rot((1))]Rj , Rj=[Rot((1))]Rj .
Simply say, if Pj doesn’t bind with Rj , then Rj dyes, else Rj
reproduces with mutations.
According to [19], the algorithm 2 implements another
variant of FIN. Namely, it is the so-called BB(n, nh)
network where several types of B-cells are generated and
stored through interactions among themselves, in spite of
the absence of any antigen.
Step 3 includes the following sub-steps:
Theorem 2.
3.1. Attach the sequence {C} to the end of the sequence
{R};
3.2. If Rk= for any k then shift Rj-1=Rj for any j>k;
3.3. Perform step 3.2 until Eq. (4) is satisfied;
3.3. Compute the length m of the resulting sequence
{R};
3.4. Make the sequence {C} empty.
For any initial population of any BB(n, nh) network only
one of the three regimes is possible:
1. Death of all B-cells;
2. Unlimited reproduction of B-cells;
3. Cyclic reproduction of the initial population
(formal immune memory).
According to [19], algorithm 1 implements the simplest
variant of FIN. Namely, it is the so-called AB(n, nh)
network, where the sequence {P} corresponds to antigens
and the sequence {C} corresponds to B-cells.
Theorem 3.
Several mathematical results can be obtained for such AB
networks [16].
Studies [16] show, that there exist variants of cyclic
regimes with several periods and lengths of populations,
including those, where the number of B-cells changes from
population to population.
Theorem 1.
If all antigens in any AB(n, nh) network are of the same
type, and at least one B-cell binds an antigen, then after
a finite number of steps, for every antigen there will be
corresponding matching B-cell.
This result affirms that even the simplest variant of FIN
models the mechanism by which antigens can control the
reproducing and the death of B-cells. Besides, we have
determined the conditions of arising and supporting of the
(formal) immune response, which implies the B-cells' desire
for acceptation of antigen's type.
Consider now processing of the initial population {R} with
empty initial population {C} by another algorithm.
Algorithm 2.
1.
2.
3.
4.
5.
Form the sequence {P: Pj-1=Rj, j=2,…m};
Compute wj between Pj and Rj by Eq. (3);
Change values of Rj and Cj according to wj and wh ;
Merge {R}{C} (see step 3 of the algorithm 1);
Repeat the steps 1-4 until {R} becomes empty or
overflows the memory limit.
For any n there exists such threshold nh that at least one
cyclic regime is possible in BB(n, nh) network.
C. Formal Grammars
Consider any vector as a coded FP [16]. Let wh=1. Then,
according to Eq. (3), any FP: S(k), k=0,...,n1, can bind only
with FP of the same type.
Consider the following initial sequences {P}, {R}, and {C}:
P : Pj  , j  n, Pj  , j  n,
R : R0  , R j  , j  0,
C : C j   j  m, C j  , j  m.
Consider processing of the sequence {P} by the following
algorithm.
Algorithm 3.
1.
2.
3.
Assign k=0;
Compute wk between Pk and Rk;
If wk=1 then change the sequence Pk, Pk+1, …, Pn
to the sequence Ck+1, …, Ck+m, Pk+1, …, Pn, and
assign n=n+m;
4.
5.
Shift the sequence Ck, Ck+1 , …, Ck+m to Ck+1, Ck+2 ,
…, Ck+m+1, and the sequence Rk, Rk+1 , …, Rk+m to
Rk+1, Rk+2 , …, Rk+m+1;
While k<n assign k=k+1 and repeat steps 2-5.
According to [19], the algorithm implements a variety of
the so-called formal T-cell. Such T-cell has a receptor, the
type of which is stored in R0. When the receptor is matched,
T-cell becomes activated and synthesizes a sequence of
FPs: P1 , …, Pm, the types of which are stored in C1 , …, Cm,
correspondingly. Then the IC inserts this sequence instead
of P0. Thus, moving along the sequence {P}, T-cell
replaces every Pj if its type is matched with the type, stored
in R0.
k 
Designate by S j the type of the vector, which is stored in
Cj. Then the function of any T-cell can be described
formally by the following rule:
S k0   S k1  ... S k m  .
(5)
Consider a correspondence between types of vectors and
symbols. For example, S(0)=’A’, S(1)=’B’, S(2)=’a’, S(3)=’b’,
etc. Let us take a set of n+1 symbols: S(0),…, S(n). Let the set
consist of two disjoint subsets: non-terminals, say S(0),…,
S(k), and terminals S(k+1),…, S(n). Point out exactly one
particular non-terminal (the so-called axiom), say S(0).
Consider now a set of the rules (5), which satisfy the
following conditions:
1.
2.
Any symbol of the left side is non-terminal;
There exists only one rule, which contains the
axiom.
According to [8, 16], such set of the rules (5) is equivalent
to a context-free (CF) grammar. Hence, behavior of the set
of corresponding T-cells can be also described by CF
grammar. It is worth to note, according to [8], that the class
of CF grammars is the most interesting class of formal
grammars both for theory and applications.
(classification) of the characteristic space. If the space is
being partitioned to the known classes (e.g. by experts),
then it is said about supervised learning. If the number of
the classes kn and the classes themselves are unknown a
priory, then it is said about unsupervised learning.
The main feature of the IC approach to pattern recognition
consists in treating an arbitrary pattern as a way of setting a
binding energy by a bilinear form in Eq. (1). A
mathematical basis of the approach is considered in details
in our previous works [12, 16]. It based essentially on the
properties of SVD of an arbitrary matrix over the field of
real numbers. According to the approach the task of pattern
recognition can be solved as follows.
1) Supervised Learning
a)
Folding vectors to matrices
Fold vector X of dimension n1 to a matrix M of dimension
nP nR=n. It has been shown strictly in [16], that such
folding increases the specificity of recognition.
b)
Learning
Form matrices M1,..., Mk for all classes c=1,...,k, and
compute their singular vectors:
{P1, R1} – for M1, ... , {Pk, Rk} – for Mk.
c)
Recognition
Compute k values of binding energy for the every input
pattern M:
w1 = – P1TMR1 , ... , wk = – PkTMRk .
Determine the class to be found by the minimal value of
binding energy, according to Eq. (2).
2) Unsupervised learning.
V. POTENTIAL APPLICATIONS
Consider examples of potential applications of the IC. They
include, but are not limited to, pattern recognition,
information security, problems solving, and modeling of
natural systems.
A. Pattern Recognition
Pattern recognition can be defined as follows. Let us treat
real values x1, …, xn as a set of characteristics. Consider an
arbitrary vector X=[ x1, …, xn]T as a pattern that belongs to
a characteristic space {X}. Consider, that the space can be
partitioned on the subsets (classes) {X}k , k=1, 2, …kn. Then
recognition of X consists in determination of such class k
that X {X}k, while learning consists in partition
Consider the matrix M= [X1... Xm] T of dimension m  n
formed by the m vectors (patterns) X1, ..., Xm. Consider the
SVD of this matrix:
 p11 
 p21 


T
M  s1  ...  R1  s2  ...  R2T  ... ,
 p1n 
 p2 n 
(6)
where s1, s2 are the first two singular values, and R1, R2 are
the right singular vectors.
Note, that every string i of the matrix M represents the
values xij of n characteristics of the pattern Xi, where
i=1,…,m and j=1,…,n. Hence, according to [16], the
components p1i, p2i of the left singular vectors P1, P2 satisfy
to the following equations:
p1i  X iT
1
1
R1 , p2i  X iT
R2 .
s1
s2
(7)
Comparison of Eq. (1) and Eq. (7) makes obvious, that the
IC is able to compute the components p1i, p2i as binding
energies w1i, w2i between Xi and R1, R2, correspondingly.
Thus, every vector Xi with n characteristics is mapped to
only two values of binding energies.
Such mapping gives a mathematically rigorous way to
represent and view all patterns, with no matter how many
characteristics, as points in two-dimensional space of
binding energies {w1, w2}. This plane could be treated also
as a shape space of the IC, according to [7]. Such
representation of patterns in the shape space of the IC
allows classify them in a very natural way by the groups
(clusters) of the neighboring points. The IC using
unsupervised learning can perform by experts using
supervised learning as well as such classifying.
The approach has already appeared to be useful in solving a
number of important practical tasks, including detection of
dangerous ballistic situations in near-Earth space [17],
complex evaluation of ecological and medical indicators in
Russia [11, 12], and prediction of danger by space-time
dynamics of the plague infection in Central Asia [20].
For example, consider a task of complex ecological
evaluation [12]. The task is the following. Let a set of
special ecological characteristics, also called indicators
(SEI), be given. It is required to find its complex ecological
characteristic, also called index (CEI). In other words, it is
required to classify this set by assigning an index (class)
denoted, usually, as a number 1,2,3, and so on.
The general solution of the task comprises two stages:
learning and recognition. The stage of learning comprises
choosing a set of typical patterns of SEI. These patterns
may be the results of monitoring areas with known CEI, or
data, determined by experts. Then, using such data, several
samples of SEI are formed for every class of CEI. At the
stage of recognition a testing set of SEI (probes) is
compared with the samples. Thus, the sample pattern,
which is the most resembling to the testing pattern, is the
CEI.
The described approach has been used to solve important
practical tasks. Among them there are detecting of the
detailed correlation and casual relationships between the
quality of environment and the children's morbidity in Tula
city [12] and computing the CEI map of Kaliningrad city
[11].
The results obtained so far [12, 17, 20] show, that this
approach to pattern recognition is rather powerful, robust
and flexible. It is able to give rather fine classification and
sharply focus attention on the most dangerous situations,
which is beyond the possibilities of the traditional statistics.
B. Information Security
Like in the natural immune system, the problem of
protecting computer systems from malicious intrusions can
similarly be viewed as the problem of distinguishing “self “
from dangerous "other" (or "non-self") and eliminating this
"other". In this case the "non-self" may be an unauthorized
user, foreign code in the form of a computer virus or worm,
unanticipated code in the form of a Trojan horse, or
corrupted data, etc. According to [10], the information
security could be completely specified based on the abstract
representation of "self" and "non-self" as sets of bit strings,
designated even as "proteins" and "peptides".
For example, "protein" could be a sequence of viral bytes in
a legitimate program, or a "signature" of computer virus. To
preserve generality, in [10] it has been proposed to
represent both the protected system (self) and infectious
agents (non-self) as dynamically changing sets of bit
strings, because in cells of the body the profile of expressed
proteins (self) changes over time. Besides, "peptide" for a
computer system is defined in terms of short sequences of
system calls executed by privileged processes in a
networked operating system. Preliminary experiments on a
limited testbed of intrusions and other anomalous behavior
[10] show that short sequences of system calls (currently
sequences of length 6) provide a compact signature for self
that distinguishes normal from abnormal behavior. By this
analogy proteins can be thought of as "the running code" of
the body while peptides serve as indicators of behavior.
Consider now that vector X represent a set of information
security indicators. For example, it can be a bit string of a
legitimate program, a signature of computer virus, a coded
sequence of system calls, statistics of current activity of the
network, etc. Consider a space {X} of such indicators,
partitioned to the k subspaces (classes) {X}1,...,{X}k. For
example, it can be simply k=2, where {X}1 is normal
behavior and {X}2 is "infection". Then, as we have a
concrete vector X, the task consists in determining it's class
c={X}c where c=1,...,k. Thus the problem is reduced to the
pattern recognition, mentioned above.
In addition, the IC could be also applied to some other
issues of information security. Consider, for example, data
hiding and encryption.
According to [4], data hiding, a form of steganography,
embeds data into digital media for the purpose of
identification, annotation and copyright. It represents a
class of processes used to embed data, such as copyright
information, into various forms of media such as image,
audio, or text with a minimum amount of degradation to the
"host" signal; i.e., the embedded data should be invisible
and inaudible to a human observer. Note that data hiding,
while similar to compression, is distinct from encryption.
Its goal is not to restrict or regulate access to the host
signal, but to ensure that embedded data remain inviolate
and recoverable.
For an example of data hiding by the IC, consider that
matrix M represents an initial data array. It could be an
image, a folded audio signal, etc. Consider the SVD of the
matrix in the form of Eq. (6). Let us add to this sum an item
in the form sr+1Pr+1RTr+1, where r is a rank of the matrix,
Pr+1, Rr+1 are unit vectors, sr is a minimal singular value of
the matrix, and sr+1<sr. According to [16], such addition
only slightly disturbs the matrix. Although such disturbance
is invisible or inaudible to a human observer, the presence
of the "hidden" addition can be surely detected in the shape
space of the IC. In this case the IC functions like the natural
immune system, which verifies identity by the presence of
peptides, or protein fragments.
Consider now data encryption. In modern cryptography,
encrypting of information is based on a widely known
algorithm and a number or string, called a key, which is
kept in secret. The key is used as a parameter to the
algorithm to encrypt and decrypt the data. Decryption with
the key is simple, but without the key is very difficult and
in some cases nearly impossible. Therefore the
"fundamental rule of cryptography" is that both the sending
and receiving sides know the method of encryption [14].
As an example of encryption by the IC, consider BBnetworks. Specifically, in the network BB(10,2) for any
type i=0,...,9 there exist populations of the type
Let algorithm 3 be also modified to implement such T-cell
with a name corresponding to a type, stored in R1, and with
receptors, the types of which are stored in R2 , …, Rm (right
side of the rule). When all receptors are matched, T-cell is
activated, synthesizes FP of the type stored in C0 , and puts
the corresponding vector into P0 (left side of the rule).
Let us add also T-cells of two specific types. Consider we
have an “initial” rule for some type k1:
S ( 0 )  S  k1  ,
(9)
and a set of “terminal” rules for some of the types k2,…,km
S k2   S k2  , ..., S km   S km  .
(10)
According to [19], S(0) can be regarded as corresponding to
an antigen, while rules (10) correspond to T-cells that
synthesize FPs independently from binding with any FP.
According to [16], a set of T-cells described by rules (8)(10) is equivalent to a special kind of attributive CF
grammar, where antigen corresponds to the axiom of the
grammar, types of R2, …, Rm – to non-terminals, types of
R1, …, Rm – to terminals, names in angle brackets – to
synthesized attributes. This method can produce some kind
of grammars for solving tasks as inference engine.
For example, consider triangle ABC in Fig. 3 with angles A,
B, C, and sides a, b, c.
S ( i  2 ) S ( i ) S ( i 2 ) S ( i ) ,
which is cyclic with the period 4, according to[16]. For
example,
1979 187800 1770991 17980 1979 ... ,
where the type S(i) is denoted by only one number i.
Consider now the numbers 10 and 2 as a key which defines
the network BB(10,2). Then the string 1979 could encrypt
the string 1770991. Knowing the key, the data can be
decrypted, say, as the string of the maximal length,
generated by the network BB(10,2) from the given string
1979. Although the example seems rather simple, it shows
the principal possibility of using the IC in cryptography.
C. Problem Solving
Fig. 3. Triangle
It is known that parameters of any triangle satisfy to the
following equations:
A + B + C =  (theorem of angles),
a
b
c


(theorem of sines),
sin A sin B sin C
a2 =b2 + c2  2 (bc)cosA (theorem of co-sines),
b2 = a2 + c2  2 (ac)cosB (theorem of co-sines),
c2 = a2 + b2  2 (ab)cosC (theorem of co-sines).
Consider a modification of the rule (5):
S k0   S k1  S k2  ... S km  .
(8)
Hence, a model of triangle for the IC could be the
following:
S(1)=A, S(2)=B, S(3)=C, S(4)=a, S(5)=b, S(6)=c,
T(1)=Tang, T(2)=Tsin, T(3)=Tcos,
S(1) <T(1)>S(2) S(3), S(1) <T(2)>S(2) S(4) S(5),
S(2) <T(1)>S(1) S(3), S(2) <T(2)>S(3) S(5) S(6),
S(3) <T(1)>S(1) S(2), S(3) <T(2)>S(1) S(6) S(4),
S(4)<T(3)>S(1) S(5) S(6),
S(5)<T(3)>S(2) S(4) S(6),
S(6)<T(3)>S(3)S(4) S(5),
D. Modeling of Natural Systems
Consider the following task: find a, when C, b, c (circled in
Fig. 1), are given:
S(0) S(4),
S(3) < S(3)>, S(5)  < S(5)>, S(6)  < S(6)>.
(11)
(12)
The IC can solve the task in the following way. Firstly,
given FPs, which correspond to the rules (12), activate Tcell that synthesizes FP of the type S(2):
S(2) <T(2)>S(3) S(5) S(6).
Secondly, this FP binds the receptor of the same type, and
together with given S(3), activates the corresponding T-cell
that synthesizes S(1):
S(1) <T(1)>S(2) S(3).
Thirdly, this FP together with given S(5) and S(6) activates
the corresponding T-cell that synthesizes S(4):
S(4)<T(3)>S(1) S(5) S(6).
Finally, this S(4) activates T-cell of the rule (11), which
gives the following solution of the task.
S(0)<T(3)><T(1)><T(2)>S(3)S(5)S(6)S(3)S(5)S(6).
In usual designation it means
a = <Tcos><Tang><Tsin>CbcCbc.
Thus, the IC has synthesized the following solution:
1.
2.
3.
Find angle B by given angle C and sides b and c
using the theorem of sines;
Find angle A by known angles B and C using the
theorem of angles;
Find side a by known angle A and sides b and c
using the theorem of co-sines.
Moreover, the IC gives the solution in the so-called “prefix
Polish notation”, which can be interpreted strictly in the
program of computations.
Although this geometrical example seems to be rather
simple, it shows the general principles of using the IC as a
problem solver. Namely, the IC would represent a kind of
engine, where inference simulates behavior of immune
networks.
It is known, that proteins represent chains consisting of 20
basic amino acids, like words consist of letters of alphabet.
Usually, it escapes attention that this number is
approximately equal to the number of letters in the
alphabets of the so-called “classical” Indo-European
languages (e.g. the Italian alphabet consists of 21 letters).
But the similar analogy induces an idea, that the IC could
be used also for linguistic modeling.
Specifically, consider T-cells, corresponding to Eq. (8),
which bind FPs by the receptors. Consider behavior of such
T-cells as a model of formation of “correct words”
(morphology) and/or “correct sentences” (syntax). Of
particular interest is the fact that there is a rather advanced
linguistic model of language, as if it was developed
especially for the proposing IC. It is L.Teniere's theory of
linguistic valence [21]. This theory stays somewhat isolated
in linguistics, because it differs strongly from the
widespread generative (or formal) grammars of N.Chomsky
[8]. Meanwhile, on the biological level nothing like “innate
grammars”, postulated by Chomsky, has been detected.
Moreover, the existence of such grammars is rather
problematical. At the same time, the theory of linguistic
valence considers ability of a word to enter into syntactic
relations with other elements based on the straight analogy
with chemical interactions, even fixed in the name of the
theory.
It is worth to note, that such way allows unite digital
computations with language representation by the IC. At
the same time, representation of linguistic knowledge is a
very serious problem for neurocomputers.
The IC could be also a perspective device for simulating the
natural immune system, including important deceases (e.g.
AIDS). This simulation is essentially based on the hardware
implementation of FIN. As it was shown above, even the
simplest variants of FIN possess the inherent properties of
immune response and immune memory. For example,
Theorems 1-3 affirm that one dimensional FIN with small
number of FP’s types is able to demonstrate such important
effects as:
1.
2.
Immune response in AB-networks under the
control of antigens;
Immune memory and generation of new immune
repertoire in the absence of outer antigens by
means of the cyclic regimes in BB-networks.
If one-dimensional FIN still yields to a pure mathematics,
then two-dimensional FIN is already much more fuzzy.
Investigation of its properties is practically impossible
without a computer simulation. Simultaneously, such FIN's
properties seem more close to the ones of the natural
immune system. No wonder that recently used biochips are
also two-dimensional [9, 13].
On the other hand, the mathematical basis of the IC relies
on the notion of FP. According to [15], the features of FP
give opportunity not to move far away from natural protein,
as artificial neuron did from its biological prototype. At any
case, modeling of the natural immune system by the IC
seems more promising then by neurocomputers or even
cellular automata with discrete states.
[2]
In addition, a promising approach to modeling continuous
states dynamics of natural systems could be connected with
the IC by the so-called Cellular Immune Networks. Such
networks have been introduced in [18] as a combination of
FIN with hybrid cellular automata. Their application for
particular task of virtual clothing gives almost three-fold
speed up comparing to traditional methods of computations
(numeric integration, finite elements method, etc.).
[7]
VI. CONCLUSION
[12]
Although the present paper gives only a sketch of the IC,
we should like to highlight three features, which make the
way towards the IC especially promising:
1.
2.
3.
Highly appropriate biological prototype of
immune networks;
Rigorous mathematical basis of FIN;
Possibility of hardware implementation by a
special immune chip.
Such implementation could raise artificial immune systems
as well as their principal applications (e.g. to information
security) on the new level of reliability, flexibility and
operating speed.
[3]
[4]
[5]
[6]
[8]
[9]
[10]
[11]
[13]
[14]
[15]
[16]
[17]
[18]
On the other hand, there matures a strong need to overcome
main disadvantages of the neural networks’ models,
including spurious patterns, small storing capacities
comparatively to the dimension of networks, non-localized
errors, etc. The matter is that they block wide application of
neurocomputers in those fields where the cost of single
error is too high (e.g. aviation, medicine, information
security). But the natural immune networks successfully
protect organism namely from such dangerous “errors” and
invaders. This allows hope that the IC in perspective would
be able to play the similar role in control systems and
computer networks.
Acknowledgement
This work was supported by the EU in the frame of the
project IMCOMP IST-2000-26016.
References
[1]
Adamatzky A. Universal computation in excitable media: the
2+medium. Advanced materials for optics and electronics, 1997, v.7,
pp.263-272.
[19]
[20]
[21]
[22]
[23]
Agnati L.F. Human brain in science and culture. Casa Editrice
Ambrociana, Milano, 1998 (in Italian).
Artificial immune systems and their applications (ed.
D.Dasgupta). Springer-Verlag, Berlin, 1999.
Bender W., Gruhl D., Morimoto N. and Lu A. Techniques for data
hiding. IBM Systems J., v.35, no.3-4, 1996, pp.313-336.
Cantor C. and Schimmel P. Biophysical chemistry. - W.H.
Freeman, San Francisco, CA, 1980.
Coutinho A. Immunology: the heritage of the past. Letters of the
Institute of L.Pasteur. Paris, 1994, no.8 , pp.26-29 (in French).
DeBoer R.J., Segel L.A. and Perelson A.S. Pattern formation in
one and two-dimensional shape space models of the immune system.J. Theoret. Biol., 1992, no.155, pp.295-333.
Ginsburg S. The mathematical theory of context-free languages.
Mc Graw-hill, NY, 1966.
Ekins R. and Chu F.W. Microarrays: their origins and
applications. Trends in Biotechnology, 1999, 17, pp.217-218.
Forrest S., Hofmeyer S. and Somayaji A. Computer immunology.
Communication of the ACM, v.40, no.10, 1997, pp.88-96.
Kuznetsov V.I., Gubanov A.F., Kuznetsov V.V., Tarakanov
A.O., Tchertov O.G. Map of complex ecological evaluation of
Kaliningrad city environment. In: Kaliningrad. Ecological atlas (11
maps), 1999 (in Russian and English).
Kuznetsov V.I., Milyaev V.B. and Tarakanov A.O.
Mathematical basis of complex ecological evaluation.- St.Petersburg
University Press, 1999.
MacBeath G. and Schreiber S.L. Printing Proteins as
Microarrays for High-Throughput Function Determination. Science,
2000, September 8; 289(5485): pp. 1760-1763.
Tannenbaum A.S. Computer networks. Prentice Hall (3rd
Edition), 1996.
Tarakanov A.O. Mathematical models of biomolecular
information processing: formal peptide instead of formal neuron.
Problems of Informatization, 1998, no.1, pp.46-51 (in Russian).
Tarakanov A.O. Mathematical models of information processing
based on the results of self-assembly. Thesis for the Doctor of
Sciences degree in physics & mathematics. St.Petersburg, Russia,
1999 (in Russian).
Tarakanov A. Formal peptide as a basic agent of immune
networks: from natural prototype to mathematical theory and
applications. Proc. of the 1st Int. Workshop of Central and Eastern
Europe on Multi-Agent Systems (CEEMAS’99), St.Petersburg,
Russia, 1999, pp.281-292.
Tarakanov A. and Adamatzky A. Virtual clothing in hybrid
cellular automata. 2000, http://www.ias.uwe.ac.uk/~aadamat/clothing/cloth_06.htm
Tarakanov A. and Dasgupta D. A formal model of an artificial
immune system. BioSystems, 2000, vol.55(1-3), pp.151-158.
Tarakanov A., Sokolova S., Abramov B. and Aikimbayev A.
Immunocomputing of the natural plague foci. Proc. of the Genetic
and Evolutionary Computation Conference (GECCO-2000),
Workshop on Artificial Immune Systems, Las Vegas, USA, 2000,
pp.38-39.
Teniere L. Basis of structural syntax.- Moscow, 1998 (in
Russian, translated from French).
Toffoli T. and Margolus N. Cellular automata machines.
London, MIT Press, 1987.
Wasserman P. Neural computing. Theory and practice. New
York: Van Nostrand Reihold, 1990.