allostery 
 plays a fundamental role in most biological processes 
 and has been suggested to be present in nearly all proteins .
one of the best - studied allosteric phenomena 
 is the activation of a receptor , which we will denote as r , by a ligand , 
 denoted as l. the most common model for allostery in this system is 
 the allosteric two - state model ( atsm ) .
we can construct 
 a thermodynamic cycle for the process of ligand - induced activation 
 of the receptor:1we refer to this as an  allosteric 
 cycle  . to describe the allostery in this system , the allosteric 
 efficacy ,  , can be calculated from the cycle as2where 
 ( assuming the volume is constant ) ,
each 
 equilibrium constant is a function of the difference in helmholtz 
 free energy , a , for the two states:3thus,4for convenience , we will 
 discuss the allosteric 
 efficacy in terms of the quantity a , which we will call 
 the thermodynamic coupling .
a 
 is symmetric at equilibrium , that is , a 
 for receptor activation conditional on ligand binding is equivalent 
 to the a for ligand binding conditional on activation . 

thus , the following two definitions of a are equivalent:5however , there is 
 no reason 
 to assume that the receptor activation is a two - state process . in 
 fact
, nmr experiments have revealed a multimodal activation process 
 in the 2-adrenergic receptor ( 2ar ) , and quantitative mass spectroscopy 
 experiments have revealed ligand - specific states in the same system .
these results , along with other evidence for 
 additional states in 2ar and other receptors , indicate that activation must be treated as either involving more 
 than two discrete states , or even as involving a continuous conformational 
 space .
. the complex behavior of an allosteric receptor 
 is thus unlikely to be well described by a single reaction coordinate ; 
 instead , the large number of potential conformational states may be 
 best described by multiple collective variables ( cvs ; variables that 
 are functions of the atomic coordinates ) that are thermodynamically 
 coupled in nontrivial ways .
it is impossible to understand the molecular 
 mechanism of ligand - induced receptor activation without explicitly 
 considering the thermodynamic coupling between the ligand binding 
 site and active site , so a minimal set of cvs should at least include 
 one cv for each one of these sites .
these cvs are intrinsic to the 
 specific receptor , and their thermodynamic coupling arises from the 
 complex network of molecular interactions that separates them spatially 
 in the receptor s structure .
this intrinsic thermodynamic coupling 
 is of great interest , as an understanding of the nature of this coupling 
 can be used to inform both the design of ligands that modulate function 
 in a highly specific manner , and the design of receptors with modified 
 allosteric properties .
we have previously represented this intrinsic 
 thermodynamic coupling using the recently developed allosteric ising 
 model ( aim ) , a two - state model of allostery 
 that implicitly includes the potential energy of interaction between 
 structural components . while the aim and other statistical mechanical 
 models of allostery , such as the ensemble allosteric model , have allowed us to derive some analytical features 
 of simple allosteric systems , a general method that does not rely 
 on the two - state assumption is still needed to study the intrinsic 
 thermodynamic coupling between structural components in real systems . 
 here
we describe such a method and illustrate its capabilities by 
 showing how allosteric coupling in the alanine dipeptide molecule 
 contributes to the overall shape of the / free energy 
 surface , and by identifying the interactions that are necessary for 
 this coupling and their contributions to the energetics .
to quantify the intrinsic thermodynamic 
 coupling between cvs , we will derive expressions analogous to the 
 allosteric efficacy for the coupled perturbation of discrete or continuous 
 cvs away from their equilibrium distributions .
let   represent the coordinates of the allosteric 
 protein and its environment that define our system , which does not 
 include any ligand that we consider here as external perturbations . 

the probability density of each microstate r is given by the 
 boltzmann distribution,6where u(r ) is the potential energy 
 function .
the numerator is the boltzmann factor denoted as7the free energy can be written as a functional 
 of the boltzmann factor function,8we define a cv as a function x(r ) 
 of the system s coordinates that can be either continuous or 
 discrete . for a continuous cv , the probability density function is9for a discrete cv
, the probability mass function , 
 p(x ) , is defined by an identical expression , but 
 is bounded to be less than 1 everywhere .
we can calculate the free 
 energy conditional on a value of the cv as10equation 10 can be 
 rewritten in terms of either f(x ) or p(x ) . 
 because we use the histogram method to estimate the probability mass 
 function of the cvs in the application following this derivation
, 
 we will assume discrete cvs described using p(x ) without loss of generality . 

the free energy becomes11consider a second cv , y(r ) , 
 with analogous 
 probability function and free energy definitions .
a joint probability 
 mass function for the two cvs can be written as12so that the analogous free energy conditional 
 on values of both cvs is13 one can imagine x(r ) 
 to describe 
 the ligand binding site and y(r ) to describe the active site 
 of the protein ; the binding of a ligand to the system then acts as 
 an external perturbation to the distributions of these cvs . to quantify 
 the intrinsic coupling between these cvs , we apply artificial perturbations 
 to the equilibrium cv distributions such that one or both cvs become 
 constrained to a given value . from the equilibrium state and these 
 artificially perturbed states , we calculate the allosteric efficacy 
 of the following thermodynamic cycle:14we will refer 
 to this class of thermodynamic 
 cycles as 
the thermodynamic 
 coupling of the perturbations at position ( x , y ) in the cv space , a(x , y ) , 
 can be calculated as15equation 15 simplifies to16this is the mathematical definition 
 we propose 
 for the central quantity a(x , y ) that we call the thermodynamic coupling function for the cvs x(r ) 
 and y(r ) . in two dimensions ,
eq 16 defines what we call the allostery 
 landscape ( see figure 1 for an example ) .
our calculations 
 reveal that 
 the allosteric coupling between  and  destabilizes the 
 high free energy regions and stabilizes the l and 
 c7ax states .
( a ) the alanine dipeptide molecule 
 with the backbone dihedral angles  and  indicated by 
 arrows .
the molecule is partitioned in three domains as indicated 
 by the black lines ( see text for details ) .
( b ) free energy surface 
 a(, ) calculated according to eq 13 , with prominent states labeled in white . 
 in all panels ,  and  are expressed in radians .
( c ) the 
 allostery landscape representing the thermodynamic coupling between 
 cvs  and  , calculated according to eq 16 .
( d ) the normalized allosteric 
 coupling , calculated according to eq 19 . in panels c and d ,
greyed - out regions represent data 
 that are not surely different from zero , based on its 95%-confidence 
 interval estimated by bootstrapping ( see methods ) .
it should be noted that the thermodynamic 
 coupling function has 
 a natural normalization when the cvs are discrete .
if the two cvs 
 are maximally coupled , constraining one cv will fully constrain the 
 other . thus , at maximum coupling,17and thus18we can then normalize ( 16 ) to this 
 upper bound 
 to define the normalized allosteric coupling ( ac),19the ac ranges from 1 to 1 and matches 
 the convention commonly used for positive and negative allostery ; 
 positive values indicate that constraining one cv reduces the free 
 energy required to constrain the other , whereas negative values indicate 
 that constraining one cv increases the free energy required to constrain 
 the other .
in essence , the magnitude of the ac describes what 
 fraction of the maximal allostery is contributing to the free energy 
 of the joint state , whereas the sign of the ac describes whether that 
 allostery is positive or negative . when applied to 
 the biophysical information transmission process 
 occurring in a receptor ( i.e. , the thermodynamic coupling between 
 the ligand binding site and the active site ) , the definitions above 
 indicate that the thermodynamic coupling function is negative if measuring 
 the active site to be in the  active  state reduces 
 the uncertainty associated with whether or not the ligand binding 
 site is in the  bound  state . the two - state ligand - induced 
 receptor activation model defined in eq 1 can be just as easily described using the thermodynamic 
 coupling function if the collective variables x(r ) and y(r ) 
 are defined to take only two discrete values ( bound / unbound and on / off , 
 respectively ) . in this context
, the two - state allosteric efficacy 
 in eq 4 can be calculated 
 from an allosteric cycle composed of four allosteric perturbation 
 cycles:20so that21thus , a large negative acycle for the allosteric 
 coupling between activation and ligand 
 binding indicates that when the receptor is in the active state , the 
 uncertainty that a ligand is bound is greatly reduced , whereas when 
 the receptor is in the inactive state , the uncertainty that a ligand 
 is not bound is greatly reduced .
the mutual information , which 
 is often used to quantify allostery , is defined as22interestingly , a(x , y ) in eq 16 is proportional to the logarithmic term in 22 , which is known as the pointwise mutual information 
 ( pmi),23like the allosteric efficacy , the pmi is symmetric 
 ( i.e. , the order of variables x and y does not matter ) . to understand 
 the pmi from the perspective of information theory
, one can consider 
 the information gained due to the reduction in uncertainty associated 
 with measuring a variable .
this information gain by measuring x(r ) 
 to be equal to x is24however , 
 if two variables are measured , and 
 those variables are dependent on each other , the amount of information 
 gained by measuring the second variable will be different from the 
 amount that would be gained if it were measured alone .
for example , 
 if y(r ) was measured to be y , the probability distribution 
 of x(r ) is now conditioned on y(r ) = y , and thus the 
 information gained by measuring x(r ) becomes25the pmi is the difference in the 
 information 
 gain,26the mutual information is the pmi 
 weighted 
 by the joint probability density function .
consequently , the mutual 
 information gives a high weight to the thermodynamic coupling of perturbations 
 of high equilibrium probability states and low weight to those of 
 low equilibrium probability .
this is important for the mechanistic 
 interpretation of allosteric couplings that are quantified only by 
 their mutual information , as functionally significant perturbations 
 do not necessarily drive the protein toward a region of its intrinsic 
 cv space that is already high probability prior to perturbation . in 
 fact
, perturbations such as ligands generally drive the system away 
 from the unbound equilibrium ( e.g. , where the inactive state is preferred 
 to the active state ) , so the mutual information would end up giving 
 larger weight to less functionally relevant states . in such cases , 
 when considering only the protein s degrees of freedom , the 
 mutual information is not a good quantification of the intrinsic thermodynamic 
 couplings that mediate the system s response to ligand binding . 

even in the simplest case of the allosteric coupling between ligand 
 binding and activation as described in the atsm , the mutual information 
 between ligand binding and activation will depend on the affinity 
 of the ligand , and will go to 0 as the affinity goes to either 0 or 
  , independent of the allosteric efficacy of the ligand . therefore , 
 we argue that it becomes preferable instead to analyze the entire 
 2-dimensional thermodynamic coupling surface , a(x , y ) , 
 which we call the  allostery landscape 
, as it contains 
 information regarding the allosteric efficacy for all possible perturbations 
 to the distribution of those cvs . the thermodynamic coupling 
 function is also related to the copula density function from probability 
 theory . the copula density function of 
 a bivariate probability distribution is27where p(x ) and 
 p(y )
any multivariate 
 distribution can be expressed as a set of marginal 
 probability distributions and a copula that defines the dependency 
 between them , and the entropy of the 
 copula distribution is equivalent to the mutual information .
thus , a multivariate thermodynamic coupling 
 function behaves like a copula , defining the information transmission 
 properties of the allosteric system .
the relationship between the 
 thermodynamic coupling function and fundamental concepts in information 
 theory and probability theory suggest that past work in these fields 
 may be able to be adapted for biophysical applications and provide 
 new insights into allostery .
having introduced 
 a quantification of the allosteric coupling between two cvs with the 
 allostery landscape , a major mechanistic question still remains .
what 
 features of the structure and energetics of a given system define 
 the thermodynamic coupling function ? to answer this question , we derive 
 the change in thermodynamic coupling function when a generic biasing 
 potential energy term ubias(r ) is added to the 
 system s total potential energy function , u(r ) . the 
 change in thermodynamic coupling , eq 16 , at 
 any point in the cv space
can be estimated using a free energy perturbation 
 approach , which we will refer to here as  biasing  to 
 avoid confusion with 
perturbing  that refers here to 
 constraining the system at ( x , y ) in the cv space in eq 14 .
the change in free energy of 
 the system when a biasing potential is added is28the 
 change in free energy of the perturbed 
 states can be similarly written as29thus , the biased thermodynamic 
 coupling function 
 is30we wish to understand the contribution to 
 the thermodynamic coupling function of a structural feature of interest , 
 or of a specific interaction between structural elements of the system . 
 assuming this feature of interest is described by a specific energy 
 term uint(r ) of the total potential energy function 

u(r ) , we can use eq 30 with a biasing potential that is equal and opposite to that 
 energy term , ubias(r ) = uint(r ) . in the next section
, we use this approach to quantify 
 the contribution of specific interactions in the alanine dipeptide 
 system by mapping the corresponding change in thermodynamic coupling , 
 a(x , y ) = a(x , y ) 
consider the special case of a particular 
 energy term of interest that is a function of a cv , z(r ) . 

this corresponds to a biasing potential of the form ubias(r ) = uint(z(r ) ) .
we have found 
 ( see appendix ) that this results in the following 
 biased thermodynamic coupling functions:1.if z(r ) is independent of either 
 x(r ) or y(r),2.if z(r ) is conditionally 
 independent 
 of y(r ) given x(r ) , or if z(r ) = x(r),31the second result is interesting 
 because in this case a(x , y ) 
 becomes a function of y only .
a corresponding result is found if x(r ) 
 and y(r ) are permuted . if z(r ) is independent of either 
 x(r ) or y(r ) , if z(r ) is conditionally 
 independent 
 of y(r ) given x(r ) , or if z(r ) = x(r),31 importantly , these findings indicate 
 that the influence of any 
 specific energy term on the thermodynamic coupling between two cvs 
 can be clearly defined from the difference between the unbiased and 
 biased thermodynamic coupling functions .
the two conditions described 
 above indicate that unbiasing by a potential energy term that mediates 
 an allosteric coupling will have a two - dimensional effect on the thermodynamic 
 coupling function ( i.e. , the second term in eq 30 is dependent on both x(r ) and y(r ) ) . we know that , 
 owing to nonadditive effects , the free energy contributions of such 
 coupled energy terms can not be rigorously deconvoluted .
therefore , the contributions obtained by biasing 
 the allosteric coupling function ( shown in figure 2 ) , can not be taken as a sensu stricto decomposition of a(x , y ) .
this method nonetheless 
 allows for the detailed analysis of the mechanism of allosteric coupling 
 and can be applied generally across any system whose conformational 
 ensemble can be sampled using methods such as molecular dynamics ( md ) . contributions 
 of specific interactions to the allosteric coupling 
 function of the alanine dipeptide .
the contribution of the specific 
 interactions identified for each panel was calculated according to eq 30 , revealing the prominent 
 role of bonded interactions between the termini and channel .
( a ) contribution 
 of the dihedral energy term corresponding to the definition of the 
 angle  ( c  n  cc ) .
the 
 corresponding interaction energy as a function of  and  , 
 biased free energy surface , the biased allosteric coupling function , 
 and the associated ac are shown in figure s2 .
( c ) contribution of the nonbonded interactions between the termini 
 and the channel .
( d ) contribution of the bonded interactions involving atoms 
 from both the termini and the channel .
greyed - out regions 
 represent data that are not surely different from zero , based on its 
 95%-confidence interval estimated by bootstrapping ( see methods section ) .
as we show below with the example of the alanine dipeptide 
 in solution , 
 md can provide the data necessary for our formalism to be implemented 
 for specific allosteric proteins of interest .
the need for this type 
 of data in order to describe allostery is foreshadowed by the discussion 
 accompanying weber s introduction of a multistate model of cooperativity in oligomers in 1972 .
he 
 noted that a thermodynamic description of allostery based on the protein s 
 intrinsic conformational equilibrium ( referred to as  tautomerization  ) 
 was  far more restricted in scope and less useful  than a model based on its oligomerization equilibrium . 

characterization of tautomerizations in 
 molecules involving as large a number of potential degrees of freedom 
 as proteins is confined to some structural detail that happens to 
 be observable  and that it appeared 
  unlikely than any method can furnish us with a precise determination 
 of the tautomerization constants necessary to characterize completely 
 the system  .
indeed , in order 
 to utilize the formalism derived above , it is necessary to observe 
 a large number of degrees of freedom simultaneously with high accuracy 
 and precision so that a free energy landscape can be estimated .
although 
 such detail had not yet been described for proteins in 1972 , the first 
 molecular dynamics ( md ) simulation of a protein was published just five years later in 1977 . using md simulations
, 
 we are now able to estimate the equilibrium distribution and corresponding 
 free energy landscape of one or more cvs in many proteins , while simultaneously 
 observing both the coordinates of all atoms in the system and their 
 interaction energies , which is essential to the application of the 
 formalism derived above .
to illustrate 
 the use and utility of the thermodynamic coupling function , we analyzed 
 the allostery landscape of the alanine dipeptide in solution .
the 
 alanine dipeptide is a popular model system for testing enhanced sampling 
 and free energy methods as the entire system can be described well 
 by only two cvs , the  and  torsion angles along the 
 bonds connecting the alanine c atom to the capped 
 n- and c - terminus , respectively ( see figure 1a ) . here , in analogy 
 to larger allosteric systems , we consider that  captures the 
 state of the n - terminal domain and  the state of the c - terminal 
 domain , and we ask the question of how the n - terminal and c - terminal 
 domains of the protein are allosterically coupled . despite the small 
 size of the system ,
the irregular features of the / 
 free energy surface ( see figure 1b ) indicate that these cvs are thermodynamically coupled 
 in a nontrivial way , and thus the alanine dipeptide is an ideal model 
 system for illustrating the power of the thermodynamic coupling function .
we constructed the 2-dimensional / probability density 
 function of alanine dipeptide in water from five independent 50 ns 
 trajectories produced with driven adiabatic free energy dynamics ( see methods ) . following a protocol that 
 we previously demonstrated to yield well - converged free energy surfaces 
 up to 40 kj / mol above the global minimum , we reconstructed the free energy landscape shown in figure 1b using the reweighted histogram 
 estimator . to investigate which features of the alanine dipeptide 
 free energy landscape are due to thermodynamic coupling between the 
 two angles
, we used eq 16 to calculate the thermodynamic coupling function shown in figure 1c .
significant allosteric 
 couplings are evident in the regions of the left - handed -helix 
 ( known as l , which should not be confused with the 
 symbol for allosteric efficacy ) and the region labeled c7ax ( shown 
 on figure 1b , slightly 
 lower than the c7ax conformation stabilized in the gas 
 phase ) .
this indicates that if  
 is driven to the ( 0 to 2 rad/0 to 120 ) region , the transition 
 of  to the ( 0 to 2 rad/0 to 120 ) and ( 1 
 to 2 rad/60 to 180 ) regions becomes 
 more favorable . in the normalized ac landscape of the alanine 
 dipeptide ,
calculated 
 according to eq 19 and shown in figure 1d , the l and 
 c7ax , regions have couplings of around 0.4 , indicating that a substantial 
 amount of the maximal theoretically possible / allostery 
 contributes to these state s stabilities .
thus , while these 
 regions have a relatively low probability , our analysis suggests that 
 the allosteric coupling accounts for the small but significant populations 
 of l and c7ax conformations that appear at equilibrium . 

in addition , there appears to be significant coupling present at the 
 transition region between l and ppii , 
 and to a lesser extent between r and c7ax , indicating 
 that these transitions may also be facilitated by allostery .
we also 
 see significant negative allosteric coupling in the high free energy 
 regions , which indicates that an unfavorable thermodynamic coupling 
 between the cvs contributes to the high free energy of these regions .
the mutual information , eq 22 , between  and  is 1.11  0.01 nats ( 95% 
 confidence interval from bootstrapping ) , or for better comparison 
 to the thermodynamic coupling , 0.29  0.03 kj / mol at 300 k. it 
 should be noted that these values are quite low compared to the numerous 
 regions of high thermodynamic coupling and normalized ac ( |a(x , y)| 
 > 6 kj / mol , |ac(x , y)| > 0.3 , see figure 1 ) , and thus utilizing the mutual information 
 alone understates the thermodynamic coupling between  and . 
 mapping the quantity summed over in eq 22 ( see figure s1 ) shows that 
 the major contributions to the mutual information come from very localized 
 regions of the cv space .
thus , using a single number to quantify the 
 coupling between  and  misses the fact that the allostery 
 landscape has significant regions of both negative and positive coupling
. 
 this can be important if , for example , one seeks to design a ligand 
 that allosterically stabilizes a lower probability state . to understand which structural features contribute 
 to the thermodynamic coupling of  and  , we decomposed 
 the alanine dipeptide into three structural components , as shown in figure 1a : ( i ) the n - terminus , 
 which includes all atoms on the n - terminal side of the c carbon , ( ii ) the c - terminus , which includes all atoms on the c - terminal 
 side of c , and ( iii ) the  channel  , 
 which includes c as well as the hydrogen and methyl 
 side chain bound to it .
these three structural components can mediate 
 the / thermodynamic coupling through three different 
 mechanisms : ( i ) direct nonbonded interaction of the termini ( estimated 
 with a dielectric constant  = 60 ) , ( ii ) indirect interaction 
 of the termini through nonbonded interactions with the channel , and 
 ( iii ) indirect interaction of the termini through bonded interaction 
 with the channel .
thus , we estimated the potential energy uint(r ) contributed by each of these groups of energy terms for 
 each frame along the trajectories and reweighted the free energy landscape 
 with an equal and opposite biasing potential ubias(r ) 
 according to eq 28 , 
 see figures s2s4 .
we then calculated 
 the contribution of uint(r ) to the thermodynamic 
 coupling landscape a(x , y ) using eqs 29 and 30 , as represented 
 on figure 2 in the form of a(x , y ) = a(x , y ) 
 
we also 
 biased using the negative of the potential energy term corresponding 
 exactly to the definition of the (r ) cv , that is , the 
 c  n  cc dihedral angle .
this 
 reweighting resulted in one - dimensional variations of the thermodynamic 
 coupling function along the  axis , as expected from eq 31 ( see figure 2a ) .
interestingly , while 
 chemical intuition may suggest that the direct interaction of the 
 termini is the major mediator of the thermodynamic coupling , we find 
 that the direct nonbonded interaction only contributes to the negative 
 thermodynamic coupling surrounding the central forbidden region , as 
 shown in figure 2b . 

the only other significant change to the thermodynamic coupling function 
 is one - dimensional bands at   1 and   
 1 rad , which indicates that the interaction itself may to 
 some extent be indirectly coupled to  through its direct dependency 
 on . figure 2d shows 
 that the bonded interactions between the termini and the channel are 
 the most significant contributors to both the positive and negative 
 thermodynamic coupling between the termini , while the nonbonded interactions 
 between the termini and the channel ( figure 2c ) do not significantly contribute to the 
 thermodynamic coupling .
these results suggest that  and  
 become thermodynamically coupled due to the energetics of the bonds , 
 angles , and dihedrals composed of atoms shared between each terminus 
 and the channel .
for example , the  and  dihedrals each 
 share the angle formed by three central atoms .
different combinations 
 of  and  frustrate this central angle to different extents , 
 leading to a thermodynamic coupling between the two .
we however note 
 that in the alanine dipeptide system , the energy terms described above 
 are tightly coupled with each other , as well as with other energy 
 terms ( such as the internal bonded energy of the channel ) . therefore , 
 the contributions represented in figure 2 do not represent an exact decomposition 
 of a(x , y ) and must be regarded as useful cues for the 
 qualitative understanding of how allosteric coupling can be established 
 between two domains of a molecular system .
we 
 have derived a thermodynamic coupling function based on the 
 allosteric efficacy that quantifies the allosteric coupling between 
 two continuous or discrete cvs .
we find that the thermodynamic coupling 
 function is related to both the pointwise mutual information and the 
 copula , and is best represented in the form of an allostery landscape , 
 in units of free energy .
such a representation reveals the allosteric 
 response to all possible perturbations of the cvs .
we showed that 
 the allostery landscape of the  and  dihedral angles 
 of the alanine dipeptide contains positive allosteric couplings that 
 appear to stabilize the l and c7ax conformations , and negative 
 allosteric couplings that coincide with the high regions of the / 
 free energy landscape .
on the basis of the formalism we developed , 
 we were able to attribute features of this thermodynamic coupling 
 function to specific interaction energy terms , thus allowing interpretation 
 of the allosteric landscape .
it is important to note that the criterion 
 introduced here for determining whether a specific interaction mediates 
 an allosteric coupling is more rigorous than our previous n - body information - based criterion . while the 3-body information between three cvs is in fact a function 
 of biased thermodynamic coupling functions ( see appendix , eq 46 ) , if z(r ) is 
 conditionally independent of x(r ) or y(r ) given the other cv , the 
 3-body information will be maximal .
consequently , the 3-body information 
 does not permit to determine definitively whether z(r ) mediates a 
 thermodynamic coupling between x(r ) and y(r ) , or if one of the cvs 
 mediates a thermodynamic coupling between z(r ) and the other .
specifically , 
 the 3-body information criterion will include some number of false 
 positives ( as we have previously described ) , whereas all structural features that correspond to a potential 
 energy term and have a two - dimensional influence on the thermodynamic 
 coupling function can be considered to be effective mediators of the 
 thermodynamic coupling .
the concepts developed here are very 
 general and are applicable 
 to larger molecular systems , provided enough sampling is available 
 and the functionally relevant cvs are known .
this second condition 
 is especially noteworthy for cases in which a complete functional 
 description involves multimolecular considerations .
for example , to 
 apply this method to a specific gpcr , one must first identify the 
 cvs that best describe the agonist binding process and the g protein 
 activation process .
however , in order to understand agonist - induced 
 activation of a g protein by a gpcr , one may need to also understand 
 the thermodynamic coupling between the agonist binding cv and additional 
 cvs that describe the process of g protein binding to the gpcr as 
 well as the process of g protein activation .
our new theoretical formalism 
 and its computational implementation remains applicable despite such 
 complications , and can serve as a powerful tool in understanding the 
 molecular mechanisms of the many proteins in which allostery is essential 
 to biological function .
it has the potential to identify novel allosteric 
 sites that modulate functionally important reaction coordinates , and 
 such capabilities can help achieve a large variety of end points . 

examples include the design of novel therapeutic agents that allosterically 
 modulate their specific targets in new ways , as well as the elucidation 
 of allosteric mechanisms to guide the design of novel , synthetic allosteric 
 proteins .
the alanine dipeptide ( n - acetyl - alanine - n-methyl amide ) was modeled with the all - atom charmm22 * 
 force field and solvated in explicit 
 tip3p water molecules .
charmm22 * was 
 chosen as it is able to reproduce an accurate alanine dipeptide free 
 energy landscape without utilizing the cmap term used by other force fields .
we chose to avoid force fields 
 using the cmap term as it induces a trivial thermodynamic coupling 
 through a direct interaction between  and  , rather than 
 allowing it to emerge from separate terms of the traditional potential 
 energy function .
molecular dynamics simulation were performed using 
 the charmm port in the gromacs 4.5 program with particle - mesh ewald treatment of electrostatics and lennard - jones interactions 
 switched off between 10 and 12  .
similarly to our previous study on dipeptides , enhanced sampling was achieved with driven adiabatic 
 free energy dynamics ( dafed ) , also known as temperature 
 accelerated molecular dynamics ( tamd ) , 
 implemented in the plumed plugin .
two 
 collective variables ( cvs ) , defined as the backbone dihedral angles 
  and  , were coupled ( harmonic constant 1000 kj / mol / rad ) to heavy fictitious particles ( pseudomass 50 amunm / rad ) held at temperature ts = 600 k by generalized gaussian moment thermostats ( order 
 2 ) .
simulations were conducted in five 
 independent replicates of 50 ns each after a standard equilibration 
 phase starting with independent initial velocities .
free energy surfaces 
 ( fess ) in the ( , ) plane were reconstructed using the reweighted histogram smoothed with 
 multivariate gaussian kernel regression in matlab ( release 2015b , 
 the mathworks , inc .
a cutoff 
 of 40 kj / mol was used for the fess , above which sampling was too poor 
 for reliable surface estimation . in principle , estimating an 
 observable from a dafed / tamd simulation 
 requires binning the observable values in the cv space , and reweighting 
 each bin by a function of the fes at this point .
however , a(x , y ) in eq 16 depends only on the probability mass at 
 300 k in the cv space , p(, ) . this can be derived directly 
 from the density obtained from the dafed / tamd simulation , padb(, ) , by rescaling and renormalizing,32due to the surface smoothing steps , propagation 
 of uncertainties is not practical for estimating confidence intervals 
 on the allostery landscape
instead , we use the bootstrapping approach . specifically , because observations from md time 
 series are notoriously not independent , we use block bootstrapping ; that is , we generate artificial samples by drawing 
 at random ( with replacement ) segments of trajectory of 1 ns in length . 

then , for each bin in the ( , ) plane , we estimate a 95% 
 confidence interval for the allosteric coupling function and for the 
 ac based on the standard deviation among the bootstrapped samples . 

if in a given bin this confidence interval includes the value zero , 
 the existence of an allosteric effect can not be assessed with certainty 
 in this bin and we represent it in a greyed - out color in panels c 
 and d of figure 1 and 
 in figure 2 .
 allostery plays a fundamental role 
 in most biological processes . 
 however , little theory is available to describe it outside of two - state 
 models . here 
 we use a statistical mechanical approach to show that 
 the allosteric coupling between two collective variables is not a 
 single number , but instead a two - dimensional thermodynamic coupling 
 function that is directly related to the mutual information from information 
 theory and the copula density function from probability theory . on 
 this basis , we demonstrate how to quantify the contribution of specific 
 energy terms to this thermodynamic coupling function , enabling an 
 approximate decomposition that reveals the mechanism of allostery . 
 we illustrate the thermodynamic coupling function and its use by showing 
 how allosteric coupling in the alanine dipeptide molecule contributes 
 to the overall shape of the / free energy surface , and 
 by identifying the interactions that are necessary for this coupling . 
Introduction
Theoretical Developments
Application: The Alanine Dipeptide
Conclusions
Methods