Transcript 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 StickyEnded Molecules
• Application for Sticky-Ended Molecular
Computers
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 largescale molecular structures
• Equilibrium and Dynamic
Order Parameter
What property changes?
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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?
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
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
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???
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
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
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P(1|0,0) = 0
P(1|0,1) = P(1|1,0) = pq
P(1|1,1) = pq(2-q)
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q : Site Occupation
probability (Concentration)
p : Bond formation
probability
t
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
• 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
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
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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
• 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