Large scale agentbased modeling of the humoral and cellular immune response
ABSTRACT The Immune System is, together with Central Nervous System, one of the most important and complex unit of our organism. Despite great advances in recent years that shed light on its understanding and in the unraveling of key mechanisms behind its functions, there are still many areas of the Immune System that remain object of active research. The development of insilico models, bridged with proper biological considerations, have recently improved the understanding of important complex systems [1,2]. In this paper, after introducing major role players and principal functions of the mammalian Immune System, we present two computational approaches to its modeling; i.e., two insilico Immune Systems. (i) A largescale model, with a complexity of representation of 10 6 − 10 8 cells (e.g., APC, T, B and Plasma cells) and molecules (e.g., immunocomplexes), is here presented, and its evolution in time is shown to be mimicking an important region of a real immune response. (ii) Additionally, a viral infection model, stochastic and lightweight, is here presented as well: its seamless design from biological considerations, its modularity and its fast simulation times are strength points when compared to (i). Finally we report, with the intent of moving towards the virtual lymph note, a costbenefits comparison among Immune System models presented in this paper.
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 10th IEEE International Conference on Bioinformatics and Bioengineering 2010, BIBE 2010. 01/2010;

Article: Immune network theory.
[Show abstract] [Hide abstract]
ABSTRACT: Theoretical ideas have played a profound role in the development of idiotypic network theory. Mathematical models can help in the precise translation of speculative ideas into quantitative predictions. They can also help establish general principles and frameworks for thinking. Using the idea of shape space, criteria were introduced for evaluating the completeness and overlap in the antibody repertoire. Thinking about the distribution of clones in shape space naturally leads to considerations of stability and controllability. An immune system which is too stable will be sluggish and unresponsive to antigenic challenge; one which is unstable will be driven into immense activity by internal fluctuations. This led us to postulate that the immune system should be stable but not too stable. In many biological contexts the development of pattern requires both activation and inhibition but on different spatial scales. Similar ideas can be applied to shape space. The principle of shortrange activation and longrange inhibition translates into specific activation and less specific inhibition. Application of this principle in model immune systems can lead to the stable maintenance of nonuniform distributions of clones in shape space. Thus clones which are useful and recognize antigen or internal images of antigen can be maintained at high population levels whereas less useful clones can be maintained at lower population levels. Pattern in shape space is a minimal requirement for a model. Learning and memory correspond to the development and maintenance of particular patterns in shape space. Representing antibodies by binary strings allows one to develop models in which the binary string acts as a tag for a specific molecule or clone. Thus models with huge numbers of cells and molecules can be developed and analyzed using computers. Using parallel computers or finite state models it should soon be feasible to study model immune systems with 10(5) or more elements. Although idiotypic networks were the focus of this paper, these modeling strategies are general and apply equally well to nonidiotypic models. Using bit string or geometric models of antibody combining sites, the affinity of interaction between any two molecules, and hence the connections in a model idiotypic network, can be determined. This approach leads to the prediction of a phase transition in the structure of idiotypic networks. On one side of the transition networks are small localized structures much as might be predicted by clonal selection and circuit ideas.(ABSTRACT TRUNCATED AT 400 WORDS)Immunological Reviews 09/1989; 110:536. · 12.91 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Biology gives us numerous examples of selfassertional systems whose essence does not precede their existence but is rather revealed through it. Immune system is one of them. The fact of behaving in order not only to satisfy external constraints as a prefixed set of possible environments and objectives, but also to satisfy internal "viability" constraints justifies a sharper focus. Adaptability, creativity and memory are certainly interesting "sideeffects" of such a tendency for selfconsistency. However in this paper, we adopted a largely pragmatic attitude attempting to find the best hybridizing between the biological lessons and the engineering needs. The great difficulty, also shared by neural net and GA users, remains the precise localisation of the frontier where the biological reality must give way to a directed design.04/2006: pages 343354;
Page 1
Large Scale AgentBased Modeling of the
Humoral and Cellular Immune Response
Giovanni Stracquadanio1, Renato Umeton2, Jole Costanza3, Viviana Annibali4,
Rosella Mechelli4, Mario Pavone3, Luca Zammataro5, and Giuseppe Nicosia3
1Department of Biomedical Engineering
Johns Hopkins University
217 Clark Hall, Baltimore, MD 21218, USA
stracquadanio@jhu.edu
2Department of Biological Engineering
Massachusetts Institute of Technology
77 Massachusetts Avenue, Cambridge, MA 02139, USA
umeton@mit.edu
3Department of Mathematics and Computer Science
University of Catania
Viale A. Doria 6, 95125, Catania, Italy
{costanza,mpavone,nicosia}@dmi.unict.it
4Neurology and Centre for Experimental Neurological Therapies (CENTERS),
S. Andrea Hospital Site, Sapienza University of Rome
Via di Grottarossa 1035, 00189, Roma, Italy
{viviana.annibali,rosella.mechelli}@uniroma1.it
5Humanitas, University of Milan
Via Manzoni 56, 20089, Rozzano, Milan, Italy
luca.zammataro@humanitasresearch.it
Abstract. The Immune System is, together with Central Nervous
System, one of the most important and complex unit of our organism.
Despite great advances in recent years that shed light on its understand
ing and in the unraveling of key mechanisms behind its functions, there
are still many areas of the Immune System that remain object of ac
tive research. The development of insilico models, bridged with proper
biological considerations, have recently improved the understanding of
important complex systems [1,2]. In this paper, after introducing major
role players and principal functions of the mammalian Immune System,
we present two computational approaches to its modeling; i.e., two in
silico Immune Systems. (i) A largescale model, with a complexity of
representation of 106− 108cells (e.g., APC, T, B and Plasma cells) and
molecules (e.g., immunocomplexes), is here presented, and its evolution
in time is shown to be mimicking an important region of a real im
mune response. (ii) Additionally, a viral infection model, stochastic and
lightweight, is here presented as well: its seamless design from biological
considerations, its modularity and its fast simulation times are strength
points when compared to (i). Finally we report, with the intent of mov
ing towards the virtual lymph note, a costbenefits comparison among
Immune System models presented in this paper.
P. Li` o, G. Nicosia, and T. Stibor (Eds.): ICARIS 2011, LNCS 6825, pp. 15–29, 2011.
c ? SpringerVerlag Berlin Heidelberg 2011
Page 2
16G. Stracquadanio et al.
1Introduction
The theory of clonal selection, formalized by Nobel Laureate F. M. Burnet (1959,
whose foundation are in common with D. Talmage’s idea (1957) of a cellular se
lection as the basis of the immune response), suggests that among all possible
cells, B and T lymphocytes, with different receptors circulating in the host or
ganism, only those who are actually able to recognize the antigen will start to
proliferate by duplication (cloning). Hence, when a B cell is activated by binding
an antigen, it produces many clones, in a process called clonal expansion. The
resulting cells can undergo somatic hypermutation, and then they can give rise
to offspring B cells with mutated receptors. During the immune activity, antigens
compete with these new B cells, with their parents and with other clones. The
higher the affinity of a B cell to bind to available antigens, the more likely it will
clone. This results in a Darwinian process of variation and selection, called affin
ity maturation. By increasing the size of those cell populations through clonal
expansion, and through the production of cells with longer lifetime expecta
tion, and then establishing a defense over time (immune memory), the immune
system (IS) assures the organism a higher specific responsiveness to recognized
antigenic attacks. In particular, on recognition, memory lymphocytes are pro
duced. Plasma B cells, deriving from stimulated B lymphocytes, are in charge of
the production of antibodies targeting the antigen. This mechanism is usually
observed in population of lymphocytes in two subsequent antigenic infections.
More in detail, the first exposition to the antigen triggers the primary response;
in this phase the antigen is recognized and the memory is developed. During the
second response, that occurs when the same antigen is found again, as a result
of the stimulation of the cells already specialized and present as memory cells, a
rapid and more abundant production of antibodies is observed. The secondary
response can be elicited from any antigen, which is similar, although not neces
sarily identical, to the original one, which established the memory. This is known
as crossreactivity.
In this article we present two new computational models capable of capturing
fundamental aspects of the IS, at two different complexity levels. A highcomplex
model, large scale agentbased, that embeds all of the entities and all of the
interaction detailed above; thanks to this computational model we have success
fully reproduced many IS processes and behaviors. In a second, lowcomplexity
model, we show how a stochastic model based on Gillespie algorithm, captures
major behaviors of the IS during a viral infection, even if we are borderline with
the definition of “wellmixed” solution. The paper is structured as follows: next
Section (S2) details the highcomplexity agentbased model; it spans from the
introduction of roleplaying entities in the model, to real simulations and discus
sion. Section 3 details the lowcomplexity IS model; there, after the introduction
of process algebra concepts adopted, it is presented a viral infection model based
on πcalculus. Conclusions (S4) end the paper and give a costbenefit comparison
of the two models in order to pave the way for the whole lymph node simulation.
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Large Scale AgentBased Modeling of the Humoral17
2AgentBased Modeling
Research on the IS dynamics, in the last two decades, has produced several
mathematical and computational models. Different approaches include differen
tial equation based models [3,4,5], cellular automata models [6,7], classifier sys
tems [8], genetic algorithms [9], network/computational models [10,11,12] and
agentbased models [13,14,15,16], which seem to be the best suited abstraction
to handle the great complexity of IS reality.
The first model here presented is based on the deterministic agentbased
paradigm; the highcomplexity of this model comes from the fact that it can
be considered a large scale model and it is extremely realistic; indeed, there
are totally 106− 108cells (e.g., APC, T, B and Plasma cells) and molecules
(e.g., immunocomplexes) involved in this model: all of the major roleplayers
of the IS are embedded and represented in our model. It is worth remarking
the centrality of the scale problem, as choosing a proper model scale have re
cently allowed for important improvements in the IS simulation [17,18,19,20].
Much like the nervous system, the IS performs pattern recognition tasks and,
additionally, it retains memory of the antigens to which it has been exposed. To
detect an antigen, the IS activates a particular recognition process that involves
many roleplayers. An overall view of IS roleplayers includes: antigens (Ag),
B lymphocytes (B), plasma B cells (PLB), antigen presenting cells (APC), T
helper lymphocytes (Th), immunocomplexes (IC) and antibodies (Ab). The Ag
is the target of the immune response. Th and B lymphocytes are responsible
for the discrimination of the selfnonself, while PLBs produce antibodies able
to label the Ags to be taken by the APCs, which represent the wide class of
macrophages. Their function is to present the phagocytised antigens to T helper
cells for activation. The ICs are Ab–Ag ties ready to be phagocytised by the
macrophages. All of the roleplaying entities here described are encoded in our
model: each agent has a type (i.e.: Ag, B, PLB, etc.) and those typical features
that characterize the type (e.g., the Ag has a unique code, or bit string, that
will determine whether there will be a bind with a complementary entity); each
agent belongs to a population (e.g., the Ags, Bs, etc.) whose size is plotted in
order to quantify group presence, affinity driven interactions, mutations, clonal
selection and all of the processes detailed above.
2.1Immuno Responses
The IS mounts two different responses against pathogenic entities: the humoral
response, mediated by antibodies, and the cellular one, mediated by cells. Along
with the aforementioned entities, the IS includes the T killer cells, the Epithelial
cells or generic virustarget cells, and various lymphokines. These components
are necessary to activate the cellular response. One can, from an abstract point
of view, envision two types of precise interaction rules: specific interactions, that
occur when an entity binds to another by means of receptors; and, nonspecific
interactions, that occur when two entities interact without any specific recog
nition process. Only a specific subset of mature lymphocytes will respond to a
Page 4
18G. Stracquadanio et al.
0
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0 50 100 150 200
Concentration
Time Steps
Antigen Dynamics
0
10000
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30000
40000
50000
60000
70000
0 20 40 60 80 100 120 140 160 180 200
Concentration
Time Steps
Antibody Dynamics
131020
245709
261828
262126
229340
253388
Fig.1. Immunization  Antigen and Antibody dynamics. Injections of antigens at time
steps 0 and 120.
0
500
1000
1500
2000
2500
0 20 40 60 80 100 120 140 160 180 200
Concentration
Time Steps
B Cell Dynamics
131020
245709
261828
262126
229340
249804
0
200
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600
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1400
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0 20 40 60 80 100 120 140 160 180 200
Concentration
Time Steps
Plasma Cell Dynamics
131020
245709
261828
262126
229340
262100
Fig.2. Immunization  B Cell and Plasma cell dynamics
given Ag, specifically those bearing the receptors that will bind the Ag. Binding
usually occurs on a small patch on the Ag (receptor or antigenic determinant
or epitope), and the antigenbinding sites on T cell receptors and antibodies
(paratopes or idiotypes). Thus, immune recognition of Ags comes from the spe
cific binding of antigentoantigen receptors on B and T cells. Hence, the immune
response derives its specificity from the fact that Ags select the clones. A non
active B or T cell that has never responded to an Ag before is called virgin or
naive. When a naive, mature lymphocyte bearing receptors of the appropriate
affinity, binds an Ag (in combination with other signals, usually cytokines), it
responds by: proliferating, i.e., cloning itself and, in turn, expanding the popu
lation of cells bearing those receptors; the produced clones will differentiate into
effector cells that will produce an appropriate response (antibody production for
a B cell; cytotoxic responses or help responses for a T cell), and memory cells
that will be ready (somewhat like the naive cell) to encounter the Ag in the
future and respond in the same way (but this time with many more cells).
Switching from theory right to simulations, we present the results of our agent
based encoding with 18 bits; its representation capability is of the order of 106
cells/molecules. Fig. 1 shows time track of antigen, and antibody population
Page 5
Large Scale AgentBased Modeling of the Humoral19
0
2000
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6000
8000
10000
12000
14000
16000
18000
0 20 40 60 80 100 120 140 160 180 200
Concentration
Time Steps
T cell Dynamics
252900
253932
251620
253412
253668
251884
0
2000
4000
6000
8000
10000
12000
0 50 100 150 200
Concentration
Time Steps
Immuno Complex Dynamics
Fig.3. Immunization  T cell and Immunocomplexes dynamics
of a system injected with the antigen at time steps 0 and 120. Figure 2 shows
primary and secondary immune responses of B lymphocyte and Plasma cell
population. Figure 3 reports T cell population and Immunocomplexes. We have
three main immune response types to a given antigen: immune response by T
killer (cytotoxic) lymphocyte; by T helper lymphocyte; and, by B lymphocyte.
In Figure 2, we can see the immune response performed by lymphocytes of class
B: the free Ag selects a B lymphocyte, whose receptors match its own. These two
entities bind together. A B lymphocyte, which has internalized and transformed
the Ag, shows on its surface fragments of Ags bound to a protein coded by MHC
molecule of class II. The mature T helper lymphocyte can, now, bind to the
complex antigenprotein, visible on the B lymphocyte. Such a binding frees the
interleukin IL2, which in turn allows the B lymphocyte to clone and differentiate.
The cellular cloning goes on as long as the B lymphocytes are stimulated by T
helper lymphocytes. Mature PLBs free their receptors, Abs, which bind free
Ags, creating ICs. In turn, they will be phagocytised by APCs (the “garbage
collectors” of the IS). Other mature B lymphocytes stay in the system as B
memory cells. Lastly, the IS comprises the hypermutation phenomena observed
during the immune responses: the DNA portion coding for Abs is subjected to
mutation during the proliferation of the B lymphocytes. This gives the IS the
ability to generate diversity. We should underline that, even if the knowledge on
the various mechanisms of the immune system is quite advanced, the relative
importance of its components with respect to each specific task is not deeply
understood.
2.2CrossReactivity and Epithelial Cell Signaling
Now we show the molecular and cellular population dynamics correlated with
two further events involved in the clonal selection principle: crossreactivity and
epithelial cell signaling. Cell concentrations give us a clear picture of the learning
and immunization processes that occurred during immune responses. To do this,
we need some recollection about a model extended by our agentbased model:
the CeladaSeiden model [7]; the manner is a robust computational model based
Page 6
20G. Stracquadanio et al.
0
10000
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30000
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0 50 100 150 200 250 300 350 400
Concentration
Time Steps
Crossreactivity  Antigen Dynamics
0011011111101111
1011011101101111
0
10000
20000
30000
40000
50000
60000
70000
80000
90000
100000
0 50 100 150 200 250 300 350 400
Concentration
Time Steps
Crossreactivity  Antibody Dynamics
262140
262044
261612
262110
261960
262093
Fig.4. CrossReactivity, Antigen and Antibody dynamics
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
0 50 100 150 200 250 300 350 400
Concentration
Time Steps
Crossreactivity  T cell Dynamics
253668
251620
188388
253925
249828
253860
0
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4000
6000
8000
10000
0 50 100 150 200 250 300 350 400
Concentration
Time Steps
Crossreactivity  Immuno Complex Dynamics
0
1
Fig.5. CrossReactivity, T cell and Immunocomplexes dynamics
on the cellular automata paradigm that has been validated with in vitro and
in vivo experiments. In particular, it has been shown that CeladaSeiden model
can reproduce real phenomena of the immune response. The model includes
the following seven entities: Ags, B cells, PLB, APC, Thelper cells, IC, and
Ab. The chemical interactions between receptors (the bindings) are mimicked as
stochastic events. The probability that two receptors interact is determined by
a bittobit matching over the bit strings representing them. Every cell can be
in one of the allowed states (e.g., a B cell can be Active, Internalized, Exposing,
Stimulated or Memory according to whether it has bound an antigen or not, if it
expose the MHC/peptide complex, if it duplicates or if it is considered a memory
B cell) and successful interactions between two entities produce a cellstate
change. The cellular automata of the CeladaSeiden’s model has an underlying
regular, hexagonal, twodimensional lattice. Each site incorporates many entities,
which interact in loci and diffuse to adjacent sites and then move randomly.
Every site of the automaton includes a large number of bit strings accounting
for the definition of the various entities and for their states (i.e., both receptors
and cellstates). In our agentbased simulation, initially there are neither PLBs
nor Abs, nor ICs in the Host, given a specific class of antigens. The plot of Fig.
4 (left plot) shows the three injections of antigens. In the first two injections,
Page 7
Large Scale AgentBased Modeling of the Humoral21
0
500
1000
1500
2000
2500
3000
0 50 100 150 200 250 300 350 400
Concentration
Time Steps
Crossreactivity  B cell Dynamics
262140
262044
261612
262110
261960
262093
0
200
400
600
800
1000
1200
1400
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1800
2000
0 50 100 150 200 250 300 350 400
Concentration
Time Steps
Crossreactivity  Plasma Cell Dynamics
262140
262044
261612
262110
261960
262093
Fig.6. CrossReactivity, B Cell and Plasma cell dynamics
we insert the same antigen, namely the binary string (0011011111101111). The
third time we inject a  mutated  antigen (two bits underwent a simulated flip
mutation), namely the binary string (1011011101101111). It is of note how the
crossreactivity, observed in nature is here reproduced: Fig. 4 (right plot), Fig.
5 and Fig. 6 present the dynamics of cell populations under control, presenting
important similarities with real immunological responses.
Moving towards larger and then more complex (but closer to the reality)
systems, we have enriched our IS simulation scenario: we indeed simulated a 20
bit encoded IS. With such an encoding, we have been able to simulate more
than 108cells and molecules. We have simulated such an environment for 400
time steps. Fig. 7(a) presents those Ags introduced in the system, while Fig.
7(b) shows the Interferon response released by lymphocytes, a countermeasure
that the IS has dynamically adopted. It is of note how the Interferon response
is triggered rightly after the introduction of Ags, with a diversified intensity.
Such intensity variation is motivated by the primary and secondary responses
and crossreactivity reaction.
0
500000
1e+06
1.5e+06
2e+06
2.5e+06
3e+06
0 50 100 150 200 250 300 350 400
Concentration
Time Steps
Antigen Dynamics
Antigen type 1
Antigen type 2
0
1e+07
2e+07
3e+07
4e+07
5e+07
6e+07
0 50 100 150 200 250 300 350 400
Concentration
Time Steps
Interferon Dynamics
Fig.7. (a) Antigen Dynamics, as observed after its introduction in the 20 bits simula
tion environment. (b) Interferon response, dynamically triggered after the Ag detection.
Page 8
22G. Stracquadanio et al.
930000
940000
950000
960000
970000
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1e+06
0 50 100 150 200 250 300 350 400
Concentration
Time Steps
Epithelial Cell Dynamics
0
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2e+06
2.5e+06
3e+06
3.5e+06
4e+06
0 50 100 150 200 250 300 350 400
Concentration
Time Steps
DSignal
Fig.8. (a) Epithelial cells in the IS; the change in time of this cell population is a key
component in the natural IS. (b) Dsignal propagated by epithelial cells to warn other
IS components thought a signaling mechanism.
In Fig. 8(a) it is presented how the number of epithelial cells changes in time:
these cells are part of our natural IS. It has been validated that these cells
represent not only a mechanical barrier in our IS, but they are also enabled to
communicate the infection through an adhoc signal as described in [21,22]; in
fact a “warning signal”, namely the Dsignal is used to propagate the information
that something uncommon is taking place. Fig. 8(b) presents the Dsignal in
object, as observed within this 20 bit simulation.
3Modeling and Simulation by Stochastic πCalculus
Biological entities, such as proteins or cells, are social entities (i.e., they act as an
organized group), and life depends on their interactions. We can then reduce a
biological system to a network of entities interacting among each others in a par
ticular way, i.e., a way in which each elementary process is coded and controlled.
The stochastic πcalculus has been recently used to model and simulate a range
of biological systems; in particular, stochasticbased models [23] and more in
general statisticsbased models [24], have recently improved our understanding
of key components of the immune system. In this section we want to show the
new features of this approach, the pros and cons, and finally the results obtained
in simulating HBV infections.
In recent years, there has been considerable research on designing program
ming languages for complex parallel computational tasks. Interestingly, some of
this research is also applicable to biological systems, which are typically highly
complex and massively parallel systems. In particular, a mathematical program
ming language known as the stochastic πcalculus [25] has been recently used
to model and simulate many biological systems. One of the main benefits of
this calculus is its ability to model large systems incrementally, by composing
simpler models of subsystems in an intuitive way (i.e., modularity). Such in sil
ico experiments can be used to formulate testable hypotheses on the behavior
of biological systems, as a guide to future experiments in vivo. Currently avail
able simulators for the stochastic πcalculus are implemented based on standard
Page 9
Large Scale AgentBased Modeling of the Humoral23
theory of chemical kinetics, using an adaptation of the Gillespie algorithm [26].
There has already been substantial research on efficient implementation tech
niques for variants of the πcalculus, in the context of programming languages
for parallel computer systems. However, this research does not take into account
some specific properties of biological systems, which differ from most computer
systems in fundamental ways. A key difference is that biological systems are
often composed of large numbers of processes with identical (or equivalent) be
havior, such as thousands of proteins of the same type. Another difference is
that the scope (the environment where the interaction can actually take place)
of private interaction channels is often limited to a relatively small number of
processes, usually to represent the formation of complexes. In general two fun
damental intuitions are shared by all of these proposals: 1. molecules (individual
biological agents, in general) can be abstracted as processes; 2. molecular inter
actions can be modeled as communications. Here we introduce SPiM [27], the
calculus simulator used in our experiments, and our simulation of infection, i.e.,
the HBV virus, based on the Perelson’s model [28].
3.1
SPiM
Proceeding from the concept that molecules are represented as processes and
interactions are communications, we can see that the syntax of processes and
environments in SPiM is a subset of the syntax of the stochastic πcalculus (SPi)
with the additional constraint that each choice of action is defined separately in
the environment. Stochastic behavior is incorporated into the system by associ
ating to each channel x its corresponding interaction rate given by ρ(x), and by
associating each delay τr with a corresponding rate r. Each rate characterizes
an exponential distribution, such that the probability of a reaction with rate r
occurring within the time t is given by F(t) = 1 − e−rt. The average duration
of the reaction is given by the mean 1/r of this distribution. A machine term
V consists of a set of private channels Z, a store S and a heap H. The heap
keeps track of the number of copies of identical species, while the store records
the activity of all the reactions in the heap. The system is executed according
to the reduction rules of the stochastic πmachine [25]. The rules rely on a con
struction operator V?P , which adds a machine process P to a machine term
For the calculus dialect implemented by SPiM, Phillips et al. proved a formal
equivalence between an adhoc graphical notation and a SPiM program [29].
This is very interesting because people without programming background can
formalize a complete and correct model in πcalculus without knowing SPiM:
for instance, biologists can detail a biological system using this graphical nota
tion and, SPiM is particularly suited to simulate a complete biological system
without building a full mathematical model of the interactions. Together with
good features there are also some drawbacks; in particular, SPiM is suitable
when all rates are known, and they do not change at running time: this is a very
strict hypothesis, because many biological systems have entities that change their
rate of interaction at running time or in some particular situations. E.g., in the
V (cf. Table 1 for detail).
Page 10
24G. Stracquadanio et al.
Table 1. Syntax of processes and environments in SPiM. The syntax is a normal form
for the stochastic πcalculus, in which each choice of actions can only occur at the top
level of a definition. For convenience, C is used to denote a restricted choice νnM and
D is used to denote the body of a definition. For each definition of the form X(m) = D
it is assumed that fn(D) ⊂ m.
P,Q ::= 0Null
Instance
Parallel
Restriction
E::=Empty
ProcessX(n)
P — Q
νxP
E,X(m)=P
E,X(m)=νnM Choice
π::= ?x(m)
!x(n)
τr
Input
Output
Delay
M::=0
π.P + M Action
Null
immune system, the rate of interaction, the affinity, between Bcell and Antigen
can change over the time because a Bcell can undergo hypermutations and then
altering its receptor and so its affinity with the Antigen [30].
3.2Modeling HBV Infection with SPiM
Here we present a stochastic picalculus model of infection caused by hepatitis
B virus (HBV). The simulation is based on the basic model of virus infection
proposed in [28], from which we report Fig. 9. The choice of the Perelson model
is justified by the fact that this model was tested in vivo and found a broad
consensus where HBV is concerned [31]. The model considers a set of cells sus
ceptible to infection (i.e., target cells), T which, through interactions with a virus
V , become infected. Infected cells I are assumed to produce new virus particles
at a constant average rate p and to die at rate δ, per cell. The average lifespan of
a productively infected cell is 1/δ, and so if an infected cell produces a total of N
virions during its lifetime, the average rate of virus production per cell, p = Nδ.
Newly produced virus particles, V , can either infect new cells or be cleared from
the Host at rate c per virion. HBV infection was modeled with these Ordinary
Differential Equations (ODE):
dT/dt = s − dT − βV T
dI/dt = βV T − δI
dV/dt = pI − cV
(1)
(2)
(3)
Target cells become infected at a rate taken proportional to both the virions
concentration and the uninfected cell concentration (βV T); while infected cells
(I) are produced at the same rate (βV T) and it is assumed that they are depleted
at rate δ per cell. Finally, it is assumed that free virions are produced at a
constant rate, p, per cell and cleared at constant rate, c. We have simulated
an HBV infection with a high number of virions in blood; we have set virions
at 20% of target cells as we used a population of 5 × 103target cells and 103
virions. This ratio corresponds to the scenario in which the infection is growing,
Page 11
Large Scale AgentBased Modeling of the Humoral25
??
?????????????????
??
?
?
????
?
???????????????????????????????????????????????????
??????
?
????
Fig.9. Perelson’s model for infection
0
100
200
300
400
500
0 50 100 150 200 250 300
Target Cells
Infected Cells
0
50
100
150
200
250
300
350
400
450
500
0 50 100 150 200 250 300
Target Cell
Virions
(a)(b)
Fig.10. Simulation of HBV infection using a stochastic πcalculus model. In (a), we
report the variation over time (xaxis) of the number of target and infected cells (y
axis); in (b), we present the variation over time (xaxis) of the number of target cells
and virions (yaxis).
rapidly, possibly degenerating in a chronic status. We have set the infection rate
at a relatively low rate, but from the analysis of the plot in Fig. 10, we can see
that the number of virions is big enough to kill target cells. Moreover, infected
cells rapidly grow with respect to healthy ones; it is clear that after a first
phase where initial virions infect target cells, thanks to the growth of infected
cells, the growth in terms of number of virions comes along. This observation
is confirmed by Perelson studies, where, after an initial steady state, virions
grow proportionally to the number of infected cells. As in Perelson model, the
immune system response is implicitly considered by setting the rate of death
for virions and infected cells; although there is an invivo assessment of this
model, in general, finegrained simulation should take into account several other
boundary conditions, like cell type and pharmacological treatment.
4 Conclusions
In this paper we have given a broad understanding of the Immune System and
we have presented two models for its simulation (i.e., a largescale agentbased
model, and a simpler stochastic one), each model has been introduced together
with its strength key points, and with its pertinence context. It is worth men
tioning in our conclusions, which are the modeling factors that have to be taken
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26 G. Stracquadanio et al.
into account in the choice of a modeling approach for the IS. If the choice is
between a more complex agentbased model versus a stochastic simpler model,
there are three factors that play a major role in the modeling outcome: (i) simula
tion time, (ii) model precision and accuracy, and (iii) model applicability. Where
simulation time is concerned, stochastic models have surely to be preferred; in
fact, models here outlined base their evolution process on the evaluation of a
stochastic function; e.g., the Gillespie algorithm used in SPiM, guarantees light
binaries and execution times in the order of seconds on a desktop computer for a
small model, such as the one we presented on HBV. As far as agentbased mod
els are concerned, execution times are generally larger of at least one order of
magnitude, moving towards minutes and, according to the number of molecules
involved in the simulations, maybe towards hours. It is also interesting how, for
agentbased and cellular automata models, there is a direct mapping (through an
opportune scaling factor) between simulated time steps and realworld timing;
the latter, can provide interesting insights about the reliability of an IS simula
tion and its biological plausibility (e.g., a model where humoral immune response
is seen within the same day of the infection has to be preferred when compared
to another model where the same response can be observed only after a simula
tion time that corresponds to one year in realworld timing). The second factor
that has to be considered in the modeling is the precision and the accuracy: if
stochastic models provide faster answers, with agentbased models we can track
the behavior of the single cell/molecule involved in the system and then we have
a significantly higher precision in the model controlling. This means, for instance,
that we can operate single element alterations at runtime (e.g., an unexpected
mutation), without the need of hardcoding this event in our model specification
(as one has to do in πcalculus and in process algebra in general). Moreover,
in agentbased models, we can tune affinity and the equivalent of reaction rate
constants at runtime. Finally, we can track the behavior of a family of agents
involved in the system and then study how different cell populations interact one
versus the other. These interesting features of agentbased models, come with
a price, that is the longer execution time discussed above. The last factor here
discussed, is the model applicability and its pertinence: with this argument we
want to highlight the fact that not all the modeling approaches can be really
extended towards the perfect virtual simulation that has a 1:1 mapping with re
ality. With respect to this, it is worth noting that when the spatial characteristic
of the IS has to be simulated, stochastic models based on the Gillespie algorithm
loose one of their theoretical axes, that is the fact that all of the molecules in
the simulation are in a “wellmixed” context. Recent extensions of the Gillespie
algorithm have been proposed to account for the spatial information [32]; in
an example in which there are areas where Antigens are the majority and the
immune response begins, then there are regions where almost nothing happens,
and finally there are other regions where naive lymphocytes are the majority, it
seems clear that the spatial information has an important role.
In conclusion, where SPiM based modeling of the IS is concerned, its features
are definitely preferable when either light computation or system modularity are
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Large Scale AgentBased Modeling of the Humoral27
more important than model precision and accuracy and when the spatial as
pect of the system does not play an important role. Finally, moving towards
the virtual lymph node, agentbased models, and in particular the 20 bit model
here presented, seems a valuable approach for the simulation of this (very)large
scale system, made of a number of cells/molecules in the order of 108, that can
interact among eachothers in a spatially aware context where different regions
are devoted to different functions. In the following we give some insights about
possible applications of such a system. Biological transferability and applicability
of a verylarge scale system are important and wide; where prevention therapies
are concerned, a verified computational model could be employed in the develop
ment of new vaccines. The idea would be to perform a simulation on a number of
new molecules (e.g., molecule [A] and molecule [B]) and study how they interact
with the Host, i.e.: f(newmolecule) and among eachothers, as the effect could be
additive (f([A] + [B]) = f([A]) + f([B])), neutral, i.e.: molecules designed for a
competing aim ignore eachothers resulting in f([A]+[B]) = f([A]) = f([B]), or
even disruptive. With a verified IS computational model, we could even study
timeseries of the simulated response – a practical application would be (i) to
modulate vaccine injection schedule in order to have and enhanced immuniza
tion; (ii) drug resistance could be studied in terms of timeseries as well. To
move towards such beautiful scenarios, we are currently considering the conver
sion of the Binary epitope into a realistic one, built on top of the amino acid
alphabet; with respect to the latter point we are investigating an alternative
epitope library employing an accepted framework [33] for the (i) Single and (ii)
MultiObjective modeling approach aimed at a design of the antibody comple
mentary determining regions that is (iii) extended with the notion of functional
Robustness [34] at the epitope binding task.
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