- Access to this full-text is provided by Hindawi.
- Learn more

Download available

Content available from Complexity

This content is subject to copyright. Terms and conditions apply.

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 eects 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 scaold 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 oen characterized by com-

plex chemical pathways whose modeling is rather challenging

and can not be recast in standard schemes [–] (see also

[–] for a dierent 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 reshuing 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-decient 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 dierent concentrations for dierent 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 coecient and can be

interpreted as the eective 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

eects underlying. eir eective action is to diminish the

overall binding capabilities of the enzyme and thus to reduce

the magnitude of its Hill coecient.

1.2. e Mechanics Background. e progressive enlargement

of a theoretical scaold to t the always increasing amount of

evidences is a common feature in the historical development

of scientic 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, conrming an ultrasensitive behavior. e tables in the bottom present the value of derived from

thebesttandtheresultingaccording to (); the estimate of the Hill coecient 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 innite. 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

andGalileantransformationsletheplacetotheKlein-

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 scaold 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 denition, 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 inuencing 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,

wewilluseanIsingspin=±1,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 diusion 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: congurations 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

=−

22+.

()

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 congurations. e

free energy is a key observable because it corresponds

to the dierence 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 coecient ; 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 dened 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 (,)

=−122, ,

(,)

=, ,()

where the average ⋅, for a generic observable depending

on the spin conguration is dened 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 (,), dened as half the

variance of the magnetization, that is,

(,)=122−2, ()

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 innite 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+122⋅. ()

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 +

−122. ()

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 inuence 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,

(,)=121+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 dened

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

coecient must be unitary; further, the stronger the coupling

, the (hyperbolically) larger the value of the Hill coecient.

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 coecient 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

eect 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 dened 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 dened 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 suciently 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 dened 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+

0L=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 eect 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 shied /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 coecient (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 −

23+. ()

e relativistic eects in chemical kinetics become transpar-

entinthisway:ifwelookattheeldfeltbythebindingsites

(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 eective Hamiltonian to

generate () that reads as

(,,)=−

<+

23

<<<;()

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 eect 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 eective 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 eective

Hamiltonian () becomes relevant. At this point, a novel,

anticooperative eect is naturally induced in the reaction and

it yields to a reduction of the Hill coecient. In the next

analysis these qualitative remarks shall be addressed in more

details.

We focus on the denition of the Hill coecient 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 coecient. 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 =122+12⇒

=141

1−.()

By plugging this result in (), we nally have

class

=1

1−.()

One can see that when =0the Hill coecient is unitary

as expected for noncooperative systems, when >0the

coecient is larger than , indicating that receptors are

interacting, and when <0the coecient 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 simplies 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, conrming 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 dierent

quantiers 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 dened as the ratio (notice that the

Koshland index is actually strongly related to the Hill

coecient (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 quantier 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=121+tanh 2× 9

10−1+12log 0.9,

1

10=121+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 dierent values of . Both models exhibit the sigmoidal shape

typical of cooperative systems; however, the former displays a slower saturation. Analogous results are obtained for dierent 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 quantier

=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

2×(9/10)−1

1+(2×(9/10)−1)2

+12log 0.9

,

1

10=12

1+tanh

2×(1/10)−1

1+(2×(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 quantier, when xing the same coupling constant

, the emerging cooperativity is weaker for the relativistic

model, as expected.

Next, let us consider the cooperativity quantier 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 Clark’s 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 fullls ∗=−1

,()

where provides another quantier for cooperativity.

Starting from the classical model, we can build the

function /,byrstgettingas 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)−2 2−1

1+(2−1)2

.()

By deriving /with respect to we get

=−(2−1)/(−1)+1/2 1− 4(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 .

Wecanobtainanestimateofthevalue∗corresponding to

the maximum by recalling ≤1and neglecting high-order

terms. In this way we get

∗≈−3+22−−9+262

2−9+2,()

and, comparing with (), we get

rel =2−9+2

−15+−9+262.()

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 quantiers 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 dierent 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 aected 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.

Inthenalpartofthissectionwewanttogetdeeperinthe

comparison between the classical and the relativistic models.

To this aim, we solved numerically (), for dierent 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 dierent 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 dierent values

of . Notice that the two parameters are related by a linear

law, whose slope is smaller than 1and decreases with .

is conrms that the relativistic model yields to a weak

cooperativity. e negative contributions in the relativistic

model get more eective 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 briey comment.

e former lie in a deeper understanding of the mathe-

matical scaold 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: dierent colors represent dierent 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 innitesimally , →

, +,;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, Hopeld neural networks [] and restricted

Boltzmann machines [], key tools in Articial Intelligence

(resp., in neural networks and machine learning), are two

types of spin glasses and it is with this last denition 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 coecient ,whichcanbemeasured

experimentally, to the interaction coupling assumed in the

model; namely, =1/(1−). is allows designing specic

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 conicts 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.

References

[] J. H. Espenson, Chemical Kinetics and Reaction Mechanisms,

McGraw-Hill, .

[] C. Mazza and M. Benaim, Stochastic Dynamics for Systems

Biology,CRCPress,.

[]E.Agliari,A.Annibale,A.Barra,A.C.C.Coolen,andD.

Tantari, “Retrieving innite numbers of patterns in a spin-glass

model of immune networks,” EPL (Europhysics Letters),vol.,

no. , Article ID , .

[] E.Agliari,A.Barra,G.DelFerraro,F.Guerra,andD.Tantari,

“Anergy in self-directed B lymphocytes: A statistical mechanics

perspective,” Journal of eoretical Biology,vol.,pp.–,

.

[] P. Andriani and J. Cohen, “From exaptation to radical niche

construction in biological and technological complex systems,”

Complexity,vol.,no.,pp.–,.

[] D. Angeli, J. E. Ferrell Jr., and E. D. Sontag, “Detection of

multistability, bifurcations, and hysteresis in a large class of bio-

logical positive-feedback systems,” Proceedings of the National

AcadamyofSciencesoftheUnitedStatesofAmerica,vol.,no.

, pp. – , .

[] E.J.Crampin,S.Schnell,andP.E.McSharry,“Mathematical

and computational techniques to deduce complex biochemical

reaction mechanisms,” Progress in Biophysics & Molecular Biol-

ogy, vol. , no. :, .

[] W. W. Chen, M. Niepel, and P. K. Sorger, “Classic and contem-

porary approaches to modeling biochemical reactions,” Genes

&Development,vol.,no.,pp.–,.

[] M. J. Gander, C. Mazza, and H. Rummler, “Stochastic gene

expression in switching environments,” Journal of Mathematical

Biology,vol.,no.,pp.–,.

[] H. Kitano, “Systems biology: a brief overview,” Science,vol.,

no.,pp.–,.

[] M. Hucka, A. Finney, H. M. Sauro et al., “e systems biology

markup language: a medium for representation and exchange

of biochemical network models,” Bioinformatics,vol.,no.,

article , .

[] C. E. Maldonado and N. A. G´

omez Cruz, “Biological hyper-

computation: A new research problem in complexity theory,”

Complexity,vol.,no.,pp.–,.

[] L.Li,J.Xu,D.Yang,X.Tan,andH.Wang,“Computational

approaches for microRNA studies: A review,” Mammalian

Genome,vol.,no.-,pp.–,.

[] M. Perc and P. Grigolini, “Collective behavior and evolutionary

games - An introduction,” Chaos, Solitons & Fractals,vol.,pp.

–, .

[] M. Perc and M. Marhl, “Dierent types of bursting calcium

oscillations in non-excitable cells, Chaos,” Sol Fraction,vol.,

no.,.

[] M. Peleg, M. G. Corradini, and M. D. Normand, “Kinetic mod-

els of complex biochemical reactions and biological processes,”

Chemie Ingenieur Technik,vol.,no.,pp.–,.

[] F. Ricci, R. Y. Lai, A. J. Heeger, K. W. Plaxco, and J. J.

Sumner, “Eect of molecular crowding on the response of an

electrochemical DNA sensor,” Langmuir,vol.,no.,pp.

–, .

[] F. Ricci, G. Adornetto, and G. Palleschi, “A review of experimen-

tal aspects of electrochemical immunosensors,” Electrochimica

Acta,vol.,pp.–,.

[] C. Valant, P. M. Sexton, and A. Christopoulos, “Orthos-

teric/allosteric bitopic ligands going hybrid with GPCRs,”

Molecular Interventions,vol.,no.,pp.–,.

[] E. Agliari, A. Barra, F. Guerra, and F. Moauro, “A ther-

modynamic perspective of immune capabilities,” Journal of

eoretical Biology,vol.,no.,pp.–,.

[] E. Agliari, M. Altavilla, A. Barra, L. Dello Schiavo, and E.

Katz, “Notes on stochastic (bio)-logic gates: Computing with

allosteric cooperativity,” Scientic Reports,vol.,articleno.

, .

[] E. Katz and V. Privman, “Enzyme-based logic systems for

information processing,” Chemical Society Reviews,vol.,no.

, pp. –, .

[] K. Lund, A. J. Manzo, N. Dabby et al., “Molecular robots guided

by prescriptive landscapes,” Nature,vol.,no.,pp.–

, .

Complexity

[] L. Qian and E. Winfree, “Scaling up digital circuit computation

with DNA strand displacement cascades,” Science,vol.,no.

, pp. –, .

[] G. Strack, M. Ornatska, M. Pita, and E. Katz, “Biocomputing

security system: Concatenated enzyme-based logic gates oper-

ating as a biomolecular keypad lock,” Journal of the American

Chemical Society,vol.,no.,pp.-,.

[] C. J. ompson, Mathematical statistical mechanics, Princeton

University Press, Princeton, NJ, USA, .

[] J. Clerk Maxwell, “Clerk-Maxwell’s kinetic theory of gases [],”

Nature,vol.,no.,p.,.

[] E. Agliari, A. Barra, R. Burioni, A. Di Biasio, and G. Uguzzoni,

“Collective behaviours: From biochemical kinetics to electronic

circuits,” Scientic Reports,vol.,articleno.,.

[] A.Biasio,E.Agliari,A.Barra,andR.Burioni,“Mean-eldcoop-

erativity in chemical kinetics,” eoretical Chemistry Accounts,

vol. , no. , pp. –, .

[] D. I. Cattoni, O. Chara, S. B. Kaufman, and F. L. G. Flecha,

“Cooperativity in binding processes: New insights from phe-

nomenological modeling,” PLoS ONE,vol.,no.,ArticleID

e, .

[] J. N. Weiss, “e Hill equation revisited: uses and misuses,” e

FASE B Journal,vol.,no.,pp.–,.

[] R. A. Cook and D. E. Koshland, “Positive and Negative Coop-

erativity in Yeast Glyceraldehyde -Phosphate Dehydrogenase,”

Biochemistry, vol. , no. , pp. –, .

[] L.H.Chao,P.Pellicena,S.Deindl,L.A.Barclay,H.Schulman,

and J. Kuriyan, “Intersubunit capture of regulatory segments

is a component of cooperative CaMKII activation,” Nature

Structural & Molecular Biology,vol.,no.,pp.–,.

[] M. Mandal, Lee. M., J. E. Barrick et al., “A glycine-dependent

riboswitch that uses cooperative binding to control gene expres-

sion,” Science,vol.,no.,.

[]J.M.Bradshaw,Y.Kubota,T.Meyer,andH.Schulman,“An

ultrasensitive Ca+/calmodulin-dependent protein kinase II-

protein phosphatase switch facilitates specicity in postsynap-

tic calcium signaling,” Proceedings of the National Acadamy of

Sciences of the United States of America,vol.,no.,pp.

–, .

[] W. Heisenberg and Werner, e revolution in Modern Science:

Physics And Philosophy,.

[] T. S. Kuhn, “e Structure of Scientic Revolutions,” e Physics

Teac h e r ,vol.,no.:,.

[] F. Mandl and G. Graham Shaw, Quantum Field eory,John

Wile y & Sons, .

[] J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields,

McGraw-Hill, .

[] A.D.Michel,L.J.Chambers,W.C.Clay,J.P.Condreay,D.

S. Walter, and I. P. Chessell, “Direct labelling of the human

PX receptor and identication of positive and negative

cooperativity of binding,” British Journal of Pharmacology,vol.

, no. , pp. –, .

[] E. Agliari et al., “Complete integrability of information process-

ing by biochemical ractions,” Scientic Reports,vol.,article

, .

[] F. Guerra, “Sum rules for the free energy in the mean eld

spin glass model,” in Mathematical physics in mathematics and

physics (Siena, 2000),vol.ofFields Institute Communications,

pp. –, .

[] A. Barra, “e mean eld Ising model through interpolating

techniques,” Journal of Statistical Physics,vol.,no.,pp.–

, .

[] A. Barra, A. D. Biasio, and F. Guerra, “Replica symmetry

breaking in mean-eld spin glasses through the Hamilton-

Jacobi technique,” Journal of Statistical Mechanics: eory and

Experiment,v

ol.,no.,ArticleIDP,.

[] A. Barra, A. Di Lorenzo, F. Guerra, and A. Moro, “On quantum

and relativistic mechanical analogues in mean-eld spin mod-

els,” Proceedings of the Royal Society A Mathematical, Physical

and Engineering S ciences,vol.,no.,ArticleID,

.

[] G. Genovese and A. Barra, “A mechanical approach to mean

eld spin models,” Journal of Mathematical Physics,vol.,no.

, Article ID , .

[] J. G. Brankov and V. A. Zagrebnov, “On the description of the

phase transition in the Husimi-Temperley model,” Journal of

Physics A: Mathematical and General,vol.,no.,pp.–

, .

[] A. Arsie, P. Lorenzoni, and A. Moro, “Integrable viscous

conservation laws,” Nonlinearity,vol.,no.,pp.–,

.

[] A. Arsie, P. Lorenzoni, and A. Moro, “On integrable conser-

vation laws,” Proceedings of the Royal Society A Mathematical,

Physical and Engineering Sciences,vol.,no.,ArticleID

, .

[] A. Barra and A. Moro, “Exact solution of the van der Waals

model in the critical region,” Annals of Physics, vol. , pp. –

, .

[] F. Giglio, G. Landol, and A. Moro, “Integrable extended van

der Waals model,” PhysicaD:NonlinearPhenomena,vol.,

pp. –, .

[] A. Merkoci et al., Comprehensive Analytical Chemistry,Elsevier,

.

[] A. Cornish-Bowden and D. E. Koshland Jr., “Diagnostic uses of

the Hill (logit and Nernst) plots,” Journal of Molecular Biology,

vol.,no.,pp.–,.

[] A.-L. Barabasi and R. Albert, “Emergence of scaling in random

networks,” American Association for the Advancementof Science:

Science,vol.,no.,pp.–,.

[] M. Mezard, G. Parisi, and M. Virasoro, Spin Glass eory and

Beyond, World Scientic Publishing, .

[] J. J. Hopeld, “Neural networks and physical systems with

emergent collective computational abilities,” Proceedings of the

NationalAcadamyofSciencesoftheUnitedStatesofAmerica,

vol. , no. , pp. –, .

[] D. Ackley, G. Hinton, and T. Sejnowski, “A learning algorithm

for Boltzmann machines,” Cognitive Science,vol.,no.,pp.

–, .

[] M. Dougoud, C. Mazza, and L. Vinckenbosch, “Ultrasensitivity

and sharp threshold theorems for multisite systems,” Journal of

Physics A: Mathematical and eoretical,vol.,no.,Article

ID , .

[] D. G. Hardie, I. P. Salt, S. A. Hawley, and S. P. Davies,

“AMP-activated protein kinase: An ultrasensitive system for

monitoring cellular energy charge,” Biochemical Journal,vol.

, no. , pp. –, .

[] D. C. LaPorte, K. Walsh, and D. E. Koshland Jr., “e branch

point eect. Ultrasensitivity and subsensitivity to metabolic

control,” e Journal of Biological Chemistry,vol.,no.,pp.

–, .

Complexity

[] D. E. Koshland Jr., G. N´

emethy, and D. Filmer, “Comparison of

experimental binding data and theoretical models in proteins

containing subunits,” Biochemistry,vol.,pp.–,.

[] B. M. C. Martins and P. S. Swain, “Ultrasensitivity in Phos-

phorylation-Dephosphorylation Cycles with Little Substrate,”

PLoS Computational Biology,vol.,no.,ArticleIDe,

.

Content uploaded by Adriano Barra

Author content

All content in this area was uploaded by Adriano Barra on Jan 20, 2018

Content may be subject to copyright.

Available via license: CC BY 4.0

Content may be subject to copyright.