ECS 232: Theory of Molecular Computation
Winter 2025

Course announcement (PDF)

Instructor

Dave Doty
doty@ucdavis.edu
Office hour: Thursday 11am-12pm
Office: Zoom link

Lectures

MWF, 11:00-11:50am, Cruess 107

Course objective

To study the fundamental abilities and limits to the engineering of automated (i.e., computational) molecular systems, in a mathematically rigorous way.

Prerequisites

ECS 120 or equivalent (familiarity with Chapters 1,3,4,7 of Introduction to the Theory of Computation by Sipser), or permission of instructor

Prior experience with probability theory is useful; in particular, Chapters 1-2 of Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis, by Mitzenmacher and Upfal.

Piazza

The Piazza page for the course can be used to ask questions about the course and homework. Use access code ecs232 to enroll. Please use Piazza instead of email unless the question is of a personal nature.

Slides

slides

Videos

I recorded videos of the lectures in Winter 2021.

Notes

lecture notes (these are not comprehensive since I often take material straight from a paper; they were used prior to Winter 2021, when I switched to slides, but they are more detailed in many proofs than the slides, while covering fewer total topics)

Homework

There is a Canvas page with the homework posted.

Project

Project ideas

Schedule

The exact dates we cover various topics may vary; here they are just listed in the order we will cover them. (One quarter the topics below were covered two per week on a Tuesday/Thursday lecture schedule, as an example of the pace.) See also the slides link above; we will go through those slides in order.

Algorithmic tile self-assembly

1. Introduction to course, introduction to abstract Tile Assembly Model (aTAM)
   aTAM video introduction
   WebTAS simulator
   
2. tile complexity of assembling squares
   O(log n) tile types for assembling an n x n square
   Ω(log n / log log n) tile types necessary to assemble an n x n square
   paper: The Program-Size Complexity of Self-Assembled Squares,
   O(log n / log log n) tile types sufficient to assemble an n x n square
   paper: Running Time and Program Size for Self-assembled Squares
3. formal definition of the aTAM
4. simulation of Turing machine with a tile assembly system
   assembling scaled-up version of any finite shape from optimal number of tile types
   paper: Complexity of Self-Assembled Shapes,
   computable shape not strictly self-assembled by any TAS
   paper: Strict Self-Assembly of Discrete Sierpinski Triangles,
   computable set not weakly self-assembled by any TAS
   paper: Computability and Complexity in Self-Assembly
5. concentration programming
   paper: Randomized self-assembly for exact shapes
   finite shape that requires more tile types to strictly self-assemble with a directed TAS than a non-directed TAS; NP^NP-completeness of computing minimum tile set strictly self-assembling a shape
   paper: The Power of Nondeterminism in Self-Assembly
6. kTAM for error analysis
   paper: Simulations of Computing by Self-Assembly
   proofreading for error-correction
   paper: Proofreading Tile Sets: Error Correction for Algorithmic Self-Assembly

• temperature 1 self assembly: powerful in 3D and probably limited in 2D
  slides (ODP), slides (PDF)
  paper (many systems limited in 2D): Limitations of Self-Assembly at Temperature 1
  paper (powerful in 3D): Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D
  paper (modest efficiency gains in 2D): Noncooperative algorithms in self-assembly
• intrinsic universality
  slides (PPT)
  paper: The tile assembly model is intrinsically universal
• temperature programming
  paper: Reducing Tile Complexity for the Self-Assembly of Scaled Shapes Through Temperature Programming
• hierarchical self-assembly
  paper (difference in power between aTAM and hierarchical model): Two Hands Are Better Than One
  paper (limitations on assembly speed): Parallelism and time in hierarchical self-assembly
• staged self-assembly
  paper: Staged Self-Assembly: Nanomanufacture of Arbitrary Shapes with O(1) Glues
• signal-passing tiles
  paper: Asynchronous Signal Passing for Tile Self-Assembly: Fuel Efficient Computation and Efficient Assembly of Shapes
• different shapes of tiles
  paper (1x2 blocks): The Power of Duples (in Self-Assembly): It's Not So Hip To Be Square
  paper (other shapes made from squares): Universal Computation with Arbitrary Polyomino Tiles in Non-Cooperative Self-Assembly
  paper (regular polygons): Computing in continuous space with self-assembling polygonal tiles

Chemical reaction networks

7. Introduction to chemical reaction networks (CRNs). Examples of stable predicate and function computation.
   paper (predicates): Computation in Networks of Passively Mobile Finite-State Sensors
   paper (functions): Deterministic Function Computation with Chemical Reaction Networks
8. Formal definition of stable predicate/function computation.
   Characterizations of stable computation in terms of reachability, probability-1, and fair executions
   Feedforward CRNs.
9. Non-competitive CRNs.
   Formal definition discrete chemical kinetic model (Gillespie model).
   paper: Exact stochastic simulation of coupled chemical reactions
10. Derivation of expected completion times for some simple CRNs.
    time complexity analysis of computing functions/predicates with CRNs
    Definition of semilinear sets and functions, claim that only they can be stably computed.
11. Every semilinear set/function can be stably computed by a CRN.
    paper (predicates): Computation in Networks of Passively Mobile Finite-State Sensors
    paper (functions): Deterministic Function Computation with Chemical Reaction Networks
12. Only semilinear sets/functions can be stably computed by a CRN.
    paper (predicates): Stably Computable Predicates are Semilinear
    paper (functions): Deterministic Function Computation with Chemical Reaction Networks
13. Statement (without proofs) of what is known about predicates/functions stably computable in sublinear time
    Brief overview of results under other modeling choices:
    • continuous CRNs with real-valued concentrations
      paper: Rate-Independent Computation in Continuous Chemical Reaction Networks
    • non-stable computation: allowing a small probability of error
      paper: Computation with Finite Stochastic Chemical Reaction Networks
    • measuring time to convergence/stabilization, how these are affected by allowing an initial leader or not
      papers:
      Fast computation by population protocols with a leader (polylogarithmic convergence with initial leader is possible for all semilinear predicates)
      Population Protocols Are Fast (sublinear convergence without initial leader is possible for all semilinear predicates)
      Deterministic Function Computation with Chemical Reaction Networks (polylogarithmic convergence with initial leader is possible for all semilinear functions)
      Hardness of computing and approximating predicates and functions with leaderless population protocols (sublinear stabilization without initial leader is possible for "most" semilinear predicates/functions)
    • allowing approximation of numerical functions rather than require exact output
      paper: Hardness of computing and approximating predicates and functions with leaderless population protocols (logarithmic stabilization without initial leader is possible for approximating linear functions with positive rational coefficients)
    Register machines.
    Turing-universality: simulation of register machines by CRNs with a small probability of error; statement of CRN time complexity lower bounds
    paper: Computation with Finite Stochastic Chemical Reaction Networks
    paper: Fast Computation by Population Protocols With a Leader

DNA strand displacement

14. Introduction to DNA strand displacement.
    Visual DSD (simulation tool)
    papers:
    • paper introducing DNA strand displacement as a tool for DNA nanotechnology: A DNA-fuelled molecular machine made of DNA
    • paper introducing the first "CRN-to-DNA compiler": DNA as a universal substrate for chemical kinetics
    • an alternate (very elegant) CRN-to-DNA compiler: Two-Domain DNA Strand Displacement
    • implementation of a multi-stack machine using all reversible (hence zero-energy dissipation) strand displacement reactions: Efficient Turing-Universal Computation with DNA Polymers
    • large Boolean circuit (computing square root of a 4-bit number) using nothing but DNA strand displacement: Scaling Up Digital Circuit Computation with DNA Strand Displacement Cascades
    • DNA strand displacement implementation (through the "two-domain" compiler above) of approximate majority algorithm from the distributed computing literature: Programmable chemical controllers made from DNA
    • first oscillating CRN to use only DNA strand displacement: Enzyme-Free Nucleic Acid Dynamical Systems
15. dual-rail logic for Boolean circuits. ABC for minimizing circuits.
    Leaks and error-prevention in DNA strand displacement.
    papers:
    • Leakless translator gates: Effective Design Principles for Leakless Strand Displacement Systems

DNA sequence design

16. DNA energy models
    polynomial-time algorithm for minimum free energy
    DNA sequence design for arbitrary constraints with stochastic local search
    papers:
    • The thermodynamics of DNA structural motifs
    • NUPACK documentation

Thermodynamic binding networks

17. Thermodynamic binding networks
    Boolean circuits
    paper: Thermodynamic binding networks
18. Thermodynamic binding networks
    Kinetic barriers
    Exponential size bound on stable polymers
    aTAM counter that is thermodynamically stable
    papers:
    • Thermodynamic binding networks
    • Programming Substrate-Independent Kinetic Barriers with Thermodynamic Binding Networks

Project presentations

19. Project presentations
20. Project presentations