Molecular Computation and Splicing Systems J.H.M. Dassen, Summarized by Dongmin Kim

© 2002, SNU BioIntelligence Lab, Introduction Molecular Computation is interesting from both a theoretical and a practical viewpoint.  Differences in what problems are tractable.  Turing machine can perform the same computation as any other devices. (Church-Turing hypothesis)  But, some implementable models may be more than polynomially faster than others.

© 2002, SNU BioIntelligence Lab, Advantages of Molecular Computation Energy-efficient Massive parallelism  A sequential computer is an approximation of a deterministic Turing machine.  A parallel computer is an approximation of a nondeterministic Turing machine.  From a practical perspective, molecular computation may redefine the limits of feasible computation. Density of information storage

© 2002, SNU BioIntelligence Lab, Limitations of Adleman ’ s approach Solves combinatorial problems only. The operations involved are very slow and highly error prone. Scalability to large problem instances is doubtful. Requires external operators But, now several universal models; some approaches do not require an external operator; and less error prone operations and probabilistic approaches are being studied.

© 2002, SNU BioIntelligence Lab, Molecular Computation Models Special purpose vs. universal  Adleman’s, Lipton’s approach.  Beaver’s and Rothemund’s simulation of Turing machines. In vitro vs. in vivo The information carrier  DNA vs. RNA or ‘unusual’ DNA structures. The operations Instructions -> data  Rothemund’s Turing machine simulation treat instructions as data

© 2002, SNU BioIntelligence Lab, Molecular Computation Models (2) Rewritability of the information carrier One-pot vs. multiple phases Error-resilience Communication  Between information carriers.  The operator works ‘blindly’ Native or not  There is no model that is ‘native’ to Molecular Computation

© 2002, SNU BioIntelligence Lab, Special Purpose Models (1) Adleman’s approach  Special purpose, in vitro model, the information carrier is not rewritten, multiple separated phases.  Limitations of the abstract model  It cannot break the exponential barrier: (Juris Hartmanis, On the weight of computations, Bulletin of the European Association for Theoretical Computer Science, 55: , 1995.) Solving a 200 node instance of DHPP would require an amount of DNA weighing more than the Earth.  The output of the initialization step fall in a limited class of languages. When the self-assembly is linear, this class is that of regular languages.

© 2002, SNU BioIntelligence Lab, Special Purpose Models (2) Lipton’s model  Solving SAT problem.  Introduces the notion of “test tube”.  Suggests using a molecular computational device as a special purpose co-processor or unit for performing exponential searches: an electronic/ molecular hybrid computer.

© 2002, SNU BioIntelligence Lab, Universal Models Does a model in Molecular Computation exist that is capable of simulating all computations:  The answer seems like to be ‘Yes’.  One was from several more or less practical proposals that simulate classes of Turing machines using Molecular Computation.  The other was from the theory of splicing systems.

© 2002, SNU BioIntelligence Lab, Universal Models (2) - Turing machines Basic model  A finite control (a transition table, a current state), a tape of potentially unlimited, symbols from a finite alphabet, and a read/ write head. Representation  ‘hardware – software’ vs. ‘constant – variable components’ Configurations  Describes the contents of the tape, the position of the read/ write- head and the state of the finite control. Determinism vs. Nondeterminism  Nondeterministic Turing machine is ‘faster’ than deterministic one.

© 2002, SNU BioIntelligence Lab, Universal Models (3) Beaver’s model  Simulates deterministic Turing machine. A new operator: context-sensitive substitution  We want to substitute substring to.  Add.  Then we have and PCR.  We have.  Destroy by S1 nuclease. Simulation  Each substitution is corresponding to a configuration of TM.  If duplicate tubes, it simulates Nondeterminism.

© 2002, SNU BioIntelligence Lab, Universal Models (4) Rothemund’s Model  The simulation is performed by implementing the single steps from one configuration to another.  Instead of developing a simulation of a universal Turing machine directly, Rothemund uses a small non-universal Turing machine and then suggests how to scale up to a universal TM. Useful enzymes  Chose to use class IIS restriction enzymes. Representing instantaneous descriptions  The contents of the tape: symbols are each assigned a sequence.  The position of the head: another sequence which indicates the recognition site of a restriction enzyme and the splicing site.  The state of the finite control: is encoded in the space between the recognition site and the current symbol.

© 2002, SNU BioIntelligence Lab, Universal Models (5) Transitions  Representing the transition table  Is encoded into four type oligonucleotides.  Implementing the transitions Estimates  Representing one mole of bits requires about 260 m 3 water.  Each transition take about 4.5 hours Problems  It does not describe how to generate the initial tapes.  Rothemund does not explain whether his scheme is suitable for Nondeterministic TM.  The scheme requires many different kind of restriction enzymes.

© 2002, SNU BioIntelligence Lab, Universal Models (6) Winfree’s model: simulating cellular automata  Blocked cellular automata  One-dimensional variation and can be universal.  The transition rule is formulated for pairs of cells.  Simulates a universal blocked cellular automaton  By designing small units of DNA that they self-assemble into two-dimensional complexes.  One direction corresponds to the state of the whole automaton, and the other shows the contents of one cell during the whole developments of the automaton.  It is unclear how practical Winfree’s approach is, but it is conceptually much simpler than previous ones.

© 2002, SNU BioIntelligence Lab, Splicing Systems (1) Abstract models for the languages generated by strands of DNA under the application of restriction enzymes and subsequent annealing and ligation.  Thomas Head, Formal language theory and DNA: an analysis of the generative capacity of specific recombinant behaviors, Bulletin of Mathematical Biology, 49(6): ,  If DNA-related problems are difficult to solve, then DNA-based primitives may enable solutions to difficult problems. The splicing operator  In general formal language theory, strings are formed by applying the concatenation of symbols.  Splicing is the operation of concatenating a prefix of one string and a suffix of another string. (e.g. splice (‘snack’, ‘tofu’) = ‘snafu’)

© 2002, SNU BioIntelligence Lab, Splicing Systems (2) Splicing rules  Just as the use of concatenation is regulated by grammatical rules, the use of splicing is regulated by splicing rules.  Is consists of four finite strings u 1, u 2, u 3, u 4  u 1, u 2 (u 3, u 4 ) determine the possible sites of the splicing in the first (second) string.  u 1, u 4 are kept but u 2, u 3 are not.  Formally, a splicing rule looks like as follows:

© 2002, SNU BioIntelligence Lab, Splicing Systems (3) H scheme  V is an alphabet, R is a set of splicing rules. H system  L is a given language. Extended H system  T is the terminal alphabet, A is the set of axioms. Classes  Both A and R are finite: regular languages.  A is finite, but R regular: recursively enumerable languages.

© 2002, SNU BioIntelligence Lab, Splicing Systems (4) Question  Are there splicing systems that can generate the recursively enumerable languages, for which a realistic implementation is possible? Requirements  The amount of initial strands and the number of different restriction enzymes is finite.  DNA strands are consumed in splicing.  The length of a recognition site of a restriction enzyme is limited  Some restrictions on the use of the splicing operator are difficult to implement.

© 2002, SNU BioIntelligence Lab, Splicing Systems (5) Candidates  Splicing systems based on multisets  K.L. Denninghoff and R.W. Gatterdam. On the undecidability of splicing systems. International Journal of computer Mathematics, 27: ,  Splicing systems for circular strings  Takashi Yokomori, Satoshi Kobayashi, and Claudio Ferretti. On the power of circular splicing systems and DNA computability. Technical Report CSIM 95-01, Univ. of Electro-Communications,  Multiset splicing system with finite axioms and radius 2  Thomas Head, Gheorghe Paun, and Dennis Pixton. Generative Mechanisms Suggested by DNA Recombination. Vol. 2 of Rozenberg and Salomaa

© 2002, SNU BioIntelligence Lab, Conclusion Practical molecular computation  Molecular Computation has great potential.  The scale-up problem is difficult.  Some models are being refined and some new ones are introduced using very different paradigms or implementations. Theory  Provides us with a new way of viewing biological and chemical processes which may prove valuable in various fields.

© 2002, SNU BioIntelligence Lab, About Our TSP H system series  implementation Another model of DNA computing  New model beating H system series (??)  Variants of H system series.  Another theoretically universal system.  More practical ones to address Turing tar-pit.