Arrhenius-kinetics evidence for quantum tunneling in microbial “social” decision rates

Abstract

Social-like bacteria, fungi, and protozoa communicate chemical and behavioral signals to coordinate their specializations into an ordered group of individuals capable of fitter ecological performance. Examples of microbial “social” behaviors include sporulation and dispersion, kin recognition, and nonclonal or paired reproduction. Paired reproduction by ciliates is believed to involve intra- and intermate selection through pheromone-stimulated “courting” rituals. Such social maneuvering minimizes survival-reproduction tradeoffs while sorting superior mates from inferior ones, lowering the vertical spread of deleterious genes in geographically constricted populations and possibly promoting advantageous genetic innovations. In a previous article, I reported findings that the heterotrich Spirostomum ambiguum can out-complete mating rivals in simulated social trials by learning behavioral heuristics which it then employs to store and select sets of altruistic and deceptive signaling strategies. Frequencies of strategy use typically follow Maxwell-Boltzmann (MB), Fermi-Dirac (FD), or Bose-Einstein (BE) statistical distributions. For ciliates most adept at social decision making, a brief classical MB computational phase drives signaling behavior into a later quantum BE computational phase that condenses or favors the selection of a single fittest strategy. Appearance of the network analogue of BE condensation coincides with Hebbian-like trial-and-error learning and is consistent with the idea that cells behave as heat engines, where loss of energy associated with specific cellular machinery critical for mating decisions effectively reduces the temperature of intracellular enzymes cohering into weak Fröhlich superposition. I extend these findings by showing the rates at which ciliates switch serial behavioral strategies agree with principles of chemical reactions exhibiting linear and nonlinear Arrhenius kinetics during respective classical and quantum computational phases. Nonlinear Arrhenius kinetics in ciliate decision making suggest transitions from one signaling strategy to another result from quantum tunneling in social information processing.

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A wide diversity of microbes have evolved capacities to communicate “social” information via secreted chemicals and motile behaviors to better cope with challenging ambient or host environments.1–7 Social signals emitted by microbes may induce cooperation and competition within or between strains, species, genera and even kingdoms. Many kinds of microbial social behaviors, such as induced “herding” defenses against predation and other stressors or swarming “forages” and “hunts” for nutrition, are now recognized by microbiologists. However, one of the earliest observed forms of microbial social communication involves the pheromoneinduced preconjugal dances of ciliates.2,7–11 Ciliates can engage in sexual-like conjugation at the end of their lifecycle. Conjugation stages begin with the release of pheromones that attract and elicit complex stereotypic motility and cellcell contacts from nonself mating types. These preconjugal dances, composed of avoidance and probing behaviors, convey classical and perhaps quantum information about mating availability and fitness to prospective mates and rivals before the onset of paired reproduction.11 As with “higher” eukaryotes, skilled use of “courting” rituals by ciliates figures to encourage contacts with fitter mates while also discouraging conjugal attempts from rivals. In general, mate selection thus minimizes shared fitness tradeoffs and assures reproduction between compatible mates (i.e., ciliates with high motility and reproductive competency), while preventing less fit ciliates from passing their inferior genes to future generations.8

I recently showed that the large contractile ciliate Spirostomum ambiguum can learn then store its repertoire of serial avoidance behaviors signaling differential fitness into isomorphic computational networks.8 Ciliates use these networks as heuristics to search, plan and execute appropriate signaling strategies in mock social trials. The most proficient ciliates accomplish strategy searches at quantum efficiencies. S. ambiguum employs Hebbian-like preferential attachment rules to build heuristics from groups of serial contraction or ciliary reversal strategies.8–10 Preferential attachment rules of complex technological networks obey classical Maxwell-Boltzmann (MB), quantum Fermi-Dirac (FD) and quantum Bose-Einstein (BE) statistics.12 Likewise, the network characteristics of ciliate behavioral heuristics and social decision making follow a fitness model framework capable of classical and quantum computations.9 Ciliates that sensitize or habituate signaling patterns to emit brief periods of either deceptive “harder-to-get” or altruistic “easier-to-get” serial escape reactions begin testing condensed on initially perceived fittest courting solutions. When these ciliates switch from their first strategy choices, the network analogue of BE condensation for strategy use sharply dissipates into a MB computational phase no longer dominated by a single fittest strategy. Recursive trialand- error strategy searches anneal strategy use back into a condensed phase consistent with performance optimization and emergence of weak Fröhlich superposition. Social decisions performed by ciliates showing no nonassociative learning are largely governed by FD statistics, resulting in degenerate distributions of strategy choices.

Hebbian-like fitness models12 of the sort used by S. ambiguum9,10 designate a unique fitness parameter, η_i, for each network node, i: {l = 1, 2, …, i}. The fitness parameter, η_i, along with the number of existing links, k_i, acquired earlier by node i describe that node's probability, Π_i, of obtaining new links, m, where: 1 ${\prod }_i=({\text{η}}_i{k}_i)/({\Sigma }_l{\text{η}}_l{k}_l).$1

Each node is also assigned a unique, fixed energy or fitness level, ε_i, defined as: 2 ${\text{ε}}_i=-(1/\text{β})\log {\text{η}}_i,$2 where: β = 1/T and absolute temperature T (or time or population size in the metaphorical treatment of temperature) acts as a control variable for transitions between BE condensate and “fit-get-rich” computational phases. Lower ε_i levels tend to amass more links. Connections between any two different nodes correspond to noninteracting, inert particles filling the energy level of each node, so that 2m particles are equally distributed at some time t_i of the network's evolution. The partition function Z_t yields an occupation number,12 3 $k_i({\text{ε}}_i,t,t_i)=m{(t/t_i)}^{ƒ({\text{ε}}_{\iota })},$3 specifying the number of connections a node arising at time t_i with energy ε_i has captured by a later time t. The rate a node increases its network links proceeds via a power law established from the exponent, 4 $ƒ({\text{ε}}_i)=e^{-\text{β}({\text{ε}}_i-{\text{μ}}_i)},$4 where µ equals the chemical potential. When ƒ(ε_i) = 1, BE statistics hold,13,14 generating the occupation number for particles or connections populating level ε_i, 5 $n({\text{ε}}_i)=1/(e^{\text{β}({\text{ε}}_i-{\text{μ}}_i)}-1).$5

These constraints compel node i to preserve its overwhelming proportion of network connections. For ƒ(ε_i) << 1, the BE statistical distribution switches to one described by classical MB statistics,13,14 resulting in the occupation number: 6 $n({\text{ε}}_i)=1/e^{\text{β}({\text{ε}}_i-{\text{μ}}_i)}.$6

Classical statistics characterize “fit-get-rich” and “first-mover-advantage” computational phases which approach zero connectivity in the thermodynamic limit.12 In addition, the occupation number defined by FD statistics,13,14 7 $n({\text{ε}}_i)=1/(e^{\text{β}({\text{ε}}_i-{\text{μ}}_i)}+1),$7 appear when ƒ(ε_i) >> 1.

Parameters included in Eqs. (1)1 ${\prod }_i=({\text{η}}_i{k}_i)/({\Sigma }_l{\text{η}}_l{k}_l).$1 through (7)7 $n({\text{ε}}_i)=1/(e^{\text{β}({\text{ε}}_i-{\text{μ}}_i)}+1),$7 satisfy conditions of social decisions planned and executed by individual S. ambiguum.8–10 Each vertex or node of a networked heuristic learned by a ciliate matches a unique serial contraction or reversal strategy signaling mating fitness. Separate heuristics with N algorithmic steps or solutions emerge for contraction and reversal behavior. New nodes may be added by a ciliate to an evolving heuristic storing fewer than eight existing nodes, the upper limit of possible serial strategies observed in a repertoire of contractions or reversals. A ciliate picks fitness parameter, η, for each node from a distribution ρ(η) containing values proportional to the net reproductive payoffs perceived or expected from honest signaling patterns exchanged between courting couples.8,9 For some node i, η_i is expressed as: 8 ${\eta }_i=\left|p_i{P}_i\right|.$8

Argument p_i in Eq. (8)8 ${\eta }_i=\left|p_i{P}_i\right|.$8 denotes the Bayesian probability that a ciliate chooses to add node i to a heuristic over other possible nodes in its behavioral repertoire. A ciliate's perceived or expected net payoffs, P_i, relate benefits and costs of reproductive effort determined from respective probabilities of single and paired offspring count, B_i and 1 − B_i and of being and not being prepared, R_i and 1 − R_i, to perform the serial behavioral strategy of node i. It follows: 9 $\begin{array}{c}{P}_i=[R_i{B}_i+R_i(1-B_i)]-\\ \text{ [(1}-R_i\text{)}{B}_i\text{+(1}-R_i\text{)(1}-B_i\text{)].}\end{array}$9

Nodes compete for a ciliate's every decision, represented as heuristic connections, to use certain serial behavioral strategies signaling different fitness content.9 Modal signals are courting solutions selected more frequently by ciliates and the nodes associated with those signals, as well as others, attract new or newly recurrent connections, m, at each algorithmic solution n in the manner articulated by Eq. (1)1 ${\prod }_i=({\text{η}}_i{k}_i)/({\Sigma }_l{\text{η}}_l{k}_l).$1 . Recurrent connections in this definition depart from previous treatments12 to allow nodes to be revisited during the formation of decision algorithms without violating Eq. (1)1 ${\prod }_i=({\text{η}}_i{k}_i)/({\Sigma }_l{\text{η}}_l{k}_l).$1 . Positive or negative exponential changes in probability, Π_i, for node i to accumulate new links indicates use of the serial strategy corresponding to node i has become sensitized or habituated.9 While Cov[Π_i,Π_j] gives a convenient measure for the strength of connectivity between any two joined heuristic nodes i and j and their respective serial behavioral strategies. Increasingly positive covariance between nodes denotes rising coupling strength brought about from learning reliant on Hebbian-like processes. Increasingly negative covariance between nodes denotes depreciating coupling strength produced from either inhibitory learning or extinction also reliant on Hebbian-like processes. Latter types of modification in coupling strength are equivalent to active processes of memory suppression rather than passive ones of normal or pathological memory decay. Moreover, because each decision made by a ciliate to stay with the same strategy or to switch to another of different fitness can be regarded as a transition between logic states,8,9 Eq. (2)2 ${\text{ε}}_i=-(1/\text{β})\log {\text{η}}_i,$2 agrees with Landauer's Principle for memory storage devices:15–17 10 ${\text{ε}}_i=-(1/\text{β})\ln 2{\text{η}}_i,$10 where β = 1/k_BT, k_B is Boltzmann's constant and T is the absolute temperature of the local ambient environment, such as culture medium.

Due to informational degrees of freedom, energy level ε_i in this form accordingly depicts the minimum, but possibly physically imprecise, energy absorbed and expended to “reset” or “erase” the strategy of node i.17 Ciliates exist as isotherms with their immediate surroundings, so alterations in energy level ε_i reflect the system's computational effort when transitioning between nodes during some time interval.8,9 This (thermal) macroscopic energy correlates with the structural entropy or missing information about the precise internal microstate of node i and can be reformulated under Landauer's principle without term T necessary for expression of energy exchange.15,18 Incongruities in the value of ε_i owing to different informational and physical degrees of freedom can be corrected with limited success by universal and holographic entropy bounds calculated for a cell and its cellular compartments (e.g., ciliature, cortical areas important for information processing and movement, etc.,) of definite physical dimensions and density.9 As a consequence of the above computational characteristics, heuristic node i with low ε_i operating under recurrent autofeedback, such as a ciliate continuing to choose the same serial behavioral strategy, collects more links (i.e., decisions) by sustaining higher Π_i, higher positive Cov[Π_i,Π_i] and closer to perfect actual thermodynamic efficiency. This convention befits over-learned choice tasks performed by animals that progressively use less metabolic energy and tune sparser structural resources than when performing comparable under-learned tasks.9

Notably, transitions between heuristic nodes accompanying social decisions can be interpreted as endothermic or exothermic chemical reactions. A courting decision to switch from a strategy of higher fitness and therefore lower ε_i, to one of lower fitness and therefore higher ε_i, absorbs energy analogous to endothermic reactions (Fig. 1A). Conversely, a courting decision to switch from a strategy of lower fitness and therefore higher ε_i, to one of higher fitness and therefore lower ε_i, dissipates energy analogous to exothermic reactions (Fig. 1B). The tight relationship between ciliate decision making and mechanochemical processes8–10 suggests that any thermodynamic-independent quantum effects in the rate of information processing during mating rituals might be revealed by nonlinear Arrhenius kinetics. The Arrhenius equation determines the rate of a first-order reaction and takes the form: 11 $k=A{e}^{-E_a/RT},$11 where k is the rate constant, A is a proportionality constant related to Boltzmann and Planck constants, E_a is the activation energy, R is the gas constant and T is the absolute temperature. For classical reactions, plotting lnk versus 1/T returns a linear function. Whereas, reactions involving quantum tunneling produce a nonlinear function plotted between the same two variables (Fig. 1 insets).19

The Arrhenius equation has been insightfully used by several researchers to characterize thermally sensitive protozoan behavior,20–23 such as the rate of peristomial ciliary beat and wave motion or of whole cell contraction and re-extension. I here apply the Arrhenius equation in a similar way to evaluate the rate at which modal serial behavioral strategies are selected by S. ambiguum from contraction or ciliary reversal repertoires. To accomplish this goal, the Arrhenius equation is restated to fit the following form: 12 $k=(k_{\text{B}}\Gamma /h)f({\text{ε}}_i),$12 where k_B is Boltzmann's constant, Γ is the “critical tunneling field strength”, h is Planck's constant and ƒ(ε_i) is the energy-dependent exponent determining heuristic node connectivity. The annealing or control parameter, Γ, replaces absolute temperature T to be consistent with previous findings.9 Applying this form of the Arrhenius equation to ciliate signaling behavior employing serial contractions shows that social decisions dominated by a classical MB statistical distribution produce linear first-order decisional kinetics (Fig. 2A). Nonlinearities in Arrhenius kinetics, however, emerge for those serial contraction (Fig. 2A) and serial reversal signaling decisions (Fig. 2B) dominated by quantum BE and FD statistical distributions, indicating the existence of quantum- like tunneling in social decision rates. These findings are the first to identify the computational analogue of quantum tunneling effects in the social information processing of live, intact macroscopic biological systems. Given the dependence of ciliate signaling behavior on mechanically activated Ca²⁺-induced Ca²⁺ reactions,24 computational analogues of quantum tunneling in social decision rates likely parallel changes in reaction kinetics that support fast intracellular Ca²⁺ waves25 and possible weak Frohlich condensation26 believed to help store and to select previously learned signaling strategies at quantum efficiencies.8–10

Figures and Tables

Figure 1 Model potential energy diagrams and Arrhenius plots for the rate or kinetics of microbial “social” decision making. (A) Under preferential attachment rules, decisions to switch from one social strategy to another of less fitness may require “energy absorption” analogous to endothermic chemical reactions. The initial fitter strategy or reactant (R_s) maintains a lower potential energy. Forward transition to a less fit strategy or product (P_s) of higher potential energy may proceed over the reaction barrier via an intermediate or transition state (T_s) or through the reaction barrier without a transition state. The former case is a classical reaction needing activation energy (E_a) and a change in the system's heat (ΔH). The latter case is a quantum reaction, tunneling through the reaction barrier. Arrhenius kinetics (inset) of the classical reaction (black) form a linear relationship between the rate of strategy switching (k) and the inverse of the system's temperature (1/T), whereas the quantum reaction (grey) produces a nonlinear function between the same variables. (B) Social decision making by microbes may also require “energy dissipation” analogous to exothermic chemical reactions. Here the initial less fit strategy (R_s) of higher potential energy transitions to a more fit strategy (P_s) of lower potential energy using the classical or quantum principles described for (A). The Arrhenius kinetics (inset) of exothermic reactions give plots identical to that of endothermic reactions.

Figure 2 Arrhenius plots for learned “courting” decisions made by Spirostomum ambiguum during simulated “social” trials. (A) S. ambiguum that respond with initial low (top panel), medium (middle panel), or high (bottom panel) responsiveness learn to encourage or discourage mating interactions by emitting serial contraction strategies of differential fitness. When the rate or kinetics of strategy switching are dominated by quantum Bose-Einstein (BE, grey diamond) or Fermi-Dirac (FD, white diamond) statistical distributions, Arrhenius plots show pronounced nonlinearity consistent with quantum-like tunneling in decision making processes. When a classical Maxwell-Boltzmann (MB, black diamond) statistical distribution dominates social decision making by S. ambiguum, plots yield a pronounced linear function consistent with classical, thermodynamic-sensitive strategy switching. (B) S. ambiguum that respond with initial medium (middle panel) or high (bottom panel) responsiveness also learn to encourage or discourage mating interactions by emitting serial ciliary reversal strategies of differential fitness. Rates of strategy switching are dominated by quantum FD (white diamond) and BE (black diamond) statistical distributions with pronounced nonlinear Arrhenius kinetics consistent with quantum-like tunneling. For (A) and (B), temperature T was substituted for the “critical tunneling field strength”, Γ, an annealing parameter directly related to the critical condensation temperature T_C by Γ_C = (T_Ck_B)/[n/ζ(3/2)]^2/3, where k_B is Boltzmann constant, ζ is the Riemann zeta function, and n is the “particle” density.

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