A Membrane System for the Leukocyte
Selective Recruitment
Giuditta Franco and Vincenzo Manca
Dipartimento di Informatica
Università di Verona, Italy
{franco, manca}@sci.univr.it.
Abstract. A formal description is developed for the phenomenon of
leukocyte recruitment that plays a critical role in the immune response.
Due to its complex nature and capability to rapidly adapt to the attack
of infectious agents, the immune system may be considered a typical
example of complex adaptive system [9].
Here the leukocyte selective recruitment, crucial in immunity, is modeled
as a dynamical system of interactions between leukocytes and endothelial
cells, where a special kind of membrane structure turns out to be a
very useful tool in the formal analysis of the recruitment process. In our
membrane system, besides the traditional rules for communication and
transformation of P systems [8], rules are allowed for the expression of
receptors, for adhesion between membranes, and for the encapsulation
of a membrane inside another membrane.
1
The Phases of Cell Migration
The first response of an inflammatory process in a given organism activates a
tissue-specific recruitment of leukocytes that relies on the complex functional
interplay between the surface molecules that are designed for specialized functions. These molecules are differently expressed on leukocytes circulating in the
blood and on endothelial cells covering the blood vessel.
Leukocyte recruitment into tissues requires extravasation from blood by a
process of transendothelial migration, and three major steps have been identified
in the process of leukocyte extravasation (each mediated by a distinct protein
family [6]): tethering-rolling of free-flowing white blood cells, activation of them,
and arrest of their movement by means of their adherence to endothelial cells.
After this arrest, diapedesis happens, that is, leukocytes from blood pass beyond
endothelial cells into the tissue.
Leukocyte cell has some ‘receptors’, put on its surface, that bind with some
‘counter-receptors’ located on the surface of endothelial cells, and these bonds
slow down the initial speed of leukocyte. Moreover, some molecules (called
chemokines) are produced by the epithelium and by bacteria that have activated
the inflammation process. Chemokines can bind with receptors expressed on a
leukocyte, producing signals inside it. Such signals generate on the leukocyte surface others and different receptors that, interacting with endothelial receptors,
greatly slow down the speed of the cell, until it does not arrest (see Figure 1).
C. Martı́n-Vide et al. (Eds.): WMC 2003, LNCS 2933, pp. 181–190, 2004.
c Springer-Verlag Berlin Heidelberg 2004
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E
C
c
E
Fig. 1. Leukocyte cell attacked by chemokines and endothelial receptors (C and c
leukocytes, E epithelium).
We call A the initial state with leukocytes quickly circulating into the blood,
B the state of rolling, C the state of activation, and D the final state of adhesion. Therefore the system has three big phases: A → B (by means of some
receptor-receptor interactions), B → C (by means of some chemokine-receptor
interactions), and C → D (by means of some receptor-receptor interactions).
Recently, in [3] it was argued that an overlapping of these phases and the
crucial role of quantitative differences provide a very significant discrimination
among behavioral patterns. In our model, after their activation the phases persist
during all the process, but they start at different times (say, an initial time
t1 forA → B, an intermediate time t2 for B → C, and a final time t3 for
C → D); this formal representation suggests that overlapping is a consequence
of a decentralized communication among the different phases.
Moreover, receptors and chemokines are produced with respect to some kinetics. For example, the production of ICAM-1, which is an endothelial receptor,
follows a Gaussian function; on the other hand, LFA-1, that is one of its affine
receptor on leukocyte, is constitutively expressed on the cell. A more accurate
analysis should take into account a finer partition of phases according to the
kind of kinetics.
A Membrane System for the Leukocyte Selective Recruitment
2
183
A Membrane System for Cell Migration
We consider a blood vessel as the region between two membranes. Inside the
external membrane many copies of n different leukocytes revolve around the
internal membrane E (of endothelial cells) at high speed. Our leukocytes are
membranes indexed by 1, . . . , n, respectively. The external membrane, indexed
by L, maintains constant the quantity of leukocytes in the region; it permits
the entrance of further leukocytes from the environment but it does not permit
the exit of them. In Figure 2 we pictorially represent the complete membrane
system.
i
4
2
n−1
j
2
3
1
E
n
i
n
L
1
Fig. 2. Cell Migration System
In order to cope with the phenomenon of receptor-receptor recognition and
bond, we suppose that the E-membrane and leukocytes’ membranes have a sort
of double surface structure which we denote with [E [E and [j [j for j = 1, . . . , n.
Inside these two membrane surfaces there is an interstice where objects can stay
and can be visible from the outside.
This feature can be considered an extension of the communication mechanism of PB systems [2] and symport/antiport P systems [7]. In fact, in PB systems
x[i y means that the membrane i can see outside of its boundary, namely, the
objects x near to its external surface. Now we allow that also semi-internal
objects are visible from outside.
The following discussion will give insights concerning such kind of rules, and
the biological motivation for their introduction. However, here we will not deal
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with the formal developments of the details involved in the membrane model
we use, because we are mainly concerned with the immunological phenomenon
we want to model.
Double surface structured membranes can receive and expel objects; moreover, we admit that the membrane [E [E can engulf membranes of the type [j [j
as a simulation of diapedesis phenomenon. In order to facilitate the comprehension of the system, here we sketch some types of rules that we will assume (the
complete scheme of the rules will be given later):
[E [E o →
[E [E o →
[j o [j , [E o [E → [j
[E o [E , [j o [j → [j
[j o [j o → [j
[E [E o
[E o [E
o o [j , [E [E
[j o , [E [E
o o [j
[j o [j → [E [E [j [j
[E [E [j [j o → [E [E [j [j o
[E [E [j [j o → [E [E [j [j
object o inside [E [E transforms into o
o is located as receptor or chemokine
affine receptors o, o match
recognition between o, o produces o
object o transforms into receptor o
o produces encapsulation of [j [j in [E [E
o inside[E [E is transformed by[j [j into o
object o inside [E [E is deleted by [j [j
where o, o , o are objects, and j = 1, . . . , n. We separate with commas membranes floating in the same region.
Some objects of our system represent the adhesion molecules that, beside
their role in mediating different steps of leukocyte-endothelial cell interaction,
participate to the generation of diversity in the leukocyte recruitment. At
present, at least 3 selectins, 5 mucins, 5 integrins, and 6 immunoglobulin-like
ligands are known to be relevant to leukocyte recruitment, and their various
combinations can be interpreted as tissue-specific recognition signals that provide the targeting of distinct leukocyte subtypes to different organs in different
inflammatory situations. So, in order to indicate these types of molecules, we
use symbols as described in Table 1.
Table 1. Some known receptors.
W
X
w1 = L-selectin x1 = PSGL-1
w2 = E-selectin x2 = ESL-1
w3 = P-selectin x3 = Gly-CAM
..
.
x4 = CD34
x5 = MAdCAM-1
..
.
Y
y1 = LFA-1
y2 = CR3
y3 = VLA-4
Z
z1 = ICAM-1
z2 = ICAM-2
z3 = ICAM-3
y4 = o 4o 7
y5 = o v o 3
..
.
z4 = VCAM-1
z5 = MadCAM-1
z6 = PECAM-1
..
.
A Membrane System for the Leukocyte Selective Recruitment
185
We call receptors the symbols of R = W ∪ X ∪ Y ∪ Z and, as we will see,
they are involved with rules that insert or manage objects into the interstice
of double membrane surfaces. Here we distinguish leukocyte receptors RL and
endothelial receptors RE , where R = RL ∪ RE .
We use a symbol b for indicating the bacteria (of a certain type), and put b
in the E-membrane to start the inflammatory process. We use the symbol ib to
express the inflammation provided by a specific b, and the symbols of a finite
set C = {c1 , . . .} (see Table 2) for indicating the numerous chemokines produced
by inflammed endothelial cells. The infectious agents produce further substances
that favor the adhesion between cells.
Table 2. Some known chemokines.
c1 = CCR2 cj = CXCL9
ck = CCL11
c2 = CCR3 cj+1 = CXCL10 ck+1 = CCL19
..
.
cj+2 = CXCL11 ck+2 = CCL21
.
ci = CXC3 cj+3 = CXCL12 ..
..
..
.
.
Several chemokines act on a single receptor and a single chemokine ‘engages’
more than one receptor, nevertheless these bonds have different binding affinities and conformations. Moreover, leukocyte receptors interact with endothelial
receptors according to a sort of affinity.
In order to express these phenomena we define Ra and Rs that are two
functions on receptors and chemokines.
Let REL = {{x, y} | x ∈ RE , y ∈ RL } and RCL = {{x, y} | x ∈ C, y ∈ RL },
and let Ra and Rs be functions such that:
Ra : REL → N,
Rs : RCL → N,
where Ra ({f, g}) = 0 iff no bond happens between the receptors f and g, and
Rs ({p, q}) = 0 iff no receptor-chemokine bond happens between p and q.
We call affine two elements x, y such that either Ra ({x, y}) > 0 or Rs ({x, y})
> 0. Thus we have two different cases of affinity: the case that x, y are two
affine receptors and the case that they are two affine chemokine and receptor.
For example, with our notation (see Tables 1 and 2), we know that y1 is affine
with cj+3 , ck+1 , ck+2 , and that y2 is affine with cj , cj+1 , cj+2 .
Finally, we have a function S which is defined on
+
RCL
= {{x, y} | {x, y} ∈ RCL , Rs ({x, y}) > 0}
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G. Franco and V. Manca
and its values are signals happening inside the leukocyte cell during own
‘activation’.
In conclusion, our alphabet is constituted by receptors RL and RE , by
+
chemokines C, by b and ib , and by the elements of S(RCL
). For j = 1, . . . , n,
the membrane enter-exit rules are written in Table 3.
Table 3. Leukocyte selective recruitment rules. 1. bacteria b produce inflammation γb 2. inflammation and endothelial cells produce chemokines and RE -receptors αb
3. RE -receptors and chemokines inside the membrane move into E-interstice 4. affine
receptors go into the leukocyte-interstice 5. chemokines o with affine RL -receptors o
generate signals S({o, o }) inside the leukocyte 6. signals inside the leukocyte produce
some RL -receptors βs .
1.
[ E [ E b → [ E [ E b γb
γb ∈ {ib }
2.
[E [E ib → [E [E αb
αb ∈ (C ∪ RE )
3.
[E [E o → [E o [E
o ∈ C ∪ RE
4.
[j o [j , [E o [E → [j o o [j , [E [E
5. [E o [E , [j o [j → [j [j S({o, o }), [E [E
6.
if Ra ({o, o }) > 0
if Rs ({o, o }) > 0
+
s ∈ S(RCL
), α, βs ∈ (RL )
[j α [j s → [j α βs [j
Biological parameters control the productions of chemokines and receptors
by the inflammation, and the productions of RL -receptors by some internal
signals (inside j-membrane) provided by chemokine-receptor interactions.
However, we note that the chemokines present in the system are specific of
the inflammation, and therefore only leukocytes that have receptors affine with
those specific chemokines are selectively recruited.
The values of Ra ({f, g}) and Rs ({p, q}) indicate the force of bonding between
the elements {f, g} and {p, q} respectively.
Let k be the number of different RL -receptors that we find inside the interstice
J of the j-membrane; for m = 1, . . . , k let hm be the quantity of rm receptors.
We can define the force of interaction of the endothelial receptor e on the jmembrane as:
k
Fj (e) =
hm Ra ({rm , e}), e ∈ RE ∩ J.
m=1
Moreover, we define the friction coefficient of an j-membrane as
Fj (e).
Hj =
e∈J
A Membrane System for the Leukocyte Selective Recruitment
187
Of course, the leukocyte j is recruited when its friction coefficient is sufficiently hight, and Hj -value during the recruitment process is inversely proportional to the velocity of the j-cell. Therefore, we have Hj ≈ 0 initially, when
the velocity of j-leukocyte is maximum because there are no bonds involving its
receptors (Fj = 0); if Mj is the threshold of force necessary to stop the circulating j-leukocyte (its value depends on the initial speed of the cell), then we can
assume that Hj ≈ Mj when the speed of j-cell is zero.
Now we can introduce with Table 4 the membrane-inserting rules related to
the leukocyte membranes, where z is the minimum number of bonds able to
produce diapedesis phenomenon, α ∈ (RL ) , and j = 1, . . . , n.
Table 4. Diapedesis and final neutralization of bacteria.
[j α [j
→ [E [E [j [j ∀j such that Hj ≈ Mj , |α| > z
[E [E [j [j b γb → [E [E [j [j b γb ∈ {ib }
[E [E [j [j b → [E [E [j [j
These three rules constituted the final part of the process: the first one
simulates the diapedesis and the other two the neutralization of bacteria by
specific leukocytes.
3
The Dynamics of Cell Migration
In this section we describe the application strategy of the above rules. The rules
are applied with respect to the three phases explained in Section 1. During the
process these phases start in three different times t1 < t2 < t3 , and after their
activation they persist during the whole process.
– Initial Condition
The initial configuration of the system is constituted by an empty E
membrane [E [E ]E ]E and by many copies of [i αi [i ]i ]i , with αi ∈ (RL ) ,
placed outside of E-membrane and inside the L-membrane (see Figure 2):
[L η1 η2 . . . ηn [E [E ]E ]E ]L
where ηi ∈ {[i αi [i ]i ]i } with αi ∈ (RL ) and i = 1, . . . , n.
We can imagine to have as many copies of [i αi [i ]i ]i as we want because they
can arrive from the environment (see Section 2).
The process starts by putting inside the endothelial cells some bacteria that
produce inflammation, a consequent production of specific chemokines, and
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G. Franco and V. Manca
a time-differentiated quantity of endothelial receptors. So the starting rule
is the following one:
[E [E → [E [E b
In other words, the presence of b in the E-membrane induces the parallel
and repeated application of the following rules (that are of the same type as
1, 2, 3 in Table 3):
[E [E b
[E [E ib
[E [E ib
[E [E o
→
→
→
→
[E [E b γb , γb ∈ {ib } ,
[E [E αb , αb ∈ C ,
[E [E δb , δb ∈ (RE ) ,
[E o [E , o ∈ C ∪ RE .
These rules are applied during the whole process in a constant way, except the third one that has a sort of reactivity dictated by the known kinetic of endothelial receptors production. Initially, there are not receptorreceptor bonds because leukocytes are not activated and the production of
RE -receptors is low.
– A → B Rolling Phase
This phase starts at the time t1 , before the activation of leukocytes, and it
persists during the whole process constantly.
Let o ∈ RL , o ∈ RE , α ∈ (RE ) . Suppose that Ra ({o, o }) > 0 and that the
length of α exceeds a prefixed value. For j = 1, . . . , n, rules of the following
type (4. in Table 3) are repeatedly applied.
[j o [j , [E o α [E → [j o o [j , [E α [E
These rules express the fact that when the leukocytes are not activated,
the rolling-phase needs a sufficient production of endothelial receptors. Of
course, after the parallel application of above rules, some Hj change their
values becoming greater than zero.
– B → C Activation of Leukocytes
This phase starts at the time t2 and it persists constantly (because we supposed a constant chemokine presentation) if the following conditions are
satisfied.
In order to simulate the activation of the j-leukocyte, that is, the abundant expression of its RL -receptors (given by internal signals generated by
chemokine-receptors interactions), we define an activation function Gj .
Let k be the number of different RL -receptors that we find inside the interstice J of the j-membrane; for m = 1, . . . , k, let hm be the quantity of rm
receptors, and
Gj (c) =
k
m=1
hm Rs ({rm , c}),
c ∈ C ∩ J.
A Membrane System for the Leukocyte Selective Recruitment
189
We consider Aj =
c∈J Gj (c) as a value that has to exceed a threshold
for the activation of a leukocyte j, that is, the condition to start the RL
production.
Note that Aj is related to several quantitative parameters that are important for the activation and for the consequent arrest of the cell: the number
of receptors expressed on the cell surface (hm ), the affinity constants for
ligand receptor interactions (Rs values) and the density of chemokines
presented on the epithelium.
If Rs ({o, o }) > 0 this rule (5. in Table 3) is applied, for j = 1, . . . , n, until
Aj does not exceed a fixed value:
[E o [E , [j o [j
→ [j [j S({o, o }), [E [E
Then, when Aj exceeds a fixed value, the following rule (6. in Table 3) is
applied:
+
[j α [j s → [j α βs [j s ∈ S(RCL
), α, βs ∈ (RL )
and the number of produced RL -receptors can be computed. Thus, the following rule (4. in Table 3) expresses the fact that during the leukocytes
activation many and different RL -receptors are produced in such a way that
the adhesive phase happens also when few RE -receptors are produced:
[j α o [j , [E o [E → [j o o α [j , [E [E
where o ∈ RE , o ∈ RL , Ra ({o, o }) > 0, α ∈ (RL ) , j = 1, . . . , n, and the
length of α exceeds a fixed value.
Note that in the previous steps the values of Hj , that are related to the
recruited leukocytes, grow; the speed of selected leukocyte decreases consequently, and the conditions for the next phase are generated.
– C → D Adhesive Phase
This phase starts at the time t3 continuing the previous one, and it persists
during the whole process constantly.
Let o ∈ RE , o ∈ RL , α ∈ (RL ) be. If Ra ({o, o }) > 0 and the length of α
exceeds a fixed value, for every j = 1, . . . , n, the following rule (4. in Table
3) is applied until Hj is not close to Mj :
[j α o [j , [E o [E → [j o o α[j , [E [E
This rule and the following ones express the fact that the adhesion happens when many RL -receptors were produced, also during a low chemokine
production.
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G. Franco and V. Manca
For those values of j such that Hj ≈ Mj and if |α| > z (see Table 4), the
following diapedesis rule is applied:
[j α [j → [E [E [j [j α ∈ (RL )
For those values of j such that diapedesis happened the following rules (see
Table 4) are applied:
[E [E [j [j b γb → [E [E [j [j b, γb ∈ {ib }
[E [E [j [j b → [E [E [j [j
The process ends as soon as no b and no ib are present in E-membrane.
4
Conclusion
The analysis developed so far intends to be an application of membrane systems
(with special features) to a real phenomenon occurring in the immunological
system.
The gain of such a formal representation is in the possibility of designing computer simulations of our model in order to observe behaviours, classify them and
predict some dynamical aspects that can be crucial from a biological viewpoint
[4,5].
This kind of research is connected to the study of dynamical system based
on strings [1,10] that we aim at developing in the next future.
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