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Complex Reaction Kinetics in Chemistry: A Unified Picture Suggested by Mechanics in Physics

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Complex biochemical pathways or regulatory enzyme kinetics can be reduced to chains of elementary reactions, which can be described in terms of chemical kinetics. This discipline provides a set of tools for quantifying and understanding the dialogue between reactants, whose framing into a solid and consistent mathematical description is of pivotal importance in the growing field of biotechnology. Among the elementary reactions so far extensively investigated, we recall the socalled Michaelis-Menten scheme and the Hill positive-cooperative kinetics, which apply to molecular binding and are characterized by the absence and the presence, respectively, of cooperative interactions between binding sites, giving rise to qualitative different phenomenologies. However, there is evidence of reactions displaying a more complex, and by far less understood, pattern: these follow the positive-cooperative scenario at small substrate concentration, yet negative-cooperative effects emerge and get stronger as the substrate concentration is increased. In this paper we analyze the structural analogy between the mathematical backbone of (classical) reaction kinetics in Chemistry and that of (classical) mechanics in Physics: techniques and results from the latter shall be used to infer properties on the former.
of analysis on cooperativity. (a) Klotz plot showing the saturation function Y versus the substrate concentration S (notice the logarithmic scale on the x-axis) for the relativistic (dashed line) and the classical (solid line) models. Data for Y are collected by solving numerically the self-consistent equations ((39) and (22), resp.) for J=0.9 and different values of S. Both models exhibit the sigmoidal shape typical of cooperative systems; however, the former displays a slower saturation. Analogous results are obtained for different values of J>0. (b) A Scatchard plot is built with the same collection of data by showing the ratio Y/S versus Y. Both models exhibit the concave-down shape typical of cooperative systems. However, for relatively small values of J the plot for the relativistic model is monotonically decreasing (see also Figure 3). (c) A Hill plot is built with the same collection of data by showing θ=Y/(1-Y) versus S; both observables are taken under the logarithm. When S is close to one (here S0=1) the relativistic and the classical model give overlapped curves, while when S is either very large or very small the two curves are shifted. (d) By further analyzing the plots in the previous panels we can derive estimates for the extent of cooperativity characterizing the systems. As explained in the main text, starting from data in (a) we measured the Kloshand quantifier κ=S0.9/S0.10 (⋄), by extrapolating the maximum value for data in (b) we get σ (∇), and by fitting the data in (c) at the half-saturation point we get nH (□). These estimates are obtained for both the relativistic (white symbol) and the classic (black symbols) models.
… 
of analysis on cooperativity. (a) Klotz plot showing the saturation function Y versus the substrate concentration S (notice the logarithmic scale on the x-axis) for the relativistic (dashed line) and the classical (solid line) models. Data for Y are collected by solving numerically the self-consistent equations ((39) and (22), resp.) for J=0.9 and different values of S. Both models exhibit the sigmoidal shape typical of cooperative systems; however, the former displays a slower saturation. Analogous results are obtained for different values of J>0. (b) A Scatchard plot is built with the same collection of data by showing the ratio Y/S versus Y. Both models exhibit the concave-down shape typical of cooperative systems. However, for relatively small values of J the plot for the relativistic model is monotonically decreasing (see also Figure 3). (c) A Hill plot is built with the same collection of data by showing θ=Y/(1-Y) versus S; both observables are taken under the logarithm. When S is close to one (here S0=1) the relativistic and the classical model give overlapped curves, while when S is either very large or very small the two curves are shifted. (d) By further analyzing the plots in the previous panels we can derive estimates for the extent of cooperativity characterizing the systems. As explained in the main text, starting from data in (a) we measured the Kloshand quantifier κ=S0.9/S0.10 (⋄), by extrapolating the maximum value for data in (b) we get σ (∇), and by fitting the data in (c) at the half-saturation point we get nH (□). These estimates are obtained for both the relativistic (white symbol) and the classic (black symbols) models.
… 
of analysis on cooperativity. (a) Klotz plot showing the saturation function Y versus the substrate concentration S (notice the logarithmic scale on the x-axis) for the relativistic (dashed line) and the classical (solid line) models. Data for Y are collected by solving numerically the self-consistent equations ((39) and (22), resp.) for J=0.9 and different values of S. Both models exhibit the sigmoidal shape typical of cooperative systems; however, the former displays a slower saturation. Analogous results are obtained for different values of J>0. (b) A Scatchard plot is built with the same collection of data by showing the ratio Y/S versus Y. Both models exhibit the concave-down shape typical of cooperative systems. However, for relatively small values of J the plot for the relativistic model is monotonically decreasing (see also Figure 3). (c) A Hill plot is built with the same collection of data by showing θ=Y/(1-Y) versus S; both observables are taken under the logarithm. When S is close to one (here S0=1) the relativistic and the classical model give overlapped curves, while when S is either very large or very small the two curves are shifted. (d) By further analyzing the plots in the previous panels we can derive estimates for the extent of cooperativity characterizing the systems. As explained in the main text, starting from data in (a) we measured the Kloshand quantifier κ=S0.9/S0.10 (⋄), by extrapolating the maximum value for data in (b) we get σ (∇), and by fitting the data in (c) at the half-saturation point we get nH (□). These estimates are obtained for both the relativistic (white symbol) and the classic (black symbols) models.
… 
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Research Article
Complex Reaction Kinetics in Chemistry: A Unified Picture
Suggested by Mechanics in Physics
Elena Agliari ,1,2 Adriano Barra ,2,3,4 Giulio Landolfi ,3,4
Sara Murciano,3,5 and Sarah Perrone6
1Dipartimento di Matematica, Sapienza Universit`
adiRoma,Rome,Italy
2GNFM-INdAM Sezione di Roma, Rome, Italy
3Dipartimento di Matematica e Fisica Ennio De Giorgi, Universit`
a del Salento, Lecce, Italy
4INFN Sezione di Lecce, Lecce, Italy
5D´
epartement de Physique, ´
Ecole Normale Sup´
erieure, Paris, France
6Dipartimento di Fisica, Universit`
adiTorino,Torino,Italy
Correspondence should be addressed to Elena Agliari; agliari@mat.uniroma.it
Received 10 September 2017; Accepted 1 January 2018; Published 29 January 2018
Academic Editor: Dimitri Volchenkov
Copyright ©  Elena Agliari et al. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Complex biochemical pathways can be reduced to chains of elementary reactions, which can be described in terms of chemical
kinetics. Among the elementary reactions so far extensively investigated, we recall the Michaelis-Menten and the Hill positive-
cooperative kinetics, which apply to molecular binding and are characterized by the absence and the presence, respectively, of
cooperative interactions between binding sites. However, there is evidence of reactions displaying a more complex pattern: these
follow the positive-cooperative scenario at small substrate concentration, yet negative-cooperative eects emerge as the substrate
concentration is increased. Here, we analyze the formal analogy between the mathematical backbone of (classical) reaction kinetics
in Chemistry and that of (classical) mechanics in Physics. We rst show that standard cooperative kinetics can be framed in terms
of classical mechanics, where the emerging phenomenology can be obtained by applying the principle of least action of classical
mechanics. Further, since the saturation function plays in ChemistrythesameroleplayedbyvelocityinPhysics,weshowthata
relativistic scaold naturally accounts for the kinetics of the above-mentioned complex reactions. e proposed formalism yields
to a unique, consistent picture for cooperative-like reactions and to a stronger mathematical control.
1. Introduction
1.1. e Chemical Kinetics Background. e mathematical
models that describe reaction kinetics provide chemists and
chemical engineers with tools to better understand, depict,
and possibly control a broad range of chemical processes
(see, e.g., [, ]). ese include applications to pharmacology,
environmental pollution monitoring, and food industry. In
particular, biological systems are oen characterized by com-
plex chemical pathways whose modeling is rather challenging
and can not be recast in standard schemes [–] (see also
[–] for a dierent perspective). In general, one tries to
split such sophisticated systems into a set of elementary
constituents, in mutual interaction, and for which a clear
formalization is available [–].
In this context, one of the best consolidated, elementary
scheme is given by the Michaelis-Menten law. is was
originally introduced by Leonor Michaelis and Maud Menten
to describe enzyme kinetics and can be applied to systems
made of two reactants, say (the binding molecule or, more
generally, the binding sites of a molecule) and (the free
ligand, i.e., the substrate), which can bind (and unbind) to
form the product .Ifwecallthe concentration of free
ligand, the saturation function (or fractional occupancy),
namely, the fraction of bound molecules (∈[0,1]), and,
accordingly, 1−the fraction of the unbound molecules,
under proper assumptions, one can write
(1−)=, ()
Hindawi
Complexity
Volume 2018, Article ID 7423297, 16 pages
https://doi.org/10.1155/2018/7423297
Complexity
where is the proportionality constant between response
and occupancy (otherwise stated, it is the ratio between the
dissociation and the association constants). In particular, as
standard, it is assumed that ()the reaction is in a steady
state, with the product being formed and consumed at the
same rate, ()thefreeligandconcentrationisinlargeexcess
over that of the binding molecules in such a way that it can
be considered as constant along the reaction, and ()all the
binding molecules are equivalent and independent. Also, the
derivation of the Michaelis-Menten law is based on the law of
mass action.
By reshuing the previous equation we get =/(+)
which allows stating that is the concentration of free ligand
at which 50% of the binding sites are occupied (i.e., when =
,then=1/2). us, denoting with 0the half-saturation
ligand concentration, we get
=
+0.()
isequationrepresentsarectangularhyperbolawithhor-
izontal asymptote corresponding to full saturation; that is,
=1; this is the typical outcome expected for systems
where no interaction between binding sites is at work [].
is model immediately settled down as the paradigm for
Chemical Kinetics, somehow similarly to the perfect gas
model (where atoms, or molecules, collisions apart, do not
interact) of the Kinetic eory in the early Statistical Physics
[]. Nevertheless, deviations from this behaviour were not
late to arrive: the most common phenomenon was the
occurrence of a positive cooperation among the binding
sites of a multisite molecule. Actually, many polymers and
proteins exhibit cooperativity, meaning that the ligand binds
in a nonindependent way: if, upon a ligand binding, the
probability of further binding (by other ligands) is enhanced,
thesystemissaidtoshowpositive cooperativity.
To x ideas, let us make a practical example and let
us consider the case of a well-known protein, that is, the
hemoglobin. is is responsible of oxygen transport through-
outthebodyanditultimatelyallowscellularrespiration.
Such features stem from hemoglobin’s ability to bind (and
to dislodge as needed) up to four molecules of oxygen in a
nonindependent way: if one of the four sites has captured
an oxygen molecule, then the probability that the remaining
three sites will capture further oxygen increases, and vice
versa. As a result, if the protein is in an environment rich
of oxygen (e.g., in the lungs), it readily binds up to four
molecules of oxygen, and, as much readily, it gets rid of
them when crossing an oxygen-decient environment. To
study quantitatively its behaviour one typically measures its
characteristic input-output relation. is can be achieved
by considering a set of elementary experiments where
these proteins, in the same amount for each experiment, are
prepared in a baker and allowed to bind oxygen, which is sup-
plied at dierent concentrations for dierent experiments
(e.g., 1<2<⋅⋅⋅<).WecanthenconstructaCartesian
plane, where on the abscissas we set the concentration of
the ligand (oxygen in this case, i.e., the input) while on
the -axes we put the fraction of protein bound sites
(the saturation function, i.e., the output). In this way, for
each experiment, once reached the chemical equilibrium,
we get a saturation level and we can draw a point in the
considered Cartesian plane; interpolating between all the
points a sigmoidal curve will emerge (see Figure ). Archibald
V. Hill formulated a description for the behavior of with
respect to : the so-called Hill equation empirically describes
the fraction of molecules binding sites, occupied by the
ligand, as a function of the ligand concentration [–]. is
equation generalizes the Michaelis-Menten law () and reads
as = 𝐻
0+𝐻,()
where is referred to as Hill coecient and can be
interpreted as the eective number of binding sites that are
interacting with each other. is number can be measured as
the slope of the curve log[/(1)]versus log(),calculated
at the half-saturation point. Of course, if =1there is no
cooperation at all and each binding site acts independently
of the others (and, consistently, Michaelis-Menten kinetics is
restored), and vice versa; if >1,thereactionissaidto
be cooperative (just like in hemoglobin), and if 1the
cooperation among binding sites is so strong that the sigmoid
becomes close to a step function and the kinetics is named
ultrasensitive.
e Michaelis-Menten law, together with the extension
by Hill, provided a good description for a bulk of chemical
reactions; however, things were not perfect yet. For instance,
some yeast’s proteins (e.g., the glyceraldehyde -phosphate
dehydrogenase []) produced novel (mild) deviations from
the Hill curve: for these enzymes, the cooperativity of
their binding sites decreases while increasing the ligand
concentration. e following work by Daniel E. Koshland
allowed understanding this kind of phenomenology by
further enlarging the theoretical framework through the
introduction of the concept of negative cooperativity. In fact,
in the previous example, beyond the positive cooperation
between the binding sites there are also negative-cooperative
eects underlying. eir eective action is to diminish the
overall binding capabilities of the enzyme and thus to reduce
the magnitude of its Hill coecient.
1.2. e Mechanics Background. e progressive enlargement
of a theoretical scaold to t the always increasing amount of
evidences is a common feature in the historical development
of scientic disciplines [, ]. is is the case also for
Mechanicsand,aswewillsee,theanalogywithChemical
Kinetics goes far beyond this feature.
Beyond Kinematics, which describes the motion of sys-
tems without considering their mass or the forces that caused
the motion, in the seventeenth century Newton gave a sharp
description of Mechanics, in the form of laws describing
howmassesdynamicallyrespondwhenstimulatedbyan
external force (or moment). Here, the input is the force
while the output is the motion of the body. e Newtonian
dynamics has been ruling for centuries and, in fact, it was
so well-consolidated that scientists, among which Giuseppe
L. Lagrange, William R. Hamilton, and Carl G. J. Jacobi,
Complexity
Y
1
0.8
0.6
0.4
0.2
0
−12 −10 −8 −6 −4 −2
FIA 
Dataset
VC II
VC I-II
CaMKII
VC II
VC I-II
CaMKII
0 ± 0.6
0.3 ± 0.1
1.0 ± 0.7
1.4 ± 0.2
3.0 ± 0.3
0.97
1.64
3 ± 0.3
JnH±Δn
HnFCN
H
0.67 ± 0.03
(a)
Y
1
0.8
0.6
0.4
0.2
0
FIA 
−8 −7 −6 −5
1.1 ± 0.3 4.4 ± 0.6
Dataset
CaMKII
JnFCN
H
CaMKII
(b)
F : ese plots show comparison between data from experiments (symbols) and best-ts through () (lines). Data refer to
noncooperative and positive-cooperative systems [, ] (a) and an ultrasensitive system [] (b). For the latter we report two ts: dashed
line is the result obtained by constraining the system to be cooperative but not ultrasensitive (i.e., ≤1), while solid line is the best t (without
constraints) which yields to 1.1, hence a “rst-order phase transition” in the language of statistical mechanics. e relative goodness of
the ts is 2
coop 0.85and 2
ultra 0.94, conrming an ultrasensitive behavior. e tables in the bottom present the value of derived from
thebesttandtheresultingaccording to (); the estimate of the Hill coecient taken from the literature is also shown for comparison.
is gure was presented in [].
later reformulated the entire theory in a powerful and elegant
variational avor. e theory was overall brilliant to explain
the perceivable reality, but with exceptions emerging in the
limit of too little or too fast.
We will focus on the latter. In the Newtonian world, if an
applied force is kept constant over a mass, this will constantly
accelerate, eventually reaching diverging velocities. is was
perfectly consistent with the general credo that the speed
of light was innite. However, this postulate broke down in
 when the famous experiment by Albert A. Michelson
and Edward Morley proved that such a velocity is actually
nite. e next years were dense of novel approaches and
ideas by many scientists, as Hendrik Lorentz and Hermann
Minkowski, and culminated with the special relativity by
Albert Einstein in . According to this theory, no mass
can exist whose velocity may diverge, the limiting speed being
the speed of light. e classical Hamilton-Jacobi equations
andGalileantransformationsletheplacetotheKlein-
Gordon formulation and Lorentz covariances and contravari-
ances (the natural metric being Minkowskian) []. Clearly,
classical mechanics were still a good reference framework
forthevastmajorityofthedatacollected(muchlikethe
positive cooperativity accounted for the bulk of the empirical
data in the chemical counterpart); however, there were rare
phenomena (e.g., a muon decay in atmosphere []) that
required a broader scaold which, in the opportune limits,
could recover the classical one.
Although this historical connection between Chemical
Kinetics and Classical Mechanics may look weird at a rst
glance, as we will prove, there is a formal analogy between
their mathematical representations. In the next section we
will summarize the main results concerning the analogy at
the classical level. More sharply, the saturation plot of classical
(positive-cooperative) chemical kinetics (namely, the input-
output relation between the saturation function and the
concentration of the substrate) can be derived by a minimum
action principle that is the same that holds in classical
mechanics, when describing a mass motion in the Hamilton-
Jacobi framework. In this parallelism, the saturation function
in Chemistry plays as the velocity in Physics: thus, exactly
as what happens in special relativity, the velocity of the
mass is bounded (by denition, the saturation function can
not exceed one). Indeed, we can follow this mathematical
equivalence and verify that there is actually a natural broader
Complexity
formulation for chemical kinetics that is exactly through
the Klein-Gordon setting (rather than its classical Hamilton-
Jacobi counterpart) and the theory as a whole is Lorentz-
invariant. Remarkably, when read with chemical glasses, this
extended relativistic setting allows for the anticooperative
correctionsthatKoshlandrevealedinthestudyoftheyeast
enzymes, resulting in a complex mixture of positive and
negative cooperation among binding sites.
2. The Standard Mathematical Scaffold for
Classical Cooperativity
As anticipated in Section ., cooperativity is a widespread
phenomenon in Chemistry and its underlying mechanisms
can be multiple: for example, if the adjacent binding sites
of a protein can accommodate charged ions, the attrac-
tion/repulsion between the ions themselves may result in
a positive/negative kinetics; in most common cases, the
bonds with the substrate modify the protein conformational
structure, by inuencing possible further links in an allosteric
way [, ]. Whatever the origin, cooperativity in Chemistry
is a typical emergent property that directly relates the micro-
scopic description of a system at the single binding-site level,
with the macroscopic properties shown by its constituent
molecules, cells, and organisms; thus the use of Statistical
Physics for its investigation appears quite natural [, ].
Usually, in Statistical Physics one is provided with (inverse)
temperature and with Hamiltonian (i.e., a cost-function)
(,,)describing the model at the microscopic level,
namely, in terms of elementary variables ,,couplings
among elementary variables  and external elds acting
over these. e goal is to obtain the free energy (,,)of
themodel,fromwhichtheaveragevalueofthemacroscopic
observables can be directly derived [].
2.1. Formulation of the Problem: e ermodynamical Free
Energy. In the following we summarize the minimal assump-
tions needed when modelling chemical kinetics from the
Statistical Physics perspective; for a more extensive treatment
of this kind of modelling we refer to [, , , , ], while
for a rigorous explanation of the underlying equivalence
between Statistical Mechanics and Analytical Mechanics we
refer to the seminal works by Guerra [], dealing with the
Sherrington-Kirkpatrick model (and then deepened in, e.g.,
[–]), and by Brankov and Zagrebnov in [], dealing
with the Husimi-Temperley model (and then deepened in,
e.g., [–]).
(i) Each binding site may or may not be occupied by a
ligand: this allows us to code its state (empty versus
full) by a Boolean variable. For the generic th site,
wewilluseanIsingspin1,where=−1
represents an empty th site, and vice versa; =+1
means that the th site is occupied. Clearly, if there are
overall binding sites, (1,...,).
(ii) It is rather inconvenient (and ultimately unneces-
sary)todealwiththewholeset,...,if we are
interested in the properties of large numbers of these
variables (i.e., in the so-called thermodynamic limit
corresponding to →∞). If we want to distinguish
between a fully empty state =1(1,...,)
(ordered case), a fully occupied state =+1∈
(1,...,)(ordered case), and a completely random
case where 1with equal probability (disordered
case), it is convenient to introduce the order parameter
forthesevariablesasthemagnetization (this term
stems from the original application of the Statistical
Mechanics model in the context of magnetism) that
readsasthearithmeticaverageofthespinstate,
namely,
=1
=1[−1,+1].()
ere is a univocal relation between the magne-
tization in Physics and the saturation function in
Chemistry, where, we recall, we denote with ∈[0,1]
the fractional occupation of the binding sites. In fact,
one has [, ]
=12(1+).()
Equation (5) constitutes the rst bridge between the
Chemistry we aim to describe (via the saturation
function ) and the Physics that we want to use (via
the magnetization ).
(iii) All the binding sites interact with the ligand by
the same strength. is is a standard assumption
in Chemical Kinetics [, , ] and it means that
the diusion of the ligands is fast enough to ensure
a homogeneous solution. e concentration of free
ligands is mapped into a one-body contribution 1
in the cost-function. is term encodes for the action
of an external magnetic eld in such a way that, if
the eld acting on th is positive, the spin will tend to
align upwards (namely, this direction is energetically
favored), and vice versa. is homogenous mixing
assumption translates into a homogeneous external
eld , and the related contribution reads as
1(,)=−
=1.()
Notice that playsasachemicalpotentialand,consis-
tently, it can be related to the substrate concentration
as =12log
0, ()
0being the value of the ligand concentration at half
saturation.
Equation (7) constitutes the second bridge between
the Chemistry we aim to describe (via the ligand
concentration ) and the Physics that we want to use
(via the magnetic eld ).
(iv) e binding sites can cooperate in a positive manner:
this can be modelled by introducing a coupling
Complexity
between the variables. e simplest mathematical
form is given by a two-body contribution 2in the
cost-function. is term encodes for the reciprocal
interactions among binding sites and it reads as
2(,)=−
<,()
where ≥0is the interaction strength and the
sum runs over all possible pairs; the normalization
factor 1/ensures the linear extensivity of the cost-
function with respect to the system size. A positive
value for implies an imitative interaction among
binding sites: congurations where spins tend to be
aligned each others (namely, where sites tend to be
either all occupied or all unoccupied) are energetically
more favoured and will therefore be more likely.
(v) Combining together the previous contributions we
get the total Hamiltonian:
(,,)=1(,)+2(,)
=−
<−
=1
=−
22+.
()
It is possible to introduce the free energy associated
with such a Hamiltonian as
,,=1
log
2𝑁
1,...,𝑁
exp −(,,)()
=1
log
2𝑁
1,...,𝑁
exp 
2
, +
, ()
where is the inverse temperature in proper units
and the sum runs over all possible congurations. e
free energy is a key observable because it corresponds
to the dierence between the internal energy
and the entropy (at given temperature), that is,
(,,)=(,,)−(,,).Ifwecouldobtain
an explicit expression for (,,)in terms of the
order parameter , we could obtain an expression
for the magnetization expected at equilibrium by
imposing (,,) = 0; in fact, this implies
that we are simultaneously asking for the minimum
energy and the maximum entropy.
Notice that, having stated the two bridges given by
() and (), other mappings between the two elds
(e.g., the relation between the coupling strength
and the Hill coecient ; see () later on) emerge
spontaneously as properties of the thermodynamic
solutions of the problems.
2.2. Resolution of the Problem: e Mechanical Action. We
want to nd an explicit expression (in terms of )forthe
free energy dened in (). To this task let us rename −=
and  = and let us think of these ctitious variables as
atimeandaspace,respectively.us,wecanwritethefree
energy as
(,)=1
log
2𝑁
1,...,𝑁
exp −
2
, +
, ()
wherewealsowrote< (1/2), , which implies
vanishing corrections in the thermodynamic limit. If we work
out the spatial and temporal derivativesofthefreeenergy()
we obtain (,)
 =−122, ,
(,)
 =, ,()
where the average ⋅, for a generic observable depending
on the spin conguration is dened as
, =exp −2+⋅
exp −2+⋅ ,()
and, posing =−and =,theBoltzmannaveragefor
the original system () is recovered and this shall be simply
denoted as ⋅
If we now introduce a potential (,), dened as half the
variance of the magnetization, that is,
(,)=122−2, ()
we see that, by construction, the free energy of this model
obeys the following Hamilton-Jacobi equation:
(,)
 +12(,)
 2+(,)=0, ()
and therefore (,)is also an action of Classical Mechanics.
We can simplify the previous equation by noticing that, for
large enough volumes, the magnetization is a self-averaging
quantity [, ]; thus in the innite volume limit the
potential must vanish; that is, lim→∞(,)=0.Here,we
are restricting to large volumes and we are therefore le with
a Hamilton-Jacobi equation describing a free propagation;
since the potential is zero, the Lagrangian Lcoupled to the
motion is just the kinetic term:
L=122,()
that is, the analogous of the classical formula L=(1/2)V2,
where the mass is set unitary (i.e., =1), and the
role of the velocity Vis played by the average magnetization
. Solving the Hamilton-Jacobi equation is then straight-
forward: the solution is formally written as
(,)==0,=0+
0
L,.()
Complexity
e evaluation of the Cauchy condition (=0,=
0)
is trivial because, at =0,thecouplingbetweenvariables
disappears (see ()), while the integral of the Lagrangian
over time reduces to the Lagrangian times time (as the
potential is zero). Pasting these two contributions together we
obtain (,)=ln 2+ln cosh 0+122⋅. ()
Finally, noticing that the equation of motion is a Galilean
trajectory as ()=0+(hence 0=−)and
recasting the solution back in the original variables, that is,
=−and =,wegetthefreeenergyassociatedwith
this general positive-cooperative reaction:
,,=ln 2+ln cosh +
122. ()
By extremizing (,,)with respect to we get
,,
=0
=tanh (+). ()
is result recovers the well-known self-consistency equation
fortheorderparameteroftheCurie-WeissmodelinStatisti-
cal Mechanics [, ].
2.3. Chemical Properties of the Physical Solution. e self-
consistent equation in () is an input-output relation for
a general system of binary elements, possibly positively
interacting, under the inuence of an external eld: the input
in the system is the external eld and the output is the
magnetization .Wecanrewrite()inachemicaljargon
by using the bridges coded in () and () and xing, for the
sake of simplicity, 0=1;thatis,
(,)=121+tanh (21)+12ln 
=2(2−1)
1+2(2−1) .()
Before proceeding, we check that if cooperation disap-
pears (i.e., binding sites are reciprocally independent), the
Michaelis-Menten scenario is recovered. Posing =0in the
equation above we get
(=0,)=
1+,()
that is (apart a constant factor that can be reintroduced by
taking 0=,ratherthan1), the Michaelis-Menten equation
(see ()).
One step forward, we now take into account the coupling
andrelateittotheHillcoecient. e latter is dened
in Chemistry as the slope of ()at half saturation (i.e., when
=1/2), and we can obtain its expression following this
prescription by using (), namely,
=1
(1−)
=1/2 =1
1−.()
We note that as →0we get, as expected, →1:
if there is no cooperation between binding sites, the Hill
coecient must be unitary; further, the stronger the coupling
, the (hyperbolically) larger the value of the Hill coecient.
In particular, for →1the kinetics get ultrasensitive
and discontinuities emerge. We remark that, with simpler
statistical mechanics model as linear chains of spins, phase
transitions are not allowed; hence ultrasensitive behavior can
not be taken into account: the present framework is the
simplest nontrivial scheme where all these phenomena can
be recovered at once (see Figure  and [] for more details
on ultrasensitive kinetics).
Also, it is worth highlighting the full consistency between
our treatment of ultrasensitive kinetics and more standard
ones as, for instance, reported in [] (see eq. 5.17therein),
wheretheexpressionfortheHillcoecientcanbetranslated
into our formulation as
=2−2
1−2.()
We see t h at for ±1the Hill coecient diverges, which
is the signature of an ultrasensitive behavior: this is perfectly
coherent with our approach where, in that limit, the input-
output relation (see the hyperbolic tangent ()) becomes a
step function.
However, as mentioned in the Introduction, this theory
has its aws, in Chemistry as well as in Mechanics. Regarding
the former, the complex picture of yeast’s enzymes evidenced
by Koshland [, ], where positive and negative coopera-
tivity appear simultaneously (and with the anticooperativity
eect getting more and more pronounced as the substrate
concentration is raised), still escapes from this mathematical
architecture. Further, from the mechanical point of view, two
weird things happen: the velocity is bounded by =
1, while in Classical Mechanics the velocity may diverge;
further, if we look at the Boltzmann factor in the free energy
(see ()), this reads as exp[(−2/2+)]and, recalling
that the kinetic energy in this mechanical analogy reads as
2/2(the mass is unitary, thus velocity and momentum
coincide), we are allowed to interpret (,,)as a real
action. From this perspective, the exponent can be thought
of as the coupling between the stress-energy tensor and the
metric tensor: a glance at the form of the Boltzmann factor
reveals that the natural underlying metric is (−1,+1)rather
than (+1,+1)as in classical Euclidean frames, or in other
words, it is of the Minkowskian type. All these details point
toward the generalization of the equivalence including special
relativity.
Plan of the next section is to follow the mechanical path
and extend the classical kinetic energy including relativistic
corrections and then to investigate its implications. We will
seethatinthebroader,relativisticframeworkforchemical
Complexity
kinetics the deviations that Koshland explained adding an
anticooperative interactions, beyond the cooperative ones, at
high ligand’s doses are the chemical analogies of the deviation
from classical mechanics at high velocities observed in special
relativity.
3. The Generalized Mathematical Scaffold for
Mixed Cooperativity
3.1. Relativistic Setting. e relativistic extension of the
Hamiltonian () is dened by Hamiltonian of the form
(,,)
=−1+2−, ()
where =(1/)
as usual. Next, we have to insert ()
into the free energy ():
(,)=1
log
2𝑁
exp −1+2+⋅, ()
where we already replaced =−and =in order to
work out their streaming that read as
(,)
 =−1+2, ,
(,)
 =, ,
2
(,)−2
(,)
=1−1+22
, +2
, ,
()
where the Boltzmann averages ⋅⋅⋅, are dened as (using
the magnetization as a trial function)
, =exp −1+2+⋅
exp −1+2+⋅ .()
As before, the averages ⋅⋅⋅, will be denoted by ⋅⋅⋅
whenever evaluated in the sense of thermodynamics (i.e.,
for =−and =). By a direct calculation,
it is straightforward to see that expression () obeys the
relativistic Hamilton-Jacobi equation:
2−2+(,)=1,
(,)=1
(,),()
where the symbol represents the D’Alembert operator
and (,)is the potential whose expression is chosen in
ordertomaketheequationvalidbyconstructionand,this
time, it is automatically Lorentz invariant. If the functional
(,)is suciently smooth (i.e., its derivatives are regular
functions of and ), in the thermodynamic limit, we have
lim→∞(,)=0; hence in this high-volume limit we are
le with 2=1, ()
which is the Klein-Gordon equation for a free relativistic
particle with unitary mass in natural units (0=1).
In relativistic mechanics, the stress energy tensor of this
particle is dened as =,V,()
where Vis the classical velocity of the particle, =1/1−V2,
and =0=is the relativistic energy. In addition, the
contravariant momentum is expressed through the action by
the following equation:
=−
=1+2, ,,. ()
Comparing () and (), it is immediate to identify the
magnetization as the relativistic dynamical variable:
, =V
1−V2,()
while the Lorentz factor is
=1+2
,.()
In the thermodynamic limit, the particle is free and its
trajectories are the straight lines =
0+V.Sincethe
relativistic Lagrangian L=−
−1 is constant along these
classical trajectories, the free energy can be computed as
(,)=0,0+
0L=0,0−
=log 2+log cosh 0−
=log 2+log cosh (−V)
1+2
,
=log 2+log cosh ,
1+2
,
1+2
, .
()
Setting =−and =, we nally get an explicit
expression for the free energy:
=log 2+log cosh 
1+2
+
1+2.()
Complexity
Requiring that the free energy is extremal with respect to
the magnetization (from a thermodynamical perspective
this condition can be seen as the simultaneous eect of
the minimum energy and the maximum entropy principles
and from a mechanical perspective as the minimum action
principle), the associated self-consistency equation becomes
=tanh 
1+2+. ()
3.2. e Classical Limit from a Chemical Perspective. Reading
the self-consistency () in chemical terms, that is, using the
bridges () and (), we obtain
(,)=12
1
+tanh (21)
1+(21)2+
2log
0
.()
We can now check whether, under suitable conditions, this
broader theory recovers the classical limit. First, we notice
that under the assumption of no interactions among binding
sites (i.e., =0) and replacing  = (1/2)log(/0),the
Michaelis-Menten behaviour is recovered. is can be shown
by rewriting () as
(,)=2(2−1)/1+(2−1)2
1+2(2−1)/1+(2−1)2,()
where we also shied /0→for simplicity. For =0the
previous equation reduces to ()=/(1+).Further,taking
theclassicallimit,atthelowestorder,wehavethefollowing
expansions: 1
1+2=1−2
2+O3,
1+2=+O3, ()
such that () reduces to (), in the physical context, and to
(), in the chemical context. Clearly, also the slope at =
1/2is preserved; hence, in the classical limit, we recover the
expected expression for Hill coecient (see ()), namely,
=1
(1−)
=1/2 =1
1−.()
3.3. Beyond the Classical Limit. To understand why we expect
variations with respect to the Hill paradigm at relatively large
values of the substrate concentration, we must check carefully
the relativistic self-consistency (). Let us assume we are
working at not too high velocities (i.e.,  < 1)andwe
can expand the argument inside the hyperbolic tangent; in
particular, approximating 1/(1+2)∼1−2/2,weget
=tanh 
23+. ()
e relativistic eects in chemical kinetics become transpar-
entinthisway:ifwelookattheeldfeltbythebindingsites
(i.e., the argument inside the hyperbolic tangent), we see that,
beyond the standard Hill term (that positively pairs
binding sites together), another term appears that, this time,
negatively pairs binding sites together. Retaining this level of
approximation, we could write an eective Hamiltonian to
generate () that reads as
(,,)=−
<+
23
<<<;()
hence, beyond the two-body positive coupling coded by
the rst term, another four-body negative coupling appears.
e latter is responsible for the deviation from the classical
paradigm and these deviations are in full agreement with
the Koshland generalization toward the concept of mixed
positive and negative cooperativity [].
In particular, we can see at work the entire reasoning of
Koshlandwhopointedouthow,atlargeenoughsubstrate
concentration, the positivity of the reaction diminishes. In
fact, for  ∼ 0no relativistic eect can be noted. By
increasing (the input in the system), we get a growth in
(the output in the system): the latter raises in response
of and it is enhanced because of the two-body term in
the eective Hamiltonian (), the four-body term still being
negligible. As keeps on growing, increasesaswell,up
to a point where it reaches high enough values such that,
from now on, also the four-body term inside the eective
Hamiltonian () becomes relevant. At this point, a novel,
anticooperative eect is naturally induced in the reaction and
it yields to a reduction of the Hill coecient. In the next
analysis these qualitative remarks shall be addressed in more
details.
We focus on the denition of the Hill coecient based on
the Hill equation:
= 𝐻
+𝐻.()
is equation accounts for the possibility that not all receptor
sites are independent: here is the average number of
interacting sites and the slope of the Hill plot. e latter is
based on a linear transformation made by rearranging ()
and taking the logarithm:
log
1−=log ()log ().()
us, one plots log /(1)versus log , ts with a linear
function and the resulting slope, calculated at the half-
saturation point, and provides the Hill coecient. As already
underlined, the Michaelis-Menten theory corresponds to
Complexity
=1and any deviations from a slope of 1tell us about
deviation from that model.
For the (classical and relativistic) models analyzed here
(coded in the Hamiltonians () and ()) we can estimate
the slope directly from the self-consistency equations ()
and (). Let us start with the classical model. We preliminary
notice that
log ()log
1−=1/2
=1
(1−)
log ()=1/2 =4 
log ()=1/2 .()
erefore, we just need to evaluate /log()in =1/2,
which reads as

log ()=1/2 =12sech2(21)+12log ()
⋅2
log ()+12=1/2 .()
Posing =/log()|=1/2 and noticing that =1when
=1/2,wehave =122+12
=141
1−.()
By plugging this result in (), we nally have
class
=1
1−.()
One can see that when =0the Hill coecient is unitary
as expected for noncooperative systems, when >0the
coecient is larger than , indicating that receptors are
interacting, and when <0the coecient is smaller than
, as expected for negative cooperativity.
Let us now move to the relativistic model. Again, we just
need to evaluate /log()in =1/2, which, recalling (),
reads as 
log ()=1/2
=12sech2
(21)
1+(21)2+12log ()
()
× 2
[2+4(−1)]3/2 
log ()+12=1/2 .()
Exploiting the fact that =1when =1/2,theprevious
expression simplies as

log ()=1/2 141
1−/27.()
us, we can write
rel
=
log ()log
1−=1/2
=4 
log ()=1/2 =1
1−/27.()
Note that class
/rel
<1, conrming that the relativistic
correction weakens the emerging cooperativity.
3.4. Further Robustness Checks. As stressed above, for a xed
interaction coupling , the relativistic model is expected to
exhibit a lower cooperativity with respect to the classical
model. In order to quantify this point we considered dierent
quantiers for cooperativity and we compared the outcomes
for the relativistic and the classical models set at the same
value of . Let us start with the Koshland measure of
cooperativity which is dened as the ratio (notice that the
Koshland index is actually strongly related to the Hill
coecient (see, e.g., [])) =0.9
0.1 ,()
where 0.9 denotes the substrate concentration correspond-
ing to a 90% saturation, while 0.1 denotes the substrate
concentration corresponding to a 10% saturation; that is,
(0.9)=0.9and (0.1)=0.1. In the noncooperative case
one has 0.9/0.1 =81and, accordingly, if the ratio is smaller
than 81(meaning that the saturation curve is relatively steep)
one has positive cooperativity, while if the ratio is larger
than 81one has negative cooperativity. e advantage in
using the index is that it can be easily measured, yet it
ignores all information that can be derived from the shape of
(). In particular, this quantier can be estimated starting
from a Klotz plot (see, e.g., Figure (a)) where the saturation
function is shown versus the logarithm of the (free) ligand
concentration; in the presence of positive cooperativity this
plot yields to a characteristic sigmoidal curve. For the models
analyzed here we can estimate 0.9/0.1 directly from the self-
consistency equations ()–(), ()–(). Starting from the
classical model and posing  = 0.9and =0.1we get,
respectively,
9
10=121+tanh 2× 9
10−1+12log 0.9,
1
10=121+tanh 2× 1
10−1+12log 0.1, ()
and, with some algebra (recalling 2atanh()=log[(1+)/(1−
)]),
log 0.9=2atanh 45−85=log (9)85
0.9 =9−8/5,
log 0.1=2atanh −45+85=−log (9)+85
0.1 =198/5;
()
 Complexity
0
0.5
1
S
10−2 100102
Y
Relativistic
Classic
(a)
010.80.60.40.2
0
0.2
0.4
0.6
Y/S
Y
Relativistic
Classic
(b)
0
5
FIA(S)
−5
−5
05
FIA()
Relativistic
Classic
(c)
010.80.60.40.2
J
2
4
6
8
Cooperativity
nH
(d)
F : Summaryofanalysisoncooperativity. (a) Klotz plot showing the saturation function versus the substrate concentration (notice
the logarithmic scale on the -axis) for the relativistic (dashed line) and the classical (solid line) models. Data for are collected by solving
numerically the self-consistent equations (() and (), resp.) for =0.9and dierent values of . Both models exhibit the sigmoidal shape
typical of cooperative systems; however, the former displays a slower saturation. Analogous results are obtained for dierent values of >0.
(b) A Scatchard plot is built with the same collection of data by showing the ratio /versus . Both models exhibit the concave-down shape
typical of cooperative systems. However, for relatively small values of the plot for the relativistic model is monotonically decreasing (see
also Figure ). (c) A Hill plot is built with the same collection of data by showing =/(1−)versus ; both observables are taken under
the logarithm. When is close to one (here 0=1) the relativistic and the classical model give overlapped curves, while when is either very
largeorverysmallthetwocurvesareshied.(d)Byfurtheranalyzingtheplotsinthepreviouspanelswecanderiveestimatesfortheextent
of cooperativity characterizing the systems. As explained in the main text, starting from data in (a) we measured the Kloshand quantier
=0.9/0.10 (), by extrapolating the maximum value for data in (b) we get (), and by tting the data in (c) at the half-saturation point
we get (). ese estimates are obtained for both the relativistic (white symbol) and the classic (black symbols) models.
that is, class =0.9
0.1 =81−16/5.()
Of course, when =0we recover the value 81,when>0
we get class <81,andwhen<0we get class >81.
Repeating analogous calculations for the relativistic
model we get
9
10=12
1+tanh
(9/10)−1
1+((9/10)−1)2
+12log 0.9
,
1
10=12
1+tanh
(1/10)−1
1+((1/10)−1)2
+12log 0.1
,
()
and, with some algebra,
log 0.9=2atanh 45− 8
41=log (9)8
41
0.9 =9−8/41,
Complexity 
log 0.1=2atanh 45+8
41=−log (9)+8
41
0.1 =198/41;()
that is, rel =0.9
0.1 =81−16/41.()
Again, one can check that when =0we recover the value 81,
when >0we get rel <81,andwhen<0we get rel >81.
Also, rel/class =−16/41+16/5 >1.ismeansthat,even
with this quantier, when xing the same coupling constant
, the emerging cooperativity is weaker for the relativistic
model, as expected.
Next, let us consider the cooperativity quantier derived
from the Scatchard plot. We recall that this plot is built
by showing the behavior of /with respect to .Infact,
according to the simplest scenario (this corresponds to the
Michaelis-Menten theory and to Clarks theory and it requires
a set of simplifying assumptions, among which the interac-
tion is reversible; all the binding molecules are equivalent and
independent; the biological response is proportional to the
number of occupied binding sites; the substrate only exists in
either a free (i.e., unbound) form or bound to the receptor),
at equilibrium, one can write
(1−)
=, ()
where is the proportionality constant between response and
occupancy (i.e., it is the ratio between the dissociation and the
association constants), and rearranging () we have
=−
+1.()
e previous expression ts the equation of a line for /
versus ,whoseslopeis−1/.eadvantagesinusing
the Scatchard plot is that it is a very powerful tool for
identifying deviations from the simple model, which, without
deviations, is represented by a straight line. In particular,
a concave-up curve may indicate the presence of negative
cooperativity between binding sites, while a concave-down
curve is indicative of positive cooperativity. Also, in the latter
case, the maxima occurs at the fractional occupancy
which fullls =−1
,()
where provides another quantier for cooperativity.
Starting from the classical model, we can build the
function /,byrstgettingas a function of ,andcan
be obtained by inverting formula (); namely,
()=exp [2atanh (21)−2(21)].()
By deriving /with respect to we get

()=−2(2−1) [1−4(1−)],()
which is identically equal to −1when =0, monotonically
decreasing with when >0and monotonically increasing
with when <0. e (possible) root therefore provides the
extremal point; that is,
=41
4 ,()
and, comparing with (), we get
class =4. ()
We now repeat analogous calculations for the relativistic
model. First, we get as a function of ,byinvertingformula
(), namely,
=exp
2atanh (21)−221
1+(21)2
.()
By deriving /with respect to we get

=−(2−1)/(−1)+1/2 14(1−)
[2−4(1−)]3/2 , ()
which is again identically equal to −1when =0,butitisno
longer monotonic when  =0.Moreprecisely,bystudying
(/)/we can derive that when is relatively small, /
does not exhibit any extremal points, but there is a ex at
intermediate values of ; for intermediate values of there is a
minimumatsmallvaluesofandamaximumatlargervalues
of ;forlargevaluesofthere is a maximum. e extremal
points can be found as roots of a 6th degree function of .
Wecanobtainanestimateofthevaluecorresponding to
the maximum by recalling ≤1and neglecting high-order
terms. In this way we get
−3+22−9+262
29+2,()
and, comparing with (), we get
rel =29+2
−15+−9+262.()
e three plots considered here (i.e., Klotz, Scatchard, and
Hill) and the related estimates for the extent of cooperativity
are presented in Figure . In particular, in (d) we compare the
cooperativity quantiers for several values of :asanticipated,
in general, for a given value of , the relativistic model gives
rise to a weaker cooperativity.
We proceed our analysis by deepening the role of the
coupling constant in the binding curves related to the two
models. In Figure  we present Klotz’s plot (a), the Scatchard
plot (b), and the Hill plot (c) for the relativistic and the classic
models, comparing the outcomes for dierent values of .As
expected, the point corresponding to =1and =1/2is
 Complexity
Y
Relativistic
Classic
S
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
10−2 100102
(a)
Y/S
Y
1
0.8
0.6
0.4
0.2
00 0.5 1
Relativistic
Classic
(b)
FIA()
FIA(S)
6
4
2
0
−2
−4
−6
−4 −2 024
Relativistic
Classic
(c)
F : e role of the interaction coupling . We resume the plots presented in (a)–(c) of Figure  and we show how they are aected by
the interaction coupling . In particular, we compare the outputs for =0.1(black), =0.5(blue), and =0.9(bright blue). Again, the
relativistic model (dashed line) and the classical model (solid line) are compared. Notice that the gap between relativistic and classical model
is larger when is relatively large.
a xed point in each plot and, in general, the gap between
the two models is enhanced when is larger (i.e., when is
closer to 1). Also, when is not too small, the Scatchard plot
for the relativistic model displays a ex at small values of
suggesting that when the saturation is small, the cooperativity
is not truly positive.
Inthenalpartofthissectionwewanttogetdeeperinthe
comparison between the classical and the relativistic models.
To this aim, we solved numerically (), for dierent values
of and of , getting a set of data (,).Wecanthinkofthis
set of data as the result of a set of measurements where we
collect the saturation value at a given substrate concentration.
Now, assuming that in this experiment we have no hints
about the underlying cooperative mechanisms, we may apply
the formulas for the plain positive cooperativity and infer
the value of . More practically, we calculate numerically
from the relativistic model for dierent values of and of
the coupling strength, referred to as rel for clarity. Next, we
manipulate the set of data (,rel)by inverting the formula
in (): as the value of isassumedtobeknown,wecan
derive the coupling strength, referred to as class, expected
within a classical framework. In this way, we can compare
the original coupling constant rel with the inferred one
class. We can translate these procedures in formulas as fol-
lows: class =atanh (21)(1/2)log ()
21
21=tanh
rel (21)
1+(21)2+12log ()
class =rel
1+(21)2≤rel ,
()
with equality holding only when =1/2.
In Figure (a) we plot class versus rel, for dierent values
of . Notice that the two parameters are related by a linear
law, whose slope is smaller than 1and decreases with .
is conrms that the relativistic model yields to a weak
cooperativity. e negative contributions in the relativistic
model get more eective when rel and are large, as further
highlighted in Figure (b).
4. Conclusions
e rewards in the overall bridge linking Chemical Kinetics
and Analytical Mechanics are several, both theoretical and
practical, as we briey comment.
e former lie in a deeper understanding of the mathe-
matical scaold for modelling real phenomena: it is far from
trivial that the description of chemical/thermodynamical
Complexity 
0.2 0.60.50.40.3
0.1
0.2
0.3
0.4
0.5
0.6
h
J=F;MM
JL?F
(a)
0.2 10.80.60.4
0.5
1
1.5
2
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
h
JL?F
J=F;MM/JL?F
(b)
F : Comparison between relativistic and classical model. We performed numerical experiments where we obtained (,rel)for the
relativistic model by solving the self-consistent equation (). From this data we inferred the expected classical coupling class by inverting
the self-consistent equation (). We repeated the same operations for several values of and rel.In (a) we show the inferred class versus the
xed rel: dierent colors represent dierent values of and the identity function is also shown for reference (dashed, black curve). Notice
that, in general class <rel and the inequality is enhanced as grows.In(b)weshowacontourplotfortheratioclass/rel versus =log()/2
and rel. Again, one can notice that, in general, class/rel <1and this inequality is enhanced for relatively large values of .
equilibrium is formally the same as the mechanical one.
In particular, the self-consistency relation () that emerges
from the thermodynamic principles (in fact, it stems from
the requirement of simultaneous entropy maximization and
energyminimization)alsoturnsouttobe,inthemechanical
analogy, a direct consequence of the least action principles
(,) = 0. is means that the stationary point corre-
sponds to a light perturbation of the evolution of the system in
the interval [0,]. Explicitly, we shi innitesimally ,
, +,;then
0=(,)=(,)
, ,
=tanh ,
1+2
, ,
1+2
,3/2
+, ,
1+2
,3/2 =0,
()
from which () is recovered (as usual by setting =−and
=), since this holds for all variations ,.
Even more exciting, still by the theoretical side, is the
realization of the complexity of systems presenting mixed
reaction (i.e., where both positive and negative cooperativity
are simultaneously at work) and the possible applications in
information processing, as we are going to discuss.
First, let us clarify that in the Literature we speak
of complex network or complex system with (mainly) two,
rather distinct, meanings: in full generality, let us consider a
Hamiltonian as (,)=
<()
and let us write the two-body coupling matrix as  =
,whereis the adjacency matrix, accounting for the
bare topology of the system (its entry  is 1if there is a
link connecting the related nodes (,), which are therefore
allowed to interact each other, and it is zero otherwise) and
is the weight matrix, accounting for the sign and the
magnitude of the links (i.e., the type of interactions among
binding sites).
Dealing with , networks where the topology is very
heterogeneous (e.g., the distribution of the number of links
stemming from a node has a power-law scaling) are called
complex networks,asitiscasefortheBarabasi-Albertmodel
[].
Dealing with , networks where the entries of the weight
matrix are both positive and negative are termed complex
systems, as the Sherrington-Kirkpatrick model [] for the so-
called spin glasses.
Crucially, spin glasses spontaneously show very general
information-processing skills and computational capabilities:
for instance, Hopeld neural networks [] and restricted
Boltzmann machines [], key tools in Articial Intelligence
(resp., in neural networks and machine learning), are two
types of spin glasses and it is with this last denition of
complexity that we now can read the information processing
capabilities of the elementary reactions we studied. For a
 Complexity
given macromolecule under consideration, we could paste
each binding site on a node and draw the links among nodes
that are interacting: if two nodes are correlated (they show
positive cooperativity), their relative interaction is positive,
whileiftwonodesareanticorrelated(theyshownegative
cooperativity), their relative interaction is negative. Dealing
with mixed reactions we have to deal with spin glasses and
we can thus assess how much information has been processed
in a given reaction by evaluating the amount of information
processed in its corresponding spin-glass representation,
using our bridge. We have already started this investigation
in [, , ].
Finally, from a practical perspective, in the classical limit
(i.e., for simple reactions) we have an explicit expression that
directly relates the Hill coecient ,whichcanbemeasured
experimentally, to the interaction coupling assumed in the
model; namely, =1/(1−). is allows designing specic
models and very simple validations (at least at this coarse-
grained level) and gives a new computational perspective by
which analyze already developed ones (see, e.g., [–]).
en, regarding complex reactions, the puzzling scenario,
evidenced by Koshland, nally nds out a simple descriptive
framework that, crucially, also recovers to the standard coop-
erative scenario in the proper limit: full coherence among
various, apparently antithetic, results is obtained within a
unique framework.
Conflicts of Interest
e authors declare that there are no conicts of interest
regarding the publication of this paper.
Acknowledgments
Elena Agliari and Adriano Barra are grateful to INdAM-
GNFM for partial support via the project AGLIARI.
Adriano Barra also acknowledges MIUR via the basal found-
ing for the research (-) and Salento University for
further support.
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... The problem of constructing suitable partial differential equations for state functions of thermodynamic systems and the study of critical properties in terms of critical asymptotics of the solutions to these equations is an active field of research which brought further insights on a variety of classical systems, see e.g. [34][35][36][37][38][39], and appears to be promising for the study of complex systems [40,41]. Studies exploiting the Lie symmetry analysis can be therefore carried out for other systems of physical interest. ...
... The problem of constructing suitable partial differential equations for state functions of thermodynamic systems and the study of critical properties in terms of critical asympotics of the solutions to these equations is an active field of research which brought further insights on a variety of classical systems, see e.g. [33][34][35][36][37][38], and appears to be promising for the study of complex systems [39,40]. Studies similar to the present work can be put forward therefore for other systems of physical interest. ...
Preprint
We consider a family of thermodynamic models such that the energy density can be expressed as an asymptotic expansion in the scale formal parameter and whose terms are suitable functions of the volume density. We examine the possibility to construct solutions for the Maxwell thermodynamic relations relying on their symmetry properties and deduce the critical properties implied in terms of the the dynamics of coexistence curves in the space of thermodynamic variables.
... Note that this cost function (4) can be expanded in an alternate-sign series as thus, focusing on the attractive contributions (beyond the classical pairwise model P = 2), it is enriched by P -spin terms (with P = 6, 10, ...) that yield to further synaptic couplings where information can be stored (as recently suggested by Hopfield himself [26]), while, focusing on the repulsive contributions, it also displays P -spin terms (with P = 4, 8, ...) that favour network's pruning (as suggested, in the past, by Hopfield himself and several other authors [18,24,28,29,35] to erase spurious states). The analysis of the information processing skills of this network has been accomplished elsewhere [6,10]: we summarize it by Fig. 1, referring to the original papers for further algorithmic details, while hereafter we deepen the mathematical aspects of its statistical mechanical foundation. ...
Article
The relativistic Hopfield model constitutes a generalization of the standard Hopfield model that is derived by the formal analogy between the statistical-mechanic framework embedding neural networks and the Lagrangian mechanics describing a fictitious single-particle motion in the space of the tuneable parameters of the network itself. In this analogy, the cost-function of the Hopfield model plays as the standard kinetic-energy term and its related Mattis overlap (naturally bounded by one) plays as the velocity. The Hamiltonian of the relativisitc model, once Taylor-expanded, results in a p-spin series with alternate signs: the attractive contributions enhance the information-storage capabilities of the network, while the repulsive contributions allow for an easier unlearning of spurious states, conferring overall more robustness to the system as a whole. Here, we do not deepen the information processing skills of this generalized Hopfield network, rather we focus on its statistical mechanical foundation. In particular, relying on Guerra's interpolation techniques, we prove the existence of the infinite-volume limit for the model free-energy and we give its explicit expression in terms of the Mattis overlaps. By extremizing the free energy over the latter, we get the generalized self-consistent equations for these overlaps as well as a picture of criticality that is further corroborated by a fluctuation analysis. These findings are in full agreement with the available previous results.
... Note that this cost function (4) can be expanded in an alternate-sign series as thus, focusing on the attractive contributions (beyond the classical pairwise model P = 2), it is enriched by P -spin terms (with P = 6, 10, ...) that yield to further synaptic couplings where information can be stored (as recently suggested by Hopfield himself [26]), while, focusing on the repulsive contributions, it also displays P -spin terms (with P = 4, 8, ...) that favour network's pruning (as suggested, in the past, by Hopfield himself and several other authors [18,24,28,29,35] to erase spurious states). The analysis of the information processing skills of this network has been accomplished elsewhere [6,10]: we summarize it by Fig. 1, referring to the original papers for further algorithmic details, while hereafter we deepen the mathematical aspects of its statistical mechanical foundation. ...
Preprint
Full-text available
The relativistic Hopfield model constitutes a generalization of the standard Hopfield model that is derived by the formal analogy between the statistical-mechanic framework embedding neural networks and the Lagrangian mechanics describing a fictitious single-particle motion in the space of the tuneable parameters of the network itself. In this analogy the cost-function of the Hopfield model plays as the standard kinetic-energy term and its related Mattis overlap (naturally bounded by one) plays as the velocity. The Hamiltonian of the relativisitc model, once Taylor-expanded, results in a P-spin series with alternate signs: the attractive contributions enhance the information-storage capabilities of the network, while the repulsive contributions allow for an easier unlearning of spurious states, conferring overall more robustness to the system as a whole. Here we do not deepen the information processing skills of this generalized Hopfield network, rather we focus on its statistical mechanical foundation. In particular, relying on Guerra's interpolation techniques, we prove the existence of the infinite volume limit for the model free-energy and we give its explicit expression in terms of the Mattis overlaps. By extremizing the free energy over the latter we get the generalized self-consistent equations for these overlaps, as well as a picture of criticality that is further corroborated by a fluctuation analysis. These findings are in full agreement with the available previous results.
Article
Full-text available
Statistical mechanics provides an effective framework to investigate information processing in biochemical reactions. Within such framework far-reaching analogies are established among (anti-)cooperative collective behaviors} in chemical kinetics, (anti-)ferromagnetic spin models in statistical mechanics and operational amplifiers/flip-flops in cybernetics. The underlying modeling -- based on spin systems -- has been proved to be accurate for a wide class of systems matching classical (e.g. Michaelis--Menten, Hill, Adair) scenarios in the infinite-size approximation. However, the current research in biochemical information processing has been focusing on systems involving a relatively small number of units, where this approximation is no longer valid. Here we show that the whole statistical mechanical description of reaction kinetics can be re-formulated via a mechanical analogy -- based on completely integrable hydrodynamic-type systems of PDEs -- which provides explicit finite-size solutions, matching recently investigated phenomena (e.g. noise-induced cooperativity, stochastic bi-stability, quorum sensing). The resulting picture, successfully tested against a broad spectrum of data, constitutes a neat rationale for a numerically effective and theoretically consistent description of collective behaviors in biochemical reactions.