Open problems of the automaton model of protein dynamics: reconstructing the protein automaton from output data and characterizing the class of protein automata

The Khrennikov-Yurova automaton model represents a protein as two coupled automata: an ontic automaton whose states are conformational states organized on a p-adic (ultrametric) state space, and an epistemic automaton of functional states obtained by quotienting the conformational states by functional (observational) equivalence. The source paper leaves open how to turn this conceptual model into a determined, testable one: reconstructing the automaton (state space including generalized states, transition function, input alphabet, initial state, and equivalence structure) from experimentally observed output sequences in the presence of non-stabilizing relative frequencies; settling whether proteins require finite-state or infinite-state automata; characterizing the precise class of automata matching protein behavior within the locally constant p-adic rational-valued automaton functions; and experimentally testing the predicted non-classical (p-adic) statistics.

Background

A protein molecule is described at two levels in molecular biophysics: its conformational states (the geometric shapes the molecule can adopt, an enormous and hierarchically organized set) and its functional states (what the protein actually does to its environment, e.g. catalysis, binding, or signal transduction). The standard structure-function paradigm ties function tightly to a single native conformation, but intrinsically disordered proteins, which function without a fixed native structure, do not fit that picture. Khrennikov and Yurova (2017) proposed modeling protein dynamics with the theory of automata, by analogy with the ontic/epistemic distinction in foundations of physics, where an underlying real state of affairs (ontic) is distinguished from what an observer can know (epistemic).

In their model, a protein is an input-output system: environmental influences are input symbols, and functional responses are output symbols. An automaton here means a discrete dynamical system specified by a state set $S$, an input alphabet $I$, a transition function $f:I\times S\to S$, and, for automata with output (Mealy machines; Moore machines are the variant where the output depends only on the current state), an output function. The ontic automaton has as states the conformational states, and its state space is modeled by $p$-adic numbers or more general ultrametric spaces (metric spaces obeying the strong triangle inequality $d(x,z)\le\max(d(x,y),d(y,z))$, under which distinct balls never partially overlap, so points cluster into a nested, treelike hierarchy). The $p$-adic integers $\mathbb{Z}_p$ (for a fixed prime $p$) are the completion of the integers under the $p$-adic norm, in which two integers are close when their difference is divisible by a high power of $p$; this encodes a natural hierarchical, treelike clustering of states into nested balls. The epistemic automaton is obtained by grouping conformational states into equivalence classes of states that are indistinguishable by their functional outputs; these classes are called generalized or functional states, and the quotient construction is exactly how one passes from the ontic to the observable level.

The mathematical setting connects to Anashin's $p$-adic theory of automata: an automaton with states in $\mathbb{Z}_p$ determines a mapping $F_A:\mathbb{Z}_p\to\mathbb{Z}_p$ (its automaton function), and the theory characterizes which mappings are realized by finite automata. In particular, among the locally constant functions (continuous functions on $\mathbb{Z}_p$ taking only finitely many values, each constant on a finite partition into balls), exactly those taking $p$-adic rational values are realized by finite automata. A standard class in automata theory is the transitive automata, which can generate an arbitrary output from an arbitrary input; Khrennikov and Yurova argue that proteins, being highly specialized machines that react concretely to specific input patterns, must instead be modeled by non-transitive automata, whose simplest representatives are locally constant mappings.

A distinctive statistical prediction arises at the epistemic level. When the automaton lingers in a generalized state, its output sequence can contain long runs of equal symbols, and the relative frequency of an output symbol can fluctuate without converging to any limit, violating the classical principle of stabilization of relative frequencies that underlies frequency probability. For such sequences the authors propose using probabilities valued in the $p$-adic numbers rather than the real interval $[0,1]$, a form of generalized probability theory. Whether such non-classical statistics actually occur for a real protein is untested.

Within this framework the source paper formulates, but does not solve, the problems of making the model concrete: the automaton and its state space, transition function, input alphabet, and initial state are not given in advance and must be recovered from observational data, and the precise class of automata matching real proteins is unknown. These questions span the boundary between $p$-adic mathematics, automata theory, and molecular biophysics.

Problem Statement

Within the Khrennikov-Yurova framework, in which a protein is an ontic automaton $A=\langle I,S,f\rangle$ on conformational states $S$ (structured as $p$-adic integers or a general ultrametric space) with input alphabet $I$ and transition function $f$, observed through an output function $g$ that is constant on classes of functionally equivalent states, determine the automaton of a real protein and the automaton class that real proteins occupy. The source states the following open problems and programmatic directions, which share one objective, converting the conceptual model into a determined and testable model of protein functional dynamics. Items (1)–(3) are all required for this objective; item (4) is a programmatic direction rather than a required deliverable:

(1) Identification from observational data. Given only the observed output sequence $v_k=g(i_k,s_k)$ (the output function $g$ applied to the input symbol $i_k$ and the state $s_k$ at step $k$) of the automaton with output $\langle I,S,V,f,g\rangle$ ($I$ the input alphabet, $S$ the state set, $V$ the output alphabet, $f$ the transition function, $g$ the output function), where the state space (possibly consisting of generalized states, i.e. equivalence classes), the transition function, and often the input alphabet are unknown, determine: the transition function when it is only partially defined; the input alphabet and the influence of input symbols on behavior; the presence and structure of equivalent states on $S$; the matching of the automaton with the experimentally observed output data; and the initial state. Construct sound statistical criteria for these tasks despite the model's pathological output statistics: long runs of equal outputs near a generalized state, and relative frequencies that need not stabilize, so that standard frequency-based criteria fail and $p$-adic-valued probabilities are the proposed remedy. These determinations and the statistical criteria are jointly required to identify the automaton.

(2) Model class. Settle whether protein behavior is adequately modeled by finite-state automata (Mealy or Moore machines) or requires infinite-state automata, i.e. whether the conformational state space is effectively finite or must be treated as continuous. As with (3), the operative sense of “adequately modeled” is not fixed in advance and is part of what this deliverable must settle.

(3) Characterization. Characterize the precise class of automata matching proteins' behavior. The source conjectures this class is a subclass of finite automata whose automaton functions are locally constant $p$-adic rational-valued mappings, necessarily non-transitive, and states that describing this subclass precisely requires taking into account the biological and molecular features of proteins. The characterization must give an explicit, checkable membership criterion for the class — conditions on an automaton, or on its automaton function, that decide whether it matches protein behavior — and identify the biological or molecular features that enter those conditions. Item (2) is the coarser dichotomy posed alongside it: a resolution of (3) settles the finite-versus-infinite question of (2) as a corollary, whereas settling (2) alone does not resolve (3). The characterization must fix the operative sense of “matching protein behavior” — the target against which the membership criterion is judged — and tie the criterion's conditions to that target.

(4) Experimental test and biological content of the hierarchy (programmatic direction). As a programmatic direction rather than a sharply formulated open problem, the source expresses the hope that the model's distinctive prediction, non-classical (p-adic-valued) probability behavior of functional-output sequences, can find real application and experimental test on a concrete protein system, and speculates that filling the hierarchical (treelike) organization of functional states with nontrivial biological content may require additional criteria, for example based on the magnitude of the total energy of transition from one functional state to another or the magnitude of the potential-energy barrier separating these states.

References

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