Artif Life Robotics (2008) 13:180–183 DOI 10.1007/s10015-008-0580-y
© ISAROB 2008
ORIGINAL ARTICLE
Kouji Harada · Yoshiteru Ishida
Antibody-based computing
Received and accepted: July 31, 2008
Abstract An application of the antibody’s flexible recognition (i.e. multi-reactivity) to antigenic epitopes to a combinatorial computing is just getting started. The present study discusses an antibody-based computation algorithm to solve a combinatorial problem: the stable marriage problem. The stable marriage problem supposes n men and n women, and each person ranks all members of the opposite sex in a strict order of preference. Under given preference lists, to detect all of “stable” n couples including no affair pairs means to solve this problem. Our algorithm replaces a man and a woman with an antigenic epitope and an antibody respectively, and re-scales a man (woman)’s preference to a woman (man) as strength of a binding affinity between an epitope to the man and an antibody to the woman. Under these settings, we demonstrate a parallel progression of immune reactions can solve the stable marriage problem. Key words Antibody · Epitope · Affinity · Bio-computing · The stable marriage problem
1 Introduction Nowadays, imaginative yet pragmatic parallel computing methods are developing, of which DNA computing by Adleman is distinguished. He operated DNA strands by typical biotechnological instruments, cataphoresis and applications of some enzymes, and demonstrated being able to find in parallel solution paths of Hamilton path problem.1 One feature of DNA enabling a certain type of combinatorial computing is the one-to-one binding of a single K. Harada (*) · Y. Ishida Toyohashi University of Technology, 1-1 Tempaku, Toyohashi, Aichi 441-8585, Japan e-mail:
[email protected] This work was presented in part at the 13th International Symposium on Artificial Life and Robotics, Oita, Japan, January 31–February 2, 2008
strand DNA and its complimentary strand. In some sort of combinatorial problems represented by a directed graph, an arc does not associate with its weight. The most popular example is Hamilton path problem, to which DNAcomputing scheme first applied. In those problems, DNA’s one-to-one binding feature is well-suited to coding those problem settings. However, in the other type of combinatorial problems supposing a weighted arc between vertices (e.g. the stable marriage problem2), representing weighted arcs by one-toone binding-based biomaterials, such as DNA strands would not be easy, hence the representation would become unnatural. On the other hand, immuno agents such as “antibody” and “antigenic epitope (or just epitope)” are appropriate for representing a weighted arc between vertices. Because an antigenic epitope which is a binding partner of an antibody is not always determined by a one-to-one relationship, but rather a one-to-many relationship.3 That is to say, an antibody has different binding affinity against each antigenic epitope. Hitherto, some seminal works discussing feasibility of an antibody-based computation have been presented.4–6 The present work also discusses feasibility of an antibody-based parallel computing method. By way of example, we propose an algorithm for solving the stable marriage problem. A proposed algorithm is based on the fact that a binding affinity of an antibody to each antigenic epitope is different. The stable marriage problem2 can be expressed by a bipartite graph. An instance of the stable marriage problem involves n men and n women, and each person ranks every member of the opposite sex in a strict order of preference. A matching “Mc” (which is n couples of a man and a woman) may include a so-called “blocking pair”, in which a man “M” and a woman “W” are not matched together such that M prefers W to his current partner and W prefers M to her current partner. A particular matching not including any blocking pair is called “stable” matching. To seek all stable matchings under given preference lists means to solve the problem. In our proposed solution algorithm of the stable marriage problem, a man and a women are replaced with a
181 (a)
Man “i”
Mi
( Ei , Ei′)
Ei
Ei′
W4
Anitbody-epitope complex
M3
Affair ?
Couple
A couple of epitopes
Couple
M2
( Ab j , Ab′j )
Ab j
Ab′j
E3
Binding
E3′
Ab4
E3
E3′
Ab4
Wj
Ab4′
W1
Ab4′ Woman “j”
(b)
Surveyed two couples
Binding
Ab1′
E2
Ab1′
E2
A couple of antibodies Fig. 1. A man and a woman represented by a couple of antigenic epitopes and a couple of antibodies
couple of antigenic epitopes and a couple of antibodies respectively, and a man (woman)’s preference to a woman (man) is represented as a binding affinity between an antigenic epitope to the man and an antibody to the woman. The present paper demonstrates a proposed algorithm is able to find in parallel all stable matchings in a given instance.
M
M1
………
1
W1
………
……… ………
2
W4
………
……… ………
Pref
1
E '1
………
E 'i
………
Ab1′
………
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Abj′
………
………
………
………
………
………
………
Ab4′ Ab3′
2
W3
………
……… ………
……… ………
……… ………
……… ………
Wj
k
……… ………
……… ………
………
n
……… ………
……… ………
n
………
………
3
M: man, W: woman, Pref: preference, E′: epitope, Ab′: antibody, Affn: affinity
Table 2. Correspondence relation between a woman’s preference list (left) and an antibody’s affinity list (right) W
Ab
M1
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……… ………
2
M4
………
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3
M2
………
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Abi
………
1
E1
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2
E4
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3
E2
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………
………
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Mj
k
………
………
Ej
………
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…
1
Ab1
…
………
…
W1
…
are given. Here, Mi and Wj represents i th man in the group M and j th woman in the group W, respectively. The setting of the stable marriage problem is mapped to a biochemical system equipped with a set of antibodies and antigenic epitopes. Here a man and a woman are put into a couple of antigenic epitopes and a couple of antibodies, respectively. An antigenic epitope and an antibody are respectively symbolized as “E” and “Ab”, and the i th man and j th woman are represented as (Ei, E′i) and (AbJ, Ab′j) respectively (Fig. 1). A set of the antibodies and that of the epitopes must be selected as they satisfy the previously given man and woman’s preference lists. Concretely if Mi ranks Wj k th in the M′i’s preference order, a selected antigenic epitipe E′i must satisfy ranking k th on strength of a binding affinity to Ab′j (Table 1). Likewise, if Wi ranks Mj k th in the Wis preference order, a selected antibody Abi must satisfy ranking k th on the strength of the binding affinity to Ej (Table 2). Also for any i, j (i, j = 1 … n), let strength of a binding affinity between Abi and E′j or Ab′i and Ej be zero.
Affn
…
W = {W1 , W2 ,………………, Wn },
………
…
M = {M1 , M2 ,………………, Mn },
Mi
3
k
In the setting of the stable marriage problem, a man group M and a woman group W,
E’
…
2.1 Representation by immuno-agents
Table 1. Correspondence relation between a man’s preference list (left) and an epitope’s affinity list (right)
…
2 Antibody-based solution of the stable marriage problem
Fig. 2. (a) Arrangement of antibody-epitope pairs corresponding to the surveyed two couples: M2 − W4 and M3 − W1 to detect if a relationship between M3 and W4 is a blocking pair. (b) Formation of an antibody-epitope complex when a relationship of M3 and W4 is a blocking pair
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n
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n
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Pref
k
Wi
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Affn
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M: man, W: woman, Pref: preference, E: epitope, Ab: antibody, Affn: affinity.
182 Affair ?
2.2 Blocking pair
2.3 Algorithm This section explains an immune reaction-based algorithm of the stable marriage problem. The problem supposes n persons to each sex, so the total number of matching mounts to factorial of n. For a matching MC, a choice of a surveyed couple is nC2 × 2 patterns. The factor “2” considers two possible blocking pairs in two couples. Here we consider all of antigen-epitope pairs corresponding to every two couples in MC, and arrange them in a single horizontal row (Fig. 3). One alignment corresponding to a matching MC represents a solution candidate of the stable marriage problem and let it be symbolized by CMC. Our proposed algorithm is a quite simple and composed of only two operations. The first operation is to prepare all of CMC and to arrange them (Fig. 4). This operation means to prepare all solution candidates. The final operation is to
M:
W4
M1
W4
W2
M3
W2
M3
…… Affair ?
Ab2′ E3
CM
E′1 Ab4 E′3
Ab2
E1
Ab4′ Ab2
E1
Ab4′
……
E′3
n
C2 × 2
Fig. 3. nC2 × 2 antigen-epitope pairs correspond to every combination of two couples in MC, and they are arranged in a single horizontal row
……
…… n
C2 × 2
n!
…… ……
If a matching MC is stable, any two couples in MC does not include a “blocking pair”. This section proposes an immune reaction-based scheme to detect if a blocking pair exists between surveyed two couples. Our scheme is devised to be able to detect a blocking pair through formation of an antibody-epitope complex. In the next place, we take two couples: (M2 − W4, M3 − W1) for example (Fig. 2(a)). When W4 and M3 is an affair relationship, we demonstrate that our proposed scheme can detect (M3, W4) is a blocking pair. As a preparation for the detection, our scheme needs to arrange definitely a layout of antibodies and epitopes corresponding to each man and woman in surveyed two couples. On the first couple: M2 − W4, the epitope E2 corresponding to M2 who is not surveyed must be grounded, and the antibody Ab4 in (Ab4, Ab′4) corresponding to the surveyed W4 must chemically bind the grounded epitope E2. By the same token, on the second couple: M3 − W1, the antibody Ab′1 corresponding to W1, who is not surveyed is grounded and the eqitope E′3 in (E3, E′3) corresponding to the surveyed M3 chemically bind the grounded antibody Ab′1. Under this setting, our scheme can detect a blocking pair. As M4 and M3 is an affair relationship, Ab4 prefers to bind E3 rather than the present binding partner E2, at the same time E′3 also prefers to bind Ab′4 rather than the present binding partner Ab′1. Therefore, Ab4 and E′3 disengage themselves from their present binding partners: E2 and Ab′1, then they bind together and form an antibody-epitope complex: ((Ab4, Ab′4), (E3, E′3)) (Fig. 2(b)). If the above-mentioned arrangement is applied to any two couples, through detection of an “antibody-epitope complex”, we can easily know if surveyed two couples have a blocking pair. In other words, if an antibody-epitope complex is not detected, the surveyed two couples is “stable.”
M1
n
C2 × 2 …… n
C2 × 2
Fig. 4. All CMCs are prepared
just wait until all antibody-epitope reactions proceeding in parallel come to the end. As already explained in the last section, in a certain alignment with a matching including blocking pairs, an antibody and an epitope corresponding to a couple with an affair relationship finally forms an antibody-epitope complex. Hence, every alignment not producing any antibody-epitope complex represents “stable matching” and is a solution of the stable marriage problem (Fig. 5).
3 Conclusions We have proposed the antibody based solution algorithm of the stable marriage problem in order to demonstrate the multi-reactivity between an antibody and an epitope is available for searching a particular relationship hided in many-to-many and weighted relationships. Our proposed algorithm can find in parallel all stable matchings if all solution candidates are previously prepared.
183
……
…… n
C2 × 2
Stable matching
n!
……
……
n
C2 × 2 …… n
C2 × 2
Fig. 5. Every alignment of not forming any antibody-epitope complexes is a “stable” matching
However, there are some problems still left. A major problem is that a concrete scheme to prepare all solution candidates in parallel is not established. It remains possible that operations to prepare the solution candidates become a bottleneck of the proposed algorithm. Now we consider
applying ideas of the antibody microarray7,8 for preparing the solution candidates with fewer steps, however a concrete solution is not found yet. This problem is an issue for the future. Acknowledgments This study has been supported by the Grant-in-Aid for Young Scientists (B) No.18700293 of the Ministry of Education, Culture, Sports, Science and Technology (MEXT) from 2006 to 2008. We are grateful for their support.
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