Abstract
In this work we aim to highlight a close analogy between cooperative behaviors in chemical kinetics and cybernetics; this is realized by using a common language for their description, that is meanfield statistical mechanics. First, we perform a onetoone mapping between paradigmatic behaviors in chemical kinetics (i.e., noncooperative, cooperative, ultrasensitive, anticooperative) and in meanfield statistical mechanics (i.e., paramagnetic, high and low temperature ferromagnetic, antiferromagnetic). Interestingly, the statistical mechanics approach allows a unified, broad theory for all scenarios and, in particular, MichaelisMenten, Hill and Adair equations are consistently recovered. This framework is then tested against experimental biological data with an overall excellent agreement. One step forward, we consistently read the whole mapping from a cybernetic perspective, highlighting deep structural analogies between the abovementioned kinetics and fundamental bricks in electronics (i.e. operational amplifiers, flashes, flipflops), so to build a clear bridge linking biochemical kinetics and cybernetics.
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Introduction
Cooperativity is one of the most important properties of molecular interactions in biological systems and it is often invoked to account for collective features in binding phenomena.
In order to investigate and predict the effects of cooperativity, chemical kinetics proved to be a fundamental tool and, also due to its broadness over several fields of biosciences, a number of cooperativity quantifiers (e.g. Hills number^{1}, Koshland cooperativity test^{2}, global dissociation quotient^{3}, weak and strong fine tunings^{4}, etc.), apparently independent or distinct, have been introduced. However, a clear, unified, theoretical scheme where all cooperative behaviors can be framed would be of great importance, especially in biotechnology research^{5,6} and for scientists dealing with interdisciplinary applications^{7,8}. To this task, statistical mechanics offers a valuable approach as, from its basic principles, it aims to figure out collective phenomena, possibly overlooking the details of the interactions to focus on the very key features. Indeed, a statistical mechanics description of reaction kinetics has already been paved through theoretical models based on linear Ising chains^{9}, spin lattices with nearest neighbors interactions^{10}, transfer matrix theory^{9,10} and structural probabilistic approaches^{11}.
In this work we expand such statistical mechanics picture toward a meanfield perspective^{12} by assuming that the interactions among the system constituents are not limited by any topological or spatial constraint, but are implicitly taken to be longranged, as in a system that remains spatially homogeneous. This approach is naturally consistent with the rateequation picture, typical of chemical kinetics investigations and whose validity is restricted to the case of vanishing correlations^{13,14} and requires a sufficiently high spatial dimension or the presence of an effective mixing mechanism (hence, ultimately, longrange interactions). In general, in the meanfield limit, fluctuations naturally decouple from the volumeaveraged quantities and can be treated as negligible noise.
By adopting a meanfield approach, we abandon a direct spatial representation of binding structures and we introduce a renormalization of the effective couplings. The reward lies in a resulting unique model exhibiting a rich phenomenology (e.g. phase transitions), which lowdimensional models typically lack, yet being still feasible for an exact solution. In particular, we obtain an analytical expression for the saturation function which is successfully compared with recent experimental findings, taken from different (biological) contexts to check robustness. Furthermore, from this theory basic chemical kinetics equations (e.g. MichaelisMenten, Hill and Adair equations) are recovered as special cases.
Further, there is a deep theoretical motivation underlying the development of a meanfield statistical mechanics approach to chemical kinetics: it can be used to code collective behavior of biosystems into a cybernetical framework. In fact, cybernetics, meant as the science dedicated to the understanding of selforganization and emergent communication among the constituents of a system, can be naturally described via (meanfield) statistical mechanics^{15,16,17}. Thus, the latter provides a shared formalism which allows to automatically translate chemical kinetics into cybernetics and vice versa. In this perspective, beyond theoretical interest, at least two concrete benefits may stem from our investigation: first, in the field of biotechnologies, logical gates have already been obtained through biological hardware (see e.g.^{5,6}) and for their proper functioning signal amplification turns out to be crucial. In this paper, cooperativity in reaction kinetics is mapped into amplification in electronics, hence offering a solid scaffold for biological amplification theory.
Then, as statistical mechanics has been successfully applied in the world of computing (for instance in neural networks^{18}, machine learning^{19} or complex satisfiability^{20}), its presence in the theory of biological processors could be of relevant interest. In particular, we discuss how to map ultrasensitive kinetics to logical switches and how to read anticooperative kinetics as the basic ingredient for memory storage in biological flipflops, whose interest resides in several biological machineries as gene regulatory networks^{21}, riboswitches^{22}, synaptic switches^{23}, autopoietic systems^{24} and more^{25,26,27}.
To summarize, a rigorous, promising link between cybernetics and collective biological systems can be established via statistical mechanics and this point will be sketched and corroborated by means of several examples throughout this paper, which is structured as follows:
First, we review the main concepts, facts and methods from both chemical kinetics and statistical mechanics perspectives. Then, we develop a proper theoretical framework able to bridge statistical mechanics and chemical kinetics; the former can also serve as a proper tool for describing and investigating cybernetics, thus, as a syllogism, chemical kinetics and cybernetics become also related. The agreement of our framework with real data, carefully extrapolated from recent biological researches, covering the various standard behaviors in chemical kinetics, is also successfully checked. Finally, results and outlooks are discussed.
Results
Collective behaviors in chemical kinetics
Many polymers and proteins exhibit cooperativity, meaning that their ligands bind in a nonindependent way: if, upon a ligand binding, the probability of further binding (by other ligands) is enhanced, like in the paradigmatic case of hemoglobin^{9}, the cooperativity is said to be positive, vice versa there is negative cooperativity when the binding of more ligands is inhibited^{28}, as for instance in some insulin receptors^{29,30} and most Gprotein coupled receptors^{23,31}. Several mechanisms can be responsible for this effect: for example, if two neighbor docking sites on a polymer can bind charged ions, the electrostatic attraction/repulsion may be the cause of a positive/negative cooperativity. However, the most common case is that the binding of a ligand somehow modifies the structure of the hosting molecule, influencing the binding over the other sites and this is the socalled allosteric mechanism^{32}.
Let us now formalize such behavior by considering a hosting molecule P that can bind N identical molecules S on its structure; calling P_{j} the complex of a molecule P with j ∈ [0,N] molecules attached, the reactions leading to the chemical equilibrium are the following
hence the time evolution of the concentration of the unbounded protein P_{0} is ruled by
where , are, respectively, the forward and backward rate constants for the state j = 1 and their ratios define the association constant and dissociation constant . Focusing on the steady state we get, iteratively,
Unfortunately, measuring [P_{j}] is not an easy task and one usually introduces, as a convenient experimental observable, the average number of substrates bound to the protein as
which is the wellknown Adair equation^{3}, whose normalized expression defines the saturation function . In a noncooperative system, one expects independent and identical binding sites, whose steady states can be written as (explicitly only for j = 1 and j = 2 for simplicity)
where K_{+} and K_{−} are the rates for binding and unbinding on any arbitrary site. Being the intrinsic association constant, we get
and, in general, K^{(j)} = (N − j + 1)K/j. Plugging this expression into the Adair equation (2) we get
which is the wellknown MichaelisMenten equation^{3}.
If interaction among binding sites is expected, the kinetics becomes far less trivial. Let us first sketch the limit case where intermediates steps can be neglected, that is
then
More generally, one can allow for a degree of sequentiality and write
which is the wellknown Hill equation^{3}, where n_{H}, referred to as Hill coefficient, represents the effective number of substrates which are interacting, such that for n_{H} = 1 the system is said to be noncooperative and the MichaelisMenten law is recovered; for n_{H} > 1 it is cooperative; for it is ultra sensitive; for n_{H} < 1 it is anti cooperative.
From a practical point of view, from experimental data for Y([S]), one measures n_{H} as the slope of log(Y/(1 − Y)) versus [S].
Meanfield statistical mechanics
One of the best known statistical mechanics model is the meanfield Ising model, namely the CurieWeiss model^{33}. It describes the macroscopic behavior of a magnetic system microscopically represented by N binary spins, labeled by i = 1, 2, …, N and whose state is denoted by σ_{i} = ±1. In the presence of an external field h and being J the N × N symmetric matrix encoding for pairwise interactions among spins, the (extensive, macroscopic) internal energy associated to a the configuration {σ} = {σ_{1}, σ_{2}, …, σ_{N}} is defined as
It is easy to see that the spin configurations leading to a lower energy are those where spins are aligned with the pertaining field, i.e. σ_{i}h_{i} > 0 and pairs (i, j) associated to positive (negative) coupling J_{ij} are parallel (antiparallel), i.e. σ_{i}σ_{j} = 1 (σ_{i}σ_{j} = −1). Notice that, in eq. 11, we implicitly assumed that any arbitrary spin possibly interacts with any other. This is a signature of the mean field approach which, basically, means that interactions among spins are longrange and/or that the timescale of reactions is longer than the typical time for particles to diffuse, in such a way that each spin/particle actually sees any other. We stress that the mean field approximation also implies that the probability distribution P({σ}) for the whole configuration is factorized into the product of the distribution for each single constituents, namely , analogously to classical chemical kinetic prescriptions^{10}.
For the analytic treatment of the system it is convenient to adopt a mesoscopic description where the phase space, made of all the 2^{N} possible distinct spin configurations, undergoes a coarsegraining and is divided into a collection of sets, each representing a mesoscopic state of a given energy E_{k}, (we dropped the dependence on the parameters J, h to lighten the notation). In this way, all the microscopic states belonging to the set share the same value of energy E_{k}, calculated according to (11). In order to describe the macroscopic behavior of the system through its microscopical degrees of freedom, we introduce a statistical ensemble , meant as the probability distribution over the sets in ; consequently, ρ_{i} ≥ 0 and must be fulfilled. Accordingly, the internal energy and the entropy read as
where K_{B} is the Boltzmann constant, hereafter set equal to 1. Being β > 0 the absolute inverse temperature of the system, we define the freeenergy
Notice that the minimum of F ensures, contemporary, the minimum for E and the maximum for S, hence it provides a definition for the thermodynamic equilibrium. As a consequence, from eq. 13 we calculate the derivative with respect to the probability distribution and require ∂F/∂ρ_{i} = 0; the solution, referred to as , reads as
and it is called the MaxwellBolzmann distribution. The normalization condition implies and this quantity is called “partition function”. We therefore have
In general, given a function f({σ}), its thermal average is .
As this system is expected to display two different behaviors, an ordered one (at low temperature) and a disordered one (at high temperature), we introduce the magnetization , which provides a primary description for the macroscopic behavior of the system. In particular, it works as the “order parameter” and it characterizes the onset of order at the phase transition between the two possible regimes. More precisely, as the parameters β, J, h are tuned (here for simplicity J_{ij} > 0, i_{, j}), the system can be either disordered (i.e. paramagnetic), where spins are randomly oriented and 〈m〉 = 0, or ordered (i.e. ferromagnetic), where spins are consistently aligned and 〈m〉 ≠ 0. The phase transition, separating regions where one state prevails against the other, is a consequence of the collective microscopic interactions.
In a uniform system where , _{i} ≠ j, J_{ii} = 0 and , all spins display the same expected value, i.e. 〈σ_{i}〉 = 〈σ〉, i, which also corresponds to the average magnetization 〈m〉. Remarkably, in this case the free energy of the system can be expressed through 〈m〉 by a straightforward calculation^{18} that yields
whose extremization w.r.t. to 〈m〉 ensures again that thermodynamic principles hold and it reads off as
which is the celebrated CurieWeiss selfconsistency. By simply solving eq. 17 (e.g. graphically or numerically) the macroscopic behavior can be inferred. Before proceeding, we fix β = 1, without loss of generality as it can be reabsorbed trivially by h → βh = h and J → βJ = J.
In the noninteracting case (J = 0), eq. 17 gets m(J = 0, h) = tanh(h), which reminds to an inputoutput relation for the system. When interactions are present (J > 0), one can see that the solution of eq. 17 crucially depends on J. Of course, 〈m(J = 0, h)〉 < 〈m(J > 0, h)〉, due to cooperation among spins, and, more remarkably, there exists a critical value J_{c} such that when J ≥ J_{c} the typical sigmoidal response encoded by 〈m(J, h)〉 possibly becomes a step function (a true discontinuity is realized only in the thermodynamic limit N → ∞, while at finite N the curve gets severely steep but still continuous). Hence, to summarize, the Curie Weiss model exhibits two phases: A smallcoupling phase where the system behaves paramagnetically and a strongcoupling phase where it behaves ferromagnetically. The ferromagnetic states are two, characterized by positive and negative magnetization, according to the sign of the external field.
Statistical mechanics and chemical kinetics
In this section we develop the first part of our formal bridge and show how the CurieWeiss model can be looked from a biochemical perspective. We will start from the simplest case of independent sites and later, when dealing with interacting sites, we will properly generalize this model in order to consistently include both positive and negative cooperativity.
The simplest framework: non interacting sites
Let us consider an ensemble of elements (e.g. identical macromolecules, homoallosteric enzymes, a catalyst surface), whose interacting sites are overall N and labelled as i = 1, 2, …, N. Each site can bind one smaller molecule (e.g. of a substrate) and we call α the concentration of the free molecules ([S] in standard chemical kinetics language). We associate to each site an Ising spin such that when the i^{th} site is occupied σ_{i} = +1, while when it is empty σ_{i} = −1. A configuration of the elements is then specified by the set {σ}.
First, we focus on noncollective systems, where no interaction between binding sites is present, while we model the interaction between the substrate and the binding site by an “external field” h meant as a proper measure for the concentration of freeligand molecules, hence h = h(α). One can consider a microscopic interaction energy given by
Note that writing instead of implicitly assumes homogeneity and thermalized reactions: each site displays the same coupling with the substrate and they all had the time to interact with the substrate.
We can think at h as the chemical potential for the binding of substrate molecules on sites: When it is positive, molecules tend to bind to diminish energy, while when it is negative, bound molecules tend to leave occupied sites. The chemical potential can be expressed as the logarithm of the concentration of binding molecules and one can assume that the concentration is proportional to the ratio of the probability of having a site occupied with respect to that of having it empty. In this simple case and in all meanfield approaches^{33}, the probability of each configuration is the product of the single independent probabilities of each site to be occupied and, applying the MaxwellBoltzmann distribution P(σ_{i} = ±1) = e^{±h}, one finds
and we can pose
The mean occupation number (close to the magnetization in statistical physics) reads off as
Therefore, using ergodicity to shift into 〈S〉 (see eq. 7), the saturation function can be written as
By using eq. 17 we get
which, substituting 2h = log α, recovers the MichaelisMenten behavior, consistently with the assumption of no interaction among binding sites (J = 0) in eq. 18.
A refined framework: twosites interactions
We now focus on pairwise interactions and, seeking for a general scheme, we replace the fullyconnected network of the original CurieWeiss model by a complete bipartite graph: sites are divided in two groups, referred to as A and B, whose sizes are N_{A} and N_{B} (N = N_{A} + N_{B}), respectively. Each site in A (B) is linked to all sites in B (A), but no link within the same group is present. With this structure we mirror dimeric interactions [Note: Notice that for the sake of clearness, we introduced the simplest bipartite structure, which naturally maps dimeric interactions, but one can straightforwardly generalize to the case of an nmer by an npartite system and of course values of ρ_{A} ≠ ρ_{B} can be considered too. We did not perform these extensions because we wanted to recover the broader phenomenology with the smaller amount of parameters, namely J, α only.], where a ligand belonging to one group interacts in a mean field way with ligands in the other group (cooperatively or competitively depending on the sign of the coupling, see below) and they both interact with the substrate. As a result, given the parameters J and h, the energy associated to the configuration {σ} turns out be
Some remarks are in order now. First, we stress that in eq. 23 the sums run over all the binding sites. As we will deal with the thermodynamic limit (N → ∞), this does not imply that we model macromolecules of infinite length, which is somehow unrealistic. Rather, we consider N as the total number of binding sites, localized even on different macromolecules and the underlying meanfield assumption implies that binding sites belonging to the same group are all equivalent, despite some may correspond to the bulk and others to the boundaries of the pertaining molecule; such differences can be reabsorbed in an effective renormalization of the couplings. In this way the system, as a whole, can exhibit (anti) cooperative effects, as for instance shown experimentally in^{34}.
Moreover, for the sake of clearness, in the following we will assume that couplings between sites belonging to different groups are all the same and equal to J and, similarly, h_{i} = h, for any i. This homogeneity assumption allows to focus on the simplest cooperative effects and can be straightforwardly relaxed.
We also notice that this twogroups model can mimic both cooperative and noncooperative systems but, while for the former case bipartition is somehow redundant as qualitatively analogous results are obtained by adopting a fullyconnected structure, for the latter case the underlying competitive interactions intrinsically require a bipartite structure.
Now, the two groups are assumed as equally populated, i.e., N_{A} = N_{B} = N/2, such that their relative densities are ρ_{A} = N_{A}/N = ρ_{B} = N_{B}/N = 1/2. The order parameter can be trivially extended as
and, according to statistical mechanics prescriptions, we minimize the free energy coupled to the cost function (23) and we get, in the thermodynamic limit, the following selfconsistencies
Through eqs. 25 and 26, the number of occupied sites can be computed as
from which we get the overall binding isotherm
From eqs. 25–28 one can see that Y (α) fulfills the following freeenergy minimum condition
This expression returns the average fraction of occupied sites corresponding to the equilibrium state for the system. We are now going to study separately the two cases of positive (J > 0) and negative (J < 0) cooperativity.
The cooperative case: Chemical kinetics
When J > 0 interacting units tend to “imitate” each other. In this ferromagnetic context one can prove that the bipartite topology does not induce any qualitative effects: results are the same (under a proper rescaling) as for the Curie Weiss model; indeed, in this case one can think of bipartition as a particular dilution on the previous fullyconnected scheme and we know that (pathological cases apart), dilution does not affect the physical scenario^{35,36,37,38}.
Differently from lowdimensional systems such as the linear Isingchains, the CurieWeiss model admits sharp (eventually discontinuous in the thermodynamic limit) transitions from an empty (〈m_{A}〉 = 〈m_{B}〉 = 0) to a completely filled (〈m_{A}〉 = 〈m_{B}〉 = 1) configuration as the field h is tuned. More precisely, eqs. 25 and 26 describe a transition at α = 1 and such a transition is second order (Y changes continuously, but its derivative may diverge) when J is smaller than the critical value J_{c} = 1, while it is first order (Y has a discontinuity) when J > J_{c}. The latter case is remarkable as a discontinuous behavior is experimentally well evidenced and at the basis of the socalled ultrasensitive chemical switches^{39}.
On the other hand, when J → 0, the interaction term disappears and we expect to recover MichaelisMenten kinetics. In fact, eq. 29 can be rewritten as
which, for J = 0, recovers the MichaelisMenten equation Y (α) = α/(1 + α) [Note: Note that we do not lose generality when obtaining eq. (7) and not Y (α) = α/(K + α) (which is the more familiar MM expression) because we can rewrite the latter as Y (α) = K^{−1}α/(1 + K^{−1}α) by shifting α → αK^{−1} and h → [log(α/K)]/2.]. In this case there is no signature of phase transition as Y (α) is continuous in any of its derivatives.
In general, the Hill coefficient can be obtained as the slope of Y (α) in eq. 29 at the symmetric point Y = 1/2, namely
where the role of J is clear: a large J, i.e. J close to 1, implies a strong cooperativity and vice versa.
One step forward, as the whole theory is now described through the functions appearing in the selfconsistency, we can expand them obtaining polynomials at all the desired orders, more typical of the standard route of chemical kinetics. In particular, expanding eq. 30 at the first order in J we obtain
which is nothing but the Adair equation (eq. 2) as far as we set and we rescale . These results and, in particular, the expression in eq. 30 are shown in complete generality in figs. 1 and 2.
The expression in eq. 30 can also be used to fit experimental data for saturation versus substrate concentration. Indeed, through an iterative fitting procedure, implied by the selfconsistency nature of our theoretical expression, we can derive an estimate for the parameter J and, from this, evaluate the Hill coefficient through eq. 31. As shown in fig. 3, fits are successful for several sets of experimental data, taken from different fields of biotechnology. The Hill coefficients derived in this way and the related estimates found in the literature are also in excellent agreement.
Cooperative kinetics and cybernetics: Amplifiers and comparators
Having formalized cooperativity through statistical mechanics, we now want to perform a further translation in cybernetic terms. In particular, we focus on the electronic declination of cybernetics because this is probably the most practical and known branch. We separate the small coupling case (J < J_{c}, cooperative kinetics) from the strong coupling case (J > J_{c}, ultrasensitive kinetics) and we mirror them to, respectively, the saturable operational amplifier and the analogtodigital converter^{40}. The plan is to compare the saturation curves (binding isotherms) in chemical kinetics with selfconsistencies in statistical mechanics and transfer functions in electronics so to reach a unified description for these systems.
Before proceeding, we recall a few basic concepts. The core of electronics is the operational amplifier, namely a solidstate integrated circuit (transistor) which uses feedback regulation to set its functions. In fig. 4 we show the easiest representation for operational amplifiers: there are two signal inputs (one positive received (+) and one negative received (−)), two voltage supplies (V_{sat}, −V_{sat}) and an output (V_{out}). An ideal amplifier is the “linear” approximation of the saturable one and essentially assumes that the voltage at the input collectors (V_{sat} and −V_{sat}) is always at the same value so that no current flows inside the transistor, namely, retaining the obvious symbols of fig. 4, i_{+} = i_{−} = 0^{40}. Obtaining its transfer function is straightforward as we can apply Kirchhoff law at the node 1 to delete the afferent currents, hence i_{1} + i_{2} + i_{−} = 0. Then, assuming R_{1} = 1 Ω (without loss of generality as only the ratio R_{2}/R_{1} matters), in the previous equation we can pose i_{1} = −V_{−}, i_{2} = (V_{out} − V_{−})/R_{2} and i_{−} = 0 (because the amplifier is ideal). We can further note that V_{−} = V_{+} and V_{+} = V_{in} so to rewrite Kirchhoff law as 0 = −V_{in} + (V_{out} − V_{in})/R_{2}, by which the transfer function reads off as
where G = 1 + R_{2} is called “gain” of the amplifier.
Therefore, as far as R_{2} > 0, the gain is larger than one and the circuit is amplifying the input (R_{2} < 0 is actually thermodynamically forbidden, suggesting that anticooperativity, from cybernetic perspective, must be accounted by the inverter configuration^{40}, see next section).
Let us emphasize some structural analogies with ferromagnetic behaviors and cooperative kinetics. First, we notice that all these systems “saturate”. Indeed, it is very intuitive to see that by applying a magnetic field h > 0 to a collection of spins, they will (at least partially, depending on the noise level) align with the field, resulting in 〈m(h)〉 > 0. However, once reached the critical value such that , any further increase in the strength of the field (i.e. any ) will produce no variations in the output of the system as all the spins are already aligned. Similarly, in reaction kinetics, once all the ligands of a given protein have bound to the substrate, any further growth in the substrate concentration will produce no net effect on the system. In the same way, given an arbitrary operational amplifier supplied with V_{sat}, then its output voltage will be a function of the input voltage V_{in}. However, there exists a critical input such that V_{out} = V_{sat} and when input is larger than no further amplification is possible; the amplifier is then said to be “saturated”. Of course, the sigmoidal shape of the hyperbolic tangent is not accounted by ideal amplifiers, yet for real amplifiers ± V_{sat} are upper bounds for the growth, hence recovering the expected behavior, as shown in the plot of fig. 6 l [Note: One may notice that the outlined amplifier is linear, while in chemical kinetics usually slopes are sigmoidal on linlog plots. This is only a technical point and to obtain the logarithmic amplifier it is enough to substitute R_{2} with a diode to use the exponential scale of the latter^{40}.]. Moreover, we notice that the transfer function is an input/output relation, exactly as the equation for the order parameter m. In fact, the latter, for small values of the coupling J (so to mirror ideal amplifier), can be written as (see eq. 17)
Thus, the external signal V_{in} is replaced by the external field h and the voltage V_{out} is replaced by the magnetization 〈m〉. By comparing eq. 33 and eq. 34 we see that R_{2} plays as J, and, consistently, if R_{2} = 0 the retroaction is lost (see fig. 3) and the gain is no longer possible. This is perfectly coherent with the statistical mechanics perspective, where, if J = 0, spins do not mutually interact and no feedback is allowed.
Analogously, in the chemical kinetics scenario, the Hill coefficient can be written as n_{H} = 1/(1 − J) ~ 1 + J, in the limit of small J (namely for J < J_{c} = 1, which is indeed the case under investigation). Therefore, again, we see that if J = 0 there is no amplification and the kinetics returns the MichaelisMenten scenario, while for positive J we obtain amplification and a cooperative behavior. This leads to the conceptual equivalence
hence the Hill coefficient in chemical kinetics plays as the gain in electronics. This implicitly accounts for a quantitative comparison between amplification in electronics and in biological devices.
One step forward, if J > J_{c} the equation 〈m(h)〉 becomes discontinuous in statistical mechanics just like the corresponding (ultrasensitive) saturation curve in chemical kinetics: The analogy with cybernetics can still be pursued, but with analogtodigital converters (ADCs), which are the corresponding limits of operational amplifiers.
The ADC, roughly speaking a switch, takes a continuous. i.e. analogue, input and has discrete (dichotomic in its basic implementation) states as outputs. The simplest ADC, namely flash converters, are built through cascades of voltage comparators^{40}. A voltage comparator is sketched in fig. 6h and it simply “compares” the incoming voltage values between the negative input and the positive one as follows: Let us use as the negative input the ground (V = 0) as a reference value (to mirror one to one the equivalence with chemical kinetics or statistical mechanics we deal with only one input, namely the substrate concentration α in the former and the magnetic field h in the latter). Then, if V_{in} is positive the output will be V_{sat} > 0, vice versa, if V_{in} is negative, the output will be −V_{sat} < 0 as reported in the plot in fig. 6m, representing the ADC transfer function.
An ADC is simply an operational amplifier in an open loop (i.e. R_{2} = ∞), hence its theoretical gain is infinite. Coherently, this corresponds to values of J → 1 that imply a theoretical divergence in the Hill coefficient, while, practically, reactions are referred to as “ultrasensitive” already at . Consistently, as J → 1 the curve 〈m(h)〉 starts to develop a discontinuity at h = 0 (see Fig. 6, panels d, e, f), marking the onset of a first order phase transition. As a last remark, despite we are not analyzing these systems in the frequency domain in this first paper, we highlight that, when using timedependent fields, for instance oscillatory input signals, full structural consistency is preserved as all these systems display hysteresis effects at high enough frequencies of the input signal.
The anticooperative case: Chemical kinetics and cybernetics
We can now extend the previous scheme for the description of a negativecooperative system, by simply taking a negative coupling J < 0. Hence, eqs. 25 and 26 still hold and we can analogously reconstruct Y (α) = (〈m_{A}〉 + 〈m_{B}〉)/N versus α, whose theoretical outcomes are still shown in figs. 1 and 2 and fit them against experimental results as shown in the plots of fig. 5.
Again, it is easy to check that there are two possible behaviors depending on the interaction strength J. If J < J_{c}, the two partial fractions n_{A}, n_{B} are always equal, but when the interaction is larger than J_{c}, the two partial fractions are different in a region where the chemical potential log α is around zero, as shown in fig. 1. In this region, due to the strong interaction and small chemical potential, it is more convenient for the system to fill sites on a subsystem and keep less molecules of ligands on the other subsystem. The critical value of the chemical potential  log α depends on the interaction strength: it vanishes when the average interaction equals J_{c} and grows from this value on. The region where the two fractions are different corresponds, in the magnetic models, to the antiferromagnetic phase. When J < J_{c}, the binding isotherm, plotted as a function of the logarithm of concentration, has a form resembling the MichaelisMenten curve, even if anticooperativity is at work. Conversely, when J > J_{c}, in the region around α = 1 the curve has a concavity with an opposite sign with respect to the MichaelisMenten one. In particular, there is a plateau around α = 1, which can be interpreted as the inhibition of the system, once it is half filled, towards further occupation.
Finally, in order to complete our analogy to electronics, let us consider the simplest bistable flipflop^{40}, built through two saturable operational amplifiers as sketched in fig. 6 i, such that the output of one of the two amplifiers is used as the inverted input of the other amplifiers, tuned by a resistor. This configuration, encoded in statistical mechanics by negative couplings among groups, makes the amplifiers reciprocally inhibiting because (and indeed they are called ”inverters” in this configuration), for instance, a large output from the first amplifier (say A) induces a fall in the second amplifier (say B) and vice versa. Since each amplifier pushes the other in the opposite state, there exist two stationary stable configurations (one amplifier with positive output and the other with negative output and vice versa). Thus, it is possible to assign a logical 0 (or 1) to one state and the other logical 1 (or 0) to the other state which can be regarded respectively as low concentration versus high concentration of bind ligands in chemical kinetics; negative or positive magnetization in ferromagnetic systems, low versus high output voltage of flipflops in electronics. In this way, as the flipflop can serve as an information storage device (in fact, the information (1 bit) is encoded by the output itself), the same feature holds also for the other systems. The behavior of the two flipflop transfer functions (one for each inverter) are also shown in fig. 6 n, where the two (opposite) sigmoidal shapes are displayed versus the input voltage.
Still in fig. 6, those behaviors are compared with experimental data from biochemical anticooperativity and their statisticalmechanics best fits with an overall remarkable agreement.
Possible extensions: Heterogeneity and multiple binding sites
Another point worth of being highlighted is the number of potential and straightforward extensions included in the statisticalmechanics modelization. In fact, as the literature of meanfield statisticalmechanics model is huge, once selfconsistencies are properly mapped into the saturation curves, one can perturb, generalize, or adjust the initial energy (cost) function and check the resulting effects.
As an example, we discuss chemical heterogeneity, which has been shown by recent experiments^{41,42} to play a crucial role in equilibrium reaction rates. To include this feature in our theory we can replace in eq. 18 with with h_{i} drawn from, e.g., a Gaussian distribution
such that for a → 1 homogeneous chemical kinetics is recovered, while for a → 0 we get standard Gaussian distribution for heterogeneity.
We fitted data from^{41} through the selfconsistencies obtained by either fixing a = 1, or by taking a as a free parameter; the results obtained are in strong agreement with the original one. In particular, the authors in^{41} found a ratio R between the ”real” Hill coefficient (assuming heterogeneity) and the standard (homogeneous) one as R ~ 0.53, while, theoretically we found R ~ 0.57 (and a ~ 0.3).
Further, we notice that n_{H} grows with a − 1, namely the higher the degree of inhomogeneity within the system and the smaller n_{H}, in agreement with several recent experimental findings^{42,43}.
As a last example of possible extension, we discuss quickly also the multiple binding site case, which can be simply encoded, at least within the cooperative case, by considering an interaction energy of the Pspin type as
which results in multiple discontinuities for the binding isotherms as for instance happens when considering surfactants onto a polymer gel^{44,45}, where the affinity of the surfactants to the gel is cooperatively altered by a conformational change of the polymer chains (and actually these systems show hysteresis with respect to the surfactant concentration, which is another typical feature of “ferromagnetism”).
Discussion
In this work, we describe collective behaviors in chemical kinetics through meanfield statistical mechanics. Stimulated by the successes of the latter in formalizing classical cybernetic subjects, as neural networks in artificial intelligence^{15,16,18} or NPcompleteness problems in logic^{46,47,48}, we successfully tested the statistical mechanics framework as a common language to read from a cybernetic perspective chemical kinetic reactions, whose complex features are at the very basis of several biological devices.
In particular, we introduced an elementary class of models able to mimic possibly heterogeneous systems covering all the main chemical kinetics behaviors, namely ultrasensitive, cooperative, anticooperative and noncooperative reactions. Predictions yielded by such theoretical frame have been tested for comparison with experimental data taken from biological systems (e.g. nervous system, plasma, bacteria), finding overall excellent agreement. Furthermore, we showed that our analytical results recover all the standard chemical kinetics, e.g. MichaelisMenten, Hills and Adair equations, as particular cases of this broader theory and confer to these a strong and simple physical background. Due to the presence of first order phase transition in statistical mechanics we offer a simple prescription to define a reaction as ultrasensitive: Its best fit is achieved through a discontinuous function, whose extremization through other routes is not simple as e.g. leastsquares can not be applied due to the discontinuity itself.
It is worth noticing that, despite we developed mean field techniques, hence we neglected any spatial structure, we get a direct mapping between statisticalmechanics and chemical kinetics formulas, in such a way that we can derive from the former a simple estimate for the Hill coefficient, namely for ”effective number” of interacting binding sites, in full agreement with experimental data and standard approaches.
One step forward, toward a unifying cybernetic perspective, we described a conceptual and practical mapping between kinetics of ultrasensitive, cooperative and anticooperative reactions, with the behavior of analogtodigital converters, saturable amplifiers and flipflops respectively, highlighting how statistical mechanics can act as a common language between electronics and biochemistry. Remarkably, saturation curves in chemical kinetics mirror transfer functions of these three fundamental electronic devices which are the very bricks of robotics.
The bridge built here inspires and makes feasible several challenges and improvements in biotechnology research. For instance, we can now decompose complex reactions into a sequence of elementary ones (modularity property^{5}) and map the latter into an ensemble of interacting spin systems to investigate potentially hidden properties of the latter such as selforganization and computational capabilities (as already done adopting spinglass models of neural networks^{49}). Moreover, we can reach further insights in the development of better performing biological processing hardware, which are currently poorer than silicomade references. Indeed, from our equivalence between Hill and gain coefficients the more power of electronic devices is clear as G can range over several orders of magnitude, while in chemical kinetics Hill coefficients higher than n_{H} ~ 10 are difficult to find. This, in turn, may contribute in developing a biological amplification theory whose fruition is at the very basis of biological computations^{6,7,8}.
We believe that this is an important, intermediary, brick in the multidisciplinary research scaffold of biological complexity.
Lastly, we remark that this is only a first step: Analyzing, within this perspective, more structured biological networks as for instance the cytokine one at extracellular level or the metabolic one at intracellular level is still an open point and requires extending the meanfield statistical mechanics of glassy systems (i.e. frustrated combinations of ferro and antiferro magnets), on which we plan to report soon.
Change history
11 July 2014
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11 July 2014
In this work we aim to highlight a close analogy between cooperative behaviors in chemical kinetics and cybernetics; this is realized by using a common language for their description, that is meanfield statistical mechanics. First, we perform a onetoone mapping between paradigmatic behaviors in chemical kinetics (i.e., noncooperative, cooperative, ultrasensitive, anticooperative) and in meanfield statistical mechanics (i.e., paramagnetic, high and low temperature ferromagnetic, antiferromagnetic). Interestingly, the statistical mechanics approach allows a unified, broad theory for all scenarios and, in particular, MichaelisMenten, Hill and Adair equations are consistently recovered. This framework is then tested against experimental biological data with an overall excellent agreement. One step forward, we consistently read the whole mapping from a cybernetic perspective, highlighting deep structural analogies between the abovementioned kinetics and fundamental bricks in electronics (i.e. operational amplifiers, flashes, flipflops), so to build a clear bridge linking biochemical kinetics and cybernetics.
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Acknowledgements
This works is supported by the FIRB grant RBFR08EKEV. Sapienza Università di Roma, INdAM through GNFM and INFN are acknowledged too for partial financial support.
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The bridge between chemical kinetics and statistical mechanics has been built by all the authors. The next bridge to cybernetics has been built by A.B. E.A., A.D.B. and G.U. performed data analysis and produced all the plots. E.A., A.B. and R.B. wrote the manuscript.
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Agliari, E., Barra, A., Burioni, R. et al. Collective behaviours: from biochemical kinetics to electronic circuits. Sci Rep 3, 3458 (2013). https://doi.org/10.1038/srep03458
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DOI: https://doi.org/10.1038/srep03458
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