arXiv subjects

Andrés Ortiz-Muñoz

Publications and source records attributed to Andrés Ortiz-Muñoz.

7 recordsLinked to original sources

arXiv2026

Equilibrium Combinatorial Self-Assembly via Generating Functions

We develop a generating-function calculus for equilibrium combinatorial self-assembly. Starting from a bond-level specification of allowable interactions, we define a symmetry-weighted species generating function whose evaluation yields equilibrium concentrations, and an ensemble generating function (an exponential transform) that packages equilibrium probabilities of mixtures. We ground these objects in both deterministic (coagulation-fragmentation) and stochastic (master equation) dynamics, showing how detailed balance leads to the equilibrium expressions and how the exponential generating function arises as a partition function. We develop a formal power series calculus -- derivatives, integrals, exponentials, composition -- where each operation acquires a precise combinatorial interpretation. The paper is organized around two regimes: cycle-free assembly, where binding equations for the species generating function are nonlinear and the ensemble equation couples to the species generating function; and assembly with cycles, where a cycle-opening term enters the species equation and the exponential transform linearizes the ensemble equation into a closed linear PDE with operator-exponential solutions. Each regime is developed with a linear polymer worked example in which we compute equilibrium concentrations, extract canonical partition functions, and derive canonical factorial moments. A cross-linked polymer example -- combining heterotypic chain bonds with homotypic cross-links -- illustrates both regimes together, yielding a factorized canonical partition function and an explicit gelation surface.

math.CO↗

arXiv2026

A Rigid Category of DNA Secondary Structures

We construct a strict pivotal monoidal category $\mathcal{D}_{\mathrm{DNA}}$ whose objects are DNA sequences (words over $\{A,C,G,T\}$) and whose morphisms are isotopy classes of typed noncrossing planar matchings, composed of through-strands and Watson-Crick-typed arcs, in a rectangle with source and target boundaries. The dual of a sequence is its reverse complement, evaluation and coevaluation are canonical duplex pairings, and the snake identities hold by planar isotopy. A bending correspondence identifies each morphism $x \to y$ with a secondary structure on the combined word $x{}^{\vee} y$; in particular, the generalized elements $\varepsilon \to w$ are exactly the non-pseudoknotted secondary structures on $w$. Composition, viewed in this straightened picture, is computed by a zip-and-transfer operation on complementary interfaces, a combinatorial rearrangement of base-pair connectivity of which toehold-mediated strand displacement is a kinetically specific instance. Because $\mathcal{D}_{\mathrm{DNA}}$ is rigid monoidal, it shares the categorical backbone of pregroup grammars and the DisCoCat framework for compositional semantics: a strong monoidal functor from a grammatical category to $\mathcal{D}_{\mathrm{DNA}}$ maps grammatical reductions to Watson-Crick base pairing and sentence meanings to secondary structures. We describe this functor and discuss connections to algorithmic self-assembly, composable strand-displacement circuits, and constructive dynamical systems.

math.CT↗

arXiv2026

Homotopy Cardinality and Entropy

We explore connections between homotopy type theory and information theory through homotopy cardinality. We define probability types and random variable types, prove that homotopy cardinality respects dependent sums under truncation and decidability hypotheses, and show that it does not respect dependent products in general. Using the power series expansion of the logarithm, expressed type-theoretically through deloopings of finite cyclic groups, we formulate Shannon entropy as the homotopy cardinality of a type and derive the chain rule for entropy under a trivial-action hypothesis.

math.CT↗

arXiv2025

The Nature of Organization in Living Systems

Living systems are thermodynamically open but closed in their organization. In other words, even though their material components turn over constantly, a material-independent property persists, which we call organization. Moreover, organization comes from within organisms themselves, which requires us to explain how this self-organization is established and maintained. In this paper we propose a mathematical and conceptual framework to understand the kinds of organized systems that living systems are, aiming to explain how self-organization emerges from more basic elemental processes. Additionally, we map our own notions to existing traditions in theoretical biology and philosophy, aiming to bring the main formal ideas into conceptual congruence.

q-bio.QM↗

arXiv2019

Combinatorial protein-protein interactions on a polymerizing scaffold

Scaffold proteins organize cellular processes by bringing signaling molecules into interaction, sometimes by forming large signalosomes. Several of these scaffolds are known to polymerize. Their assemblies should therefore not be understood as stoichiometric aggregates, but as combinatorial ensembles. We analyze the combinatorial interaction of ligands loaded on polymeric scaffolds, in both a continuum and discrete setting, and compare it with multivalent scaffolds with fixed number of binding sites. The quantity of interest is the abundance of ligand interaction possibilities---the catalytic potential $Q$---in a configurational mixture. Upon increasing scaffold abundance, scaffolding systems are known to first increase opportunities for ligand interaction and then to shut them down as ligands become isolated on distinct scaffolds. The polymerizing system stands out in that the dependency of $Q$ on protomer concentration switches from being dominated by a first order to a second order term within a range determined by the polymerization affinity. This behavior boosts $Q$ beyond that of any multivalent scaffold system. In addition, the subsequent drop-off is considerably mitigated in that $Q$ decreases with half the power in protomer concentration than for any multivalent scaffold. We explain this behavior in terms of how the concentration profile of the polymer length distribution adjusts to changes in protomer concentration and affinity. The discrete case turns out to be similar, but the behavior can be exaggerated at small protomer numbers because of a maximal polymer size, analogous to finite-size effects in bond percolation on a lattice.

q-bio.QM↗

arXiv2018

Stochastic Chemical Reaction Networks for Robustly Approximating Arbitrary Probability Distributions

We show that discrete distributions on the $d$-dimensional non-negative integer lattice can be approximated arbitrarily well via the marginals of stationary distributions for various classes of stochastic chemical reaction networks. We begin by providing a class of detailed balanced networks and prove that they can approximate any discrete distribution to any desired accuracy. However, these detailed balanced constructions rely on the ability to initialize a system precisely, and are therefore susceptible to perturbations in the initial conditions. We therefore provide another construction based on the ability to approximate point mass distributions and prove that this construction is capable of approximating arbitrary discrete distributions for any choice of initial condition. In particular, the developed models are ergodic, so their limit distributions are robust to a finite number of perturbations over time in the counts of molecules.

cs.CC↗

arXiv2017

Chemical Boltzmann Machines

How smart can a micron-sized bag of chemicals be? How can an artificial or real cell make inferences about its environment? From which kinds of probability distributions can chemical reaction networks sample? We begin tackling these questions by showing four ways in which a stochastic chemical reaction network can implement a Boltzmann machine, a stochastic neural network model that can generate a wide range of probability distributions and compute conditional probabilities. The resulting models, and the associated theorems, provide a road map for constructing chemical reaction networks that exploit their native stochasticity as a computational resource. Finally, to show the potential of our models, we simulate a chemical Boltzmann machine to classify and generate MNIST digits in-silico.

q-bio.MN↗