Ishida (2008) - Antibody-Based Computing Stable Marriage

7/31/2019 Ishida (2008) - Antibody-Based Computing Stable Marriage

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ORIGINAL ARTICLE

Y. Ishida (*)Department of Knowledge-Based Information Engineering, andIntelligent Sensing System Research Center, Toyohashi University ofTechnology, Tempaku, Toyohashi 441-8580, JapanTel. +81-53-244-6895; Fax +81-53-244-6873e-mail: [email protected]

This work was presented in part at the 12th International Symposiumon Artificial Life and Robotics, Oita, Japan, January 2527, 2007

Artif Life Robotics (2008) 12:125128 ISAROB 2008DOI 10.1007/s10015-007-0452-x

Yoshiteru Ishida

Antibody-based computing: an application to the stable marriage problem

On the other hand, the post-genome era proved that thesequence information of the human genome alone is notsufficient, but a higher knowledge of their function must be

revealed. The post-genome age naturally proceeds to studyimmune systems,7 focusing not only on components suchas antibodies and MHC, but also on their systemicorganization.

After the demonstration of DNA-based computing byAdelman, protein-based computing was proposed by Hugand Schuler2 and extended by Balan and Krithivasan.8 Thispaper, in a similar spirit of synthetic biology and systembiology, tries to construct an information processing de-vice incorporating the specific recognition capabilities ofantibodies. The stable marriage problem9,10 is used as abenchmark problem for demonstrating an array formatimplementation of antibody-based computation.

2 Antibody-based computing: an introduction

One of the major components propelling the informationprocessing of the immune system is the specific recognitionof antigens. The immune system is capable of recognizingeven artificially synthesized substances. Also, it can furtherclassify substances into the self (those derived from theindividual) and nonself. Among those bearing recognitioncapabilities, antibodies are undoubtedly important. Further,antibodies have been studied in great detail not only in

theoretical biology, but also in bio-engineering. Antibody-based computing directly focuses on the recognition capa-bility, and integrates it for problem-solving, includingcombinatorial problems.

2.1 Antibody-based computing andDNA-based computing

Similar to DNA-based computing, antibody-based comput-ing utilizes complementary matching between macromole-cules: antibodies. Since the computational capabilities ofDNA-based computing could be inherited by antibody-

Abstract Antibodies, among other things, are importantcomponents of the immune system. This paper proposesusing the specific recognition capability exhibited by anti-

bodies for computation, in particular, for solving the stablemarriageproblem, which has been studied as a combinato-rial computational problem. Antibody-based computationis proposed by integrating the recognition capabilities ofantibodies. The computation is carried out on an array formthat is suitable not only for expressing stable marriageproblems, but also for further integration to antibodymicroarrays.

Key words Antibody-based computing Stable marriageproblem Problem solving DNA-based computing Microarray

1 Introduction

After Adelman1 pioneered DNA-based computing in hisseminal work, many researchers established that not onlyDNA, but also other macromolecules could have computa-tional capability comparable to that of DNA.2 Elicits andLeibler3 and Sprinzak and Elowitz4 even demonstrated thata genetic circuit may not be a dream. Even synthetic multi-cellular systems have recently been studied,5 leading to syn-thetic biology. Possible drug production by engineeringyeast6 has had a great impact on the area. These new bio-

engineering technologies have provided bioinformatics notonly with new tools, but also with systemic views.

Received and accepted: July 23, 2007


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based computing, we rather focus on the difference betweenthem. Roughly, in the context of solving combinatorialproblems, the difference may be clarified by the correspon-dence between a complementary matching (as in the mar-riagetheorem) and a stable pairing based on preference (asin the stablemarriageproblem).

Affinity between antigens and antibodies can be mea-sured and their intensities can be ordered (as formatted in

an affinity matrix). That is, in contrast to Matching(DNA1,DNA2) = 1 (matched) 0 (not matched), Affinity(Antigen1,

Antibody2) could vary from 0 (no agglutination) to 1(highest agglutination). This difference would suggest thatantibody-based computing could potentially implement anerror tolerance that could not be implemented in DNA-based computing. The difference would further suggest thatantibody-based computing may be extended to a problemsolver if the adaptive mechanism of the immune systemrealized by antibodies and their maintenance is alsoinvolved.

3 Solving a combinatorial problem

3.1 Stable marriage problem

In a naive form, the problem assumes n men and n women,with each member having preference lists for members ofthe opposite sex. A pair of a man Mi and a woman Wj iscalled a blocking pair if they are not a pair in the currentsolution, but Mi prefers Wj to his current partner, and Wjprefers Mi to her current partner as well. A matchingbetween men and women with no such blocking pair iscalled stable.

Having stated the stable marriage problem, it would benatural to think of the algorithm for antibody-based com-puting. That is, the stable marriage problem (SMP) may bemapped to an antigenantibody reaction, so that the prefer-ence order of each person in the SMP will be reflected inthe affinity intensity between an antibody and an antigen.After antibodies and antigens are so arranged, the solutionof the SMP will emerge by observing the concentration ofthe agglutination. It should be remarked that the agglutina-tion process could be any agglutination (not necessarilybetween antibodies and antigens) if their intensities aremeasurable and ordered.

3.2 Mapping a stable marriage problem toantibody-based computing

As stated above, mapping a combinatorial problem toantibody-based computing can be done by composingantigenantibody compounds corresponding to a problementity. Antibodies and antigens for a compound corre-sponding to a particular individual will be determined byconsidering her (his) preference list of men (women).

Landsteiners ABO blood group system11 may be afamiliar yet simple example. His blood type system is basedon antigens (as agglutinogen) on red blood cells and anti-

bodies (as agglutinin) in the blood serum. Table 1 shows theagglutinogen and agglutinin of each blood type. The affinitybetween antibody and antigen is shown in Table 2. Table 3indicates the well-known incompatible transfusion amongthe blood types A, B, AB, and O.

In this example, we map the relation that the woman Wi(the man Mi) prefers the man Mj (the woman Wj) to anyother to the relation of whether the blood ofWi (Mi) would

be agglutinate when the blood of Mj (Wj) was transfused.That is, if the woman Wi prefers the man Mj, most bloodtypes should be so assigned that the type for Wi comprisesantibody AbWi and antigen AgWi, and that for Mj of anti-body AbMj and antigen AgMj, and the affinity Aff(AbWi,

AgMj) is highest.For the trivial case when the preference lists of men and

women are as in Table 4, a simple assignment would suffice:a man to type A and another man to type B for the womanwho likes a man with type A to type B and for anotherwoman type A (Fig. 1). It should be noted that the assign-ment to A for two men and to B for two women would not

Table 1. Landsteiners ABO blood group system

Blood type

A B AB O

Antigen (agglutinogen) A B A, B NoneAntibody (agglutinin) None ,

Table 2. Affinity matrix. The circle indicates that the antibodyantigenreaction would occur if the antibodies in the column meet with antigensin the row

Antibody Antigen

A B

(anti-A) (anti-B)

Table 3. Agglutination when the blood type in the column is trans-fused with the blood type in the row. A circle indicates that the bloodtype of the column would agglutinate when transfused to that of thecolumn

Blood type

A B AB O

A

B AB O

Table 4. A trivial preference list for the two two stable marriageproblem

M1 M2

W1 1 2W2 2 1

W1 W2

M1 1 2M2 2 1


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work, since the assignment does not reflect the preferenceof the men and women.

In the nontrivial preference list shown in Table 5, one

assignment would be type O to M1 and W1, type A to M2,and type B to W2 (Fig. 2). For other two preference lists(with the graph topologically different from those shown inFigs. 1 and 2), it is not possible to map the blood type withthe above correspondence, and other compounds should besynthesized to realize the preference lists.

These examples suggest a scheme for synthesizingantigenantibody compounds that realize mapping fromgiven preference lists to the compounds. If the woman Wiprefers the man Mj to other men, the compound corre-sponding to Wi contains antibody AbWi and the compoundcorresponding to Mj contains antigen AgMj, which satisfiesAff(AbWi, AgMj) as the highest among other AgMj

(j= 1 . . . n).If Mj is the second in the preference list of Wi, then

Aff(AbWi, AgMj) must be second highest and so on. AgMjmust realize the orders from women Wk other than Wi,hence the affinity Aff(AbWk, AgMj) must realize the orderaccordingly. (IfAgMj alone cannot realize the order, thena new antigen realizing the order must be added to the cor-responding compound. In fact, the above example also sug-gests that compounds can be composed of a set of antigensand a set of antibodies.) Constraints for selecting antibodiesand antigens for a compound corresponding to a person canbe summed up as follows:

Aff(AbW

i,AgM

j)>

Aff(AbW

i,AgM

k) if the womanW

iprefers Mj to Mk in her preference list for all WiW,and for all distinct pairs Mj, MkM;

Aff(AbMi,AgWj) > Aff(AbMi,AgWk) if the man Miprefers Wj to Wk in his preference list for all Mi M,and for all distinct pairs Wj, WkW.

3.3 Solving a stable marriage problem withan array format

An illustrative example with Landsteiners blood groupsuggests the following algorithm to solve the SMP with an

array format. In the array, row i and column j correspondto the compound for man i (i.e., AbMi and AgMi) and thatfor woman j (i.e., AbWi and AgWi). In other words, at thecross-point ij, two antigenantibody reactions between

AbMi and AgWj (reflecting the man is preference) andbetween AbWj and AgMi (reflecting the woman js prefer-ence) will take place.

Under the assumption that the concentration observed

at each cross-point is proportional to both Aff(AbMi,AgWj)and Aff(AbWj,AgMi), the array can find a stable matchingby selecting one cross-point with the highest concentrationfrom each row and column. This matching is certainly astable one, for otherwise there must be a blocking pair Mkand Wlsuch that Aff(AbMk,AgWl) > Aff(AbMk,AgWp(Mk))and Aff(AbWl, AgMk) > Aff(AbWl, AgMp(Wl)), where

p(Mk) denotes a partner of Mk in the current matching.Then both concentrations at the cross-point kl are higherthan those of kp(Mk) and those ofp(Wl)l, reflecting theaffinities.

Although obaining a stable matching shows some com-putational power, it can be solved in O(N2) time, where N

is the number of men (and women). Gale and Shapley9

have invented a well-known algorithm for giving stablematches for man-oriented matching or woman-sided one.By further assuming that the concentration observed at across-point can reflect the amount of antibodies imposed,the array is capable of obtaining any stable matching inthe array from the man-oriented (man optimal and womanpessimal) matching to the woman-oriented (woman optimaland man pessimal) one. By equally increasing all the anti-bodies AbMi (i = 1 . . . n) (or equivalently antigens AgWj

j = 1 . . . n) from a unit to , the matching would comeclose to the man-oriented one. Similarly, an increase of

AbWi (i = 1 . . . n) will bias the matching towards the

woman-oriented one. Since there are many variants of thestable marriage problem which are NP-hard, devising thearray to solve these problems is challenging. Another chal-lenge is to devise the array as a component of a problemsolver that can deal not only with a particular problem,but also with similar problems, as the immune system hasdone.

4 Conclusion

We have shown that antibodies, a macromolecule of theimmune system, with a specific recognition capability canbe used for computation as a macromolecule DNA is usedfor DNA computing. Focusing on the stable marriageproblem, and extending the ABO blood group system, it isshown that the antibody-based computation can be imple-mented on an antibody microarray.

Acknowledgments This work was supported in part by Grants-in-Aid for Scientific Research (B) 16300067, 2004. This work was alsosupported by the Global COE Program Frontiers of IntelligentSensing, from the Ministry of Education, Culture, Sports, Science andTechnology.

A

B

B

AM1

M2

W1

W2

Fig. 1. A blood-type assignmentreflecting the preferences

B

O

A

OM1

M2

W1

W2

Fig. 2. A blood-type assignmentreflecting the preferences

Table 5. A nontrivial preference list for the two two stable marriageproblem

M1 M2

W1 2 1W2 2 1

W1 W2

M1 2 1M2 2 1


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References

1. Adleman LM (2004) Molecular computation of solutions to com-binatorial problems. Science 266:10211024

2. Hug H, Schuler R (2001) Strategies for the development of apeptide computer. Bioinformatics 17:364368

3. Elicits MB, Leiblerr SA (2000) Synthetic oscillatory network oftranscriptional regulators. Nature 403:335338

4. Sprinzak D, Elowitz MB (2005) Reconstruction of genetic circuits.

Nature 438:245. Basu S, Gerchman Y, Collins CH, et al. (2005) A synthetic multi-

cellular system for programmed pattern formation. Nature 434:11301134

6. Ro DK, Paradise EM, Ouellet M, et al. (2006) Production of theantimalarial drug precursor artemisinic acid in engineered yeast.Nature 440:940943

7. Guo Z, Hood L, Malkki M, et al. (2006) Long-range multi-locushaplotype phasing of the MHC. Proceedings of the NationalAcademy of Sciences, National Academy of Sciences, USA,pp 103118

8. Balan MS, Krithivasan K (2004) Parallel computation of simplearithmetic using peptideantibody interactions. Biosystems76:303307

9. Gale D, Shapley L (1962) College admissions and the stability ofmarriage. Am Math Month 69:915

10. Irving RW, Leather P, Gusfield D (1987) An efficient algorithm

for the optimal stable marriage. J ACM 34:53254311. Landsteiner K (1900) Zur Kenntnis der antifermentativen,lytischen und agglutinierenden Wirkungen des Blutserums und derLymphe. Zentralbl Bakteriol 27:357362

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Ishida (2008) - Antibody-Based Computing Stable Marriage