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Phase Transitions in Molecular Computing

Phase Transitions in Molecular Computing. Qualitative Difference in Properties of System as a Parameter is varied Tool for Analyzing Mixtures of Sticky-Ended Molecules Application for Sticky-Ended Molecular Computers. Approach for Analysis. Map Molecular Operations onto CA Model

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Phase Transitions in Molecular Computing

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  1. Phase Transitions inMolecular Computing • Qualitative Difference in Properties of System as a Parameter is varied • Tool for Analyzing Mixtures of Sticky-Ended Molecules • Application for Sticky-Ended Molecular Computers

  2. Approach for Analysis • Map Molecular Operations onto CA Model • Apply Renormalization Group Methods to identify Critical Point, scaling exponents. • Conduct Monte Carlo Simulations to verify RG Results • Goal: Identify phase transition in MC; Reaction Conditions for Formation of large-scale molecular structures • Equilibrium and Dynamic

  3. Order ParameterWhat property changes? • Clusters of Bound Molecules • How large do they get? • Non-Percolating to Percolating • Non-Computing to Computing • Halting to Non-Halting • On average, when do these events occur?

  4. Approach for Applications • Map System under Study to MC System • Design Sticky Ends • Simulating Systems with complex interactions in disordered environments • Irreversible Dynamics • What long-term and large-scale order arises in these systems for different conditions

  5. Possible Applications • Physical Systems: Spin Glasses, Alloys, MBE, Fluid Flow • Biological Systems: Neural Networks, Protein and RNA folding • BioTech: DNA Chip design, Anti Sense, DNA Vaccines

  6. Anti Sense Antisense Drugs are short pieces of synthetic DNA or RNA that affect disease at the genetic level. Virtually all diseases are associated with inadequate or inappropriate production or performance of proteins. Traditionally drugs are designed to interact with disease causing proteins and inhibit their function. In contrast, antisense technology permits design of drugs, called antisense oligonucleotides, that intervene at the genetic level and stop the production of disease causing proteins. Antisense oligonucleotides agents designed based on genetic information are more specific with lower toxicity and side effects than traditional drugs. Target Selection???

  7. Advantages • Specify Local Interactions (sticky ends) • Self-Organization (spontaneous emergence of order as parameters vary) • Thermodynamics says disorder favored • Complex, Local Interactions (mutually, conflicting constraints) that are irreversible • Narrow space of configurations • Predict Unforseen Interactions

  8. Example: Adleman • Map Adleman’s Architecture onto Probabilistic CA (DKCA) • Time Evolution of Adleman • Simplifying Assumptions: All concentrations and Sticky End Strengths Equal • 1 = Oligo Present, 0 = Oligo absent

  9. i t 0 2 4 6 8 10 1 3 5 7 9 P(1|0,0) = 0 P(1|0,1) = P(1|1,0) = pq P(1|1,1) = pq(2-q) q : Site Occupationprobability (Concentration) p : Bond formation probability

  10. Renormalization Group • Point where matter changes from one state to another • Seemingly Unrelated Transitions Follow Same Rules • Scaling: Divide system into boxes and have atoms in boxed communicate with nearest neighbors

  11. Properties as a function of scale • Recognize Similarity Across Scales • Critical Properties are Constant as the Scale Changes • Self-Similar • Critical Behavior is characterized by loss of Scale • System Fluctuates Strongly at all Scales • Example: Critical Opalescence - Liquid/Gas Transition and Difference in Densities • Light Strongly Scattered

  12. Steps • Decimation or Coarse Graining: Average out subset of Degrees of Freedom, typically those that vary on very short scales • Rescaling: Redefine the unit length. Scaling Factor is ratio of Coarse-grained unit of length to the original unit length. • Repeat

  13. H’ = R H • R: Renormalization group Operator • Goal is to find Fixed Points of R • H* = R H* • Analyze the flows through parameter space • All systems that flow close to the fixed point will exhibit same critical exponents • Exponents determined by the eigenvalues of linear transformation matrix at the fixed point ---- Universality

  14. Solve for Fixed Point • Expand with small perturbations around fixed point • Taylor Expansion - Linearize the Transformation in a small vicinity of fixed point • Determine Eigenvalues of Transformation to determine properties of fixed point • Determine Critical Exponents

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