Thermodynamic Binding Networks in Molecular Computation

• Thermodynamic Binding Networks are an abstract model of molecular computation that determines stable configurations through maximizing bonds and favoring entropy via separate polymers.
• The framework emphasizes saturation—forming all possible complementary bonds—and stability achieved by maximizing the number of disconnected, self-saturated polymers.
• TBNs underpin equilibrium-based computation, bridging combinatorial analysis with practical SAT and IP verification methods for simulating molecular systems.

Searching arXiv for papers on Thermodynamic Binding Networks to ground the article in the primary literature. Thermodynamic Binding Networks (TBNs) are an abstract model of molecular computation in which the thermodynamically favored outcomes of a system are determined solely by two combinatorial quantities: how many complementary bonds are formed and how many connected complexes, or polymers, exist in the resulting configuration. The model was introduced to study whether the intended outputs of molecular systems can be made consistent with thermodynamic equilibrium rather than only with a desired kinetic pathway (Doty et al., 2017). In the TBN limit used in the foundational work, binding is prioritized infinitely more strongly than entropy, but among maximally bonded states the preferred configurations are those with more separate polymers (Breik et al., 2017). This makes TBNs a geometry-free framework for analyzing strand displacement cascades, algorithmic tile assembly, and related synthetic molecular systems in which equilibrium correctness is a central design constraint (Doty et al., 2017).

1. Formal model and thermodynamic criterion

A standard presentation defines a TBN as a triple

$\mathcal{T} = (D, *, \mathbf{T}),$

where $D$ is a finite set of site types, $*:D\to D$ is an involution with no fixed points, and $\mathbf{T}$ is a finite multiset of monomer types, each monomer type itself being a finite multiset over $D$ (Breik et al., 2017). Closely related presentations instead write $\mathcal{T}=(D,M)$, with $D$ the set of primary domain types and $M$ a finite set of monomer types over $D\cup D^*$ (Doty et al., 2017). The 2018 kinetic-barrier work explicitly notes that multiple equivalent ways of defining or packaging the model do not change the essential thermodynamic or computational behavior (Breik et al., 2018).

In all of these formulations, site or domain types come in complementary pairs such as $a$ and $D$0, and bonds may form only between complementary types (Doty et al., 2017). Monomers are treated as unstructured multisets of sites rather than geometric objects. A configuration is therefore determined by connectivity, not by shape, orientation, or steric constraints (Breik et al., 2017). This geometry-free character is deliberate: TBNs ignore spatial layout and sequence detail in order to isolate the combinatorial consequences of bond formation and configurational entropy (Breik et al., 2018).

The thermodynamic objective is lexicographic. In the foundational formulation, a configuration is saturated if no complementary pair remains both unbound, equivalently if it maximizes the number of bonds among all configurations of the same monomer collection (Doty et al., 2017). A configuration is stable if it is saturated and, among all saturated configurations, has the maximum number of connected components (Doty et al., 2017). The same idea is sometimes expressed as a free-energy tradeoff of the form

$D$1

where $D$2 is the number of bonds and $D$3 is the number of complexes, with the understanding that the model studies the regime in which bond maximization dominates and the complex count breaks ties among bond-maximal states (Breik et al., 2018).

A recurring misconception is that TBN stability is determined only by bond count. The model explicitly rejects that simplification: two saturated configurations with the same number of bonds can differ in thermodynamic preference because the one with more separate polymers has higher configurational entropy and is preferred in the TBN limit (Breik et al., 2017).

2. Configurations, polymers, and saturation

In the explicit-matching formalization, a configuration chooses pairings between sites such that each site is paired with at most one complementary site and only complementary types may pair (Breik et al., 2017). Two monomers bind if any site on one pairs with a site on the other, and a polymer is a connected component under this binding relation (Breik et al., 2017). The number of polymers is denoted $D$4 (Breik et al., 2017).

The 2020 integer-programming formulation reorganizes the same thermodynamic notion in polymer-centric terms. There a configuration is a partition of the monomers into polymers, and a polymer is self-saturated if it has no exposed starred sites; a configuration is saturated if every polymer is self-saturated (Haley et al., 2020). Under the star-limiting convention, where the total supply of each starred site is at most the supply of its unstarred complement, saturation is equivalent to forming all possible bonds (Haley et al., 2020). The 2017 complexity paper gives a related characterization: a configuration is saturated iff every limiting site is paired, where a site type is limiting if its complement occurs at least as often as it does (Breik et al., 2017).

Later work made the entropy term more operational. The 2020 formulation shows that among saturated configurations, stable ones are exactly those minimizing the number of polymer merge operations needed to reach them from the all-singleton state (Haley et al., 2020). If $D$5 is a configuration and $D$6 is the set of its non-singleton polymers, the merge count is

$D$7

so stability can be read equivalently as saturated plus minimum merge count (Haley et al., 2020). The 2023 signal-amplification paper adopts the same perspective via the all-melted configuration $D$8, the merginess $D$9, and the starriness $*:D\to D$0, with saturation characterized by $*:D\to D$1 (Petrack et al., 2023).

This family of equivalent viewpoints is significant because it permits both explicit combinatorial reasoning about pairings and higher-level optimization over polymers. A plausible implication is that TBNs are best understood not as a single syntactic formalism but as a thermodynamic principle that admits several mathematically convenient representations.

3. Equilibrium computation and design

The original purpose of TBNs was to study whether computation can be encoded directly into thermodynamically favored final states rather than only into prescribed time evolutions (Doty et al., 2017). The foundational paper gives constructions for Boolean AND/OR formulas and for a self-assembling binary counter whose desired outputs are exactly the stable configurations (Doty et al., 2017). It distinguishes weak output convention, where output $*:D\to D$2 means there exists a stable configuration with free output monomer, from strong output convention, where output $*:D\to D$3 means every stable configuration has the output monomer free and output $*:D\to D$4 means every stable configuration has it bound (Doty et al., 2017).

A key robustness notion in those constructions is distance to stability, the entropy gap between the best stable state and the best saturated state with the wrong output (Doty et al., 2017). For trees of AND gates of depth $*:D\to D$5 with redundancy $*:D\to D$6, the paper proves that if at least one input is absent, then the distance to stability for any saturated configuration with output $*:D\to D$7 is at least $*:D\to D$8, and notes a stronger refined bound of $*:D\to D$9 (Doty et al., 2017). For translator cascades of depth $\mathbf{T}$0 and redundancy $\mathbf{T}$1, any saturated configuration with output $\mathbf{T}$2 has entropy $\mathbf{T}$3, whereas the intended output-$\mathbf{T}$4 configuration has entropy $\mathbf{T}$5, giving an entropy gap of $\mathbf{T}$6 (Doty et al., 2017).

The computational scope of the model was extended in "Thermodynamically Favorable Computation via Tile Self-assembly" (Chalk et al., 2018). That work defines what it means for a TBN to simulate an $\mathbf{T}$7-space-bounded Turing machine or a Boolean circuit in equilibrium, gives constructions robust to variability in monomer counts and to multiple copies of the computation forming in parallel, and shows that a naïve strategy of porting abstract Tile Assembly Model systems directly into TBNs can fail because the lack of geometry permits splicing and rewiring (Chalk et al., 2018). To repair that deficiency, it introduces Geometric Thermodynamic Binding Networks (GTBNs), in which monomers are rigid geometric objects and saturation is replaced by effective saturation, allowing unbound complementary domains only when geometry makes binding impossible (Chalk et al., 2018).

The same paper shows that GTBNs can simulate Turing machines with $\mathbf{T}$8 domain and monomer type complexity, comparable to the aTAM, whereas the ordinary TBN simulation requires location-specific monomers and is therefore much less efficient (Chalk et al., 2018). This establishes a central limitation of ordinary TBNs: abstracting away geometry yields broad generality and strong equilibrium reasoning, but it can also erase the spatial constraints needed for efficient faithful simulation of kinetic assembly systems.

4. Complexity of stable-state analysis

The first systematic complexity study defines

$\mathbf{T}$9

the maximum number of polymers achievable in a saturated configuration (Breik et al., 2017). This quantity is central because it determines whether a given saturated configuration is stable and underlies reductions for other questions (Breik et al., 2017).

Problem	Decision question	Complexity
$D$0	Does a TBN have a saturated configuration with at least $D$1 polymers?	NP-complete
$D$2	Can monomer $D$3 be free in some stable configuration?	$D$4-complete
$D$5	Can monomers $D$6 be in the same polymer in some stable configuration?	$D$7-complete

The NP-hardness of $D$8 is obtained by reduction from Exact Cover (Breik et al., 2017). The construction yields a TBN $D$9 such that $\mathcal{T}=(D,M)$0 if $\mathcal{T}=(D,M)$1 and $\mathcal{T}=(D,M)$2 otherwise (Breik et al., 2017). The paper strengthens this to a gap construction $\mathcal{T}=(D,M)$3 with $\mathcal{T}=(D,M)$4 in the yes-case and $\mathcal{T}=(D,M)$5 otherwise, implying that for any $\mathcal{T}=(D,M)$6, no polynomial-time $\mathcal{T}=(D,M)$7-approximation exists for $\mathcal{T}=(D,M)$8 unless $\mathcal{T}=(D,M)$9 (Breik et al., 2017).

The problems $D$0 and $D$1 are motivated by molecular readout conventions. A monomer free in a stable configuration can model output release in strand displacement; two monomers in the same polymer can model co-localization-based readout (Breik et al., 2017). Their completeness proofs use graph-theoretic reductions through maximum independent sets and minimum vertex covers (Breik et al., 2017). The paper also gives black-box reductions connecting these problems back to $D$2; for example, if monomer $D$3 can be free, then

$D$4

and $D$5 is stably free iff $D$6 (Breik et al., 2017).

These results establish that worst-case equilibrium verification is intrinsically hard. The significance of later algorithmic work is therefore not that it eliminates hardness, but that it identifies representations in which many practically relevant instances remain tractable.

5. Verification methods, integer programming, and algebraic structure

The first practical verification method for TBNs reduces stability questions to SAT. The encoding uses Boolean variables $D$7 for site pairings, $D$8 for monomer binding, $D$9 for polymer representatives, and $M$0 for counting representatives (Breik et al., 2017). It enforces saturation using the characterization that every limiting site must be paired, ensures polymer connectivity via transitivity of $M$1, and imposes $M$2 through a simplified Sinz-style cardinality encoding (Breik et al., 2017). The reported encoding size is $M$3 clauses of size $M$4 for saturation, $M$5 constant-size clauses for polymer structure, and $M$6 clauses for the $M$7-polymer constraint (Breik et al., 2017).

The 2020 paper replaces explicit monomer copies and explicit bond matchings with an integer-programming formulation over monomer-type counts per polymer slot (Haley et al., 2020). If no better bound is known, it takes $M$8, the number of limiting monomers, as an upper bound on the number of non-singleton polymers (Haley et al., 2020). With integer variables $M$9 giving the count of monomer type $D\cup D^*$0 in polymer $D\cup D^*$1 and Boolean variables $D\cup D^*$2 indicating nonemptiness, the objective is

$D\cup D^*$3

which is exactly the total merge count over nonempty polymers (Haley et al., 2020). The formulation incorporates conservation, self-saturation, nonemptiness, big-$D\cup D^*$4 constraints for exact emptiness when enumerating optima, and lexicographic symmetry-breaking constraints (Haley et al., 2020).

This IP formulation avoids two major symmetries of the SAT approach: permutation symmetry among identical monomer copies and bond-configuration symmetry within a polymer (Haley et al., 2020). It also handles infinite counts of nonlimiting monomers, which the paper emphasizes SAT-based approaches cannot do directly (Haley et al., 2020). The authors report using SCIP to find the optimal objective value quickly and CP-SAT to enumerate all optimal solutions (Haley et al., 2020).

The same paper gives a structural theorem: if $D\cup D^*$5 is the monomer matrix whose columns are monomer net-count vectors, then self-saturated polymers are precisely the integer points in the pointed cone

$D\cup D^*$6

and the polymer basis is the Hilbert basis of this cone (Haley et al., 2020). The polymer basis consists of those polymers that appear in some saturated configuration and cannot be partitioned into two or more self-saturated polymers (Haley et al., 2020). This identifies the “fundamental components” out of which locally minimal energy configurations are composed and enables reasoning not only about stable equilibria but also about entire kinetic pathways (Haley et al., 2020).

6. Kinetic barriers, amplification, and concentration-level interpretations

Although TBNs were introduced as an equilibrium model, later work used them to reason about kinetic structure induced by thermodynamic constraints. "Programming Substrate-Independent Kinetic Barriers with Thermodynamic Binding Networks" extends the model with a barrier notion defined by the maximum excess free energy along a path of configurations and shows that one can engineer barriers that are independent of the substrate being processed (Breik et al., 2018). Its constructions use repeated modules so that any path between designated states must pass through an unfavorable intermediate whose penalty scales with the number of repeats; the paper states that for every $D\cup D^*$7, there exists a TBN family with barrier at least $D\cup D^*$8, independent of the substrate (Breik et al., 2018). The examples include catalytic systems, translator constructions, and hashgate modules (Breik et al., 2018).

The 2023 paper studies a different question: how much the stable-state landscape can change when a single analyte monomer is added (Petrack et al., 2023). It defines a distance $D\cup D^*$9 between TBNs by minimizing the $a$0 distance between polymer-count vectors of stable configurations (Petrack et al., 2023). Its main theorem states that for every $a$1 and $a$2, there exists a TBN $a$3 and analyte $a$4 such that $a$5 and $a$6 each have exactly one stable configuration, those stable configurations are at distance at least $a$7, both systems have entropy gap $a$8, and the construction uses $a$9 total monomer types, $D$00 domain types, and $D$01 domains per monomer (Petrack et al., 2023). In the analyte-absent stable state, all reporter monomers are bound; after adding one analyte copy, those reporters are unbound (Petrack et al., 2023).

The same paper also proves a limitation theorem: if a TBN has $D$02 domain types, $D$03 monomer types, and at most $D$04 domains on each monomer, with $D$05, then adding one copy of a monomer changes the stable-state landscape by at most

$D$06

This doubly exponential upper bound is derived through an integer-programming representation over polymer-basis variables and sensitivity results from integer programming (Petrack et al., 2023). The translator-gadget refinement $D$07 reduces the largest polymer size to $D$08 while preserving exponential separation between analyte-absent and analyte-present stable landscapes (Petrack et al., 2023).

A final extension connects the discrete TBN viewpoint to real-valued equilibrium concentrations in athermic chemistry. "Computing and Bounding Equilibrium Concentrations in Athermic Chemical Systems" studies monomer–polymer systems in which all interactions preserve enthalpy and equilibrium is governed purely by entropy, a regime described as consistent with saturated configurations in the TBN model (Akef et al., 17 Jul 2025). It proves that if an on-target set of polymers is stable in the appropriate ratio sense, then there exist monomer concentrations realizing a detailed-balance equilibrium in which each on-target polymer $D$09 has concentration $D$10, while every off-target polymer has exponent strictly larger than $D$11 and therefore concentration strictly below $D$12 for $D$13 (Akef et al., 17 Jul 2025). For TBN-stability closed sets, the paper shows that the on-target polymers can be taken uniform with $D$14, and derives a corollary in which each on-target polymer has concentration exactly $D$15 and each off-target polymer has concentration at most $D$16, where $D$17 is determined by entropy-loss-to-novelty ratios of canonical reactions (Akef et al., 17 Jul 2025).

Taken together, these developments show that TBNs have evolved from a geometry-free equilibrium abstraction into a broader framework spanning equilibrium computation, hardness theory, SAT and IP verification, Hilbert-basis structure, substrate-independent kinetic barriers, thermodynamic signal amplification, and concentration-level design. Their central invariant has remained unchanged: a state is favored only when it is maximally bonded and, subject to that constraint, as entropically unconstrained as possible.