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Probing the Thermodynamics of Competitive Ion Binding Using

Minimum Energy Structures

David M. Rogers and Susan B. Rempe

Center for Biological and Materials Sciences, MS 0895, Sandia National Laboratories,

Albuquerque, New Mexico 87185, USA

Abstract

Ion binding is known to affect the properties of biomolecules and is directly involved in many

biochemical pathways. Because of the highly polar environments where such ions are found, a

quantum-mechanical treatment is preferable for understanding the energetics of competitive ion

binding. Due to computational cost, a quantum mechanical treatment may involve several

approximations, however, whose validity can be difficult to determine. Using thermodynamic

cycles, we show how intuitive models for complicated ion binding reactions can be built up from

simplified, isolated ion-ligand binding site geometries suitable for quantum mechanical treatment.

First the ion binding free energies of individual, minimum energy structures determine their

intrinsic ion selectivities. Next, the relative propensity for each minimum energy structure is

determined locally from the balance of ion-ligand and ligand-ligand interaction energies. Finally,

the environment external to the binding site exerts its influence both through long-ranged

dispersive and electrostatic interactions with the binding site as well as indirectly through shifting

the binding site compositional and structural preferences. The resulting picture unifies field-

strength, topological control, and phase activation viewpoints into a single theory that explicitly

indicates the important role of solute coordination state on overall reaction energetics. As an

example, we show that the Na+ →K+ selectivities can be recovered by correctly considering the

conformational contribution to the selectivity. This can be done even when constraining

configuration space to the neighborhood around a single, arbitrarily chosen, minimum energy

structure. Structural regions around minima for K+- and Na+-water clusters are exhibited that

display both rigid/mechanical and disordered/entropic selectivity mechanisms for both Na+ and

K+. Thermodynamic consequences of the theory are discussed with an emphasis on the role of

coordination structure in determining experimental properties of ions in complex biological

environments.

1 Introduction

Practical models of solute behavior commonly involve the concept of a division between

local and non-local response of the solution medium. The ubiquitous application of these

ideas in such diverse fields as inorganic chemistry of coordination complexes, ligand

binding in biological systems, solvent cage-mediated chemical reactions, proton conduction,

spectroscopic response of dissolved solutes, and other condensed phase problems is well

known. The common element shared by these models is the (often hidden) assumption that

the exact configuration of the local solvation structure can be regarded as fixed. Such an

assumption appears directly at odds with the statistical mechanical picture asserting that

“Anything that can happen will (with a probability related to the free energy required).”

Correspondence to: slrempe@sandia.gov.

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Published in final edited form as:

J Phys Chem B. 2011 July 28; 115(29): 9116–9129. doi:10.1021/jp2012864.

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Despite this apparent contradiction, the local configuration point of view has played an

historically important role. Two foundational questions for such a theory are the calculations

of reaction free energies for inserting or transforming ionic solute molecules in fixed

chemical environments and its twin, transforming chemical environments with fixed solute.

The former appear as solute selectivities for particular environments, while the latter appear

as weighting factors for prediction of solute properties in heterogeneous solutions. Methods

for calculating ionic stability constants will increase our capacity for design and mechanistic

analysis of new materials.

The evolution of local configuration methods addressing these questions can be traced back

to a pairing of the rigid rotor, harmonic oscillator approximation of quantum mechanics

(QM) with continuum solvation models.1 Next, methods for including continuum solvation

directly into the quantum calculations2,3 were developed. In order to progress further and

correctly capture stabilization due to charge transfer or ligand field splitting effects, it

becomes necessary to include explicitly the molecules comprising the first solvation shell.

However the entropic consequences of choosing explicit ligand positions in this approach

have remained poorly understood, as discussed recently in Ref. 4. Even more recently, it has

become possible to incorporate a local treatment of quantum effects via a hybrid quantum

mechanical/molecular mechanical potential energy function,5,6 or to correct the results of a

classical molecular dynamics simulation using high-level QM calculations on sampled

structures.7–10 Clearly, a careful consideration of the thermodynamic consequences of

altering structural constraints is required in order to transition between traditional views on

coordination complexes and recent selectivity arguments for labile coordination structures.

The apparent contradiction between thermodynamic analyses based on local configurations

compared to mechanisms avoiding such structural references has also sparked debate.

Several general mechanisms have been proposed by which an isolated binding site

embedded in a dielectric environment (including vacuum) may be designed to achieve

selectivity for a larger potassium over a smaller sodium ion. A structureless, statistical field-

strength viewpoint can be constructed by analyzing changes in ion selectivity induced by

changes in the balance of ion-protein, ion-water, water-water, water-protein, and protein-

protein interactions.11 The resulting mechanism emphasizes shifting chemical interaction

strengths to control selective ion binding. However, the balance between these effects is not

intuitive and must be arrived at from separate analysis of each binding site composition.

This analysis leads to strongly selective sites in only a handful of cases.1 A structureless

statistical viewpoint can lead to emphasis on a particular energetic contribution (for example

the dipole moment contribution from carbonyl chemistry13) or the importance of binding

energy fluctuations.14 These may also be relevant only in specific circumstances.

Another set of general mechanistic arguments for designing selective ion binding sites have

been constructed based on a local configuration viewpoint. While this view shows the

importance of local chemical structure, it encounters the opposite difficulty of rigorously

and explicitly including application-specific long-range environmental effects. For example,

a local configuration approach has shown that both water and carbonyl chemistries can be

potassium selective if equivalent coordination constraints are imposed and that the absence

of such constraints annihilates selectivity.15 But applying this result in practice requires

computing the external field determining the binding site composition and geometry16,17

and presents a significant challenge. Qualitative models of the external field used recently in

combination with minimum energy binding site models have shown the important role of a

1Of the 1077 binding sites considered in Ref. 12, the average selectivity is near zero with a standard deviation of about 2 kcal/mol.

While strongly K+ selective sites were correctly identified by weak ion-ligand attraction and strong ligand-ligand repulsion, a similar

comparison for Ba2+ selectivity was notably missing.

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local pocket characterized by decreased external dielectric in muting the free energy cost for

concentrating ligands into a binding site. Such local environments provide a source of

constraint on binding site composition and structure that shifts the binding site equilibrium

toward more selective states, characterized by over-coordination or specific cavity size.

Conversely, an environment characterized by higher dielectric imposes a free energy cost for

concentrating ligands into a binding site. It can thus produce ion binding architectures like

those preferred by each ion in bulk water, reducing selectivity.18–20 While qualitative

models of the environment have lead to general hypotheses for tuning selectivity in binding

sites with externally specified ligand chemical potential, detailed analysis of the

environments around specific binding sites still proves difficult.

Despite the challenging requirement for computing the external influence on binding site

composition and geometry, the local configuration method has been applied in several

quantitative studies. Consideration of minimum energy ion-water clusters in gas phase has

led to highly accurate free energies of aqueous hydration for several ions.21–23 Statistical

configurational sampling in all-atom ab initio molecular dynamics simulations has

demonstrated the overall thermodynamic equivalence between local structural and non-

structural approaches to ion solvation in homogeneous water environments.24,25 The work

described above18 represents initial efforts in method development and QM-based

application. In a joint article,26 several groups have outlined a rigorous and computationally

feasible method for extending these results to inhomogeneous phases such as protein

binding sites, surfaces, and solution interfaces.

To permit investigations of the behavior of coordination complexes, we aim to resolve the

above issues in applying local configuration methods. The simplest path forward is to

separate out the contributions of field strength, local chemical structure, and environmental

influences in a rigorous way. In this report we provide such a decomposition, in the process

achieving a bird’s eye view of coordination complex formation and solvation through the

use of an appropriate thermodynamic cycle. This allows us to identify the influence of each

factor on the interplay of structure and energetics that determine chemical design principles

and solution behavior of metal coordination complexes in inhomogeneous environments.

1.1 Force Field Considerations

As a preliminary step for the computational study of ion coordination structures, it is

necessary to verify the molecular energy function used for ion-ligand clusters. Traditional

fixed charge force field models fail in this regard,27 consistently over-estimating the number

of ligands able to occupy cation-water first solvation shells.28 In this report, we employ the

AMOEBA polarizable force field.29 This force field has been shown to capture the structural

and energetic features of principal ion-water coordination structures.30

Over-crowded inner shell structures typically observed using traditional force field models

can lead to qualitatively different conclusions about structure vs. function relationships. For

an example, consider the set of gas phase minima containing waters inside the first solvation

shell of Na+ and K+ ions. For a compromise between the first minimum of the respective

ion-oxygen radial distribution functions, we count only configurations with water oxygens

closer than 3.1Å to the ion. One of the most widely used non-polarizable force field models

for ion-protein interactions, CHARMM,31 finds [1,1,1,1,1,9,4, and 3] different minima for

Na+-water clusters with 1–8 waters (respectively), whereas AMOEBA finds [1,1,1,1,4,4,0,

and 0]. The corresponding numbers for K+-water clusters with oxygens inside a 3.1Å cutoff

are [1,1,1,1,3,9,6, and 7] (CHARMM) vs. [1,1,1,2,5,8,4, and 4] (AMOEBA). In addition,

there is accumulating evidence for the argument that ligand polarization and possibly even

charge transfer are required for a correct description of ion-ligand interaction potential

energy surfaces. These effects are important for competitive ion binding thermodynamics,

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especially when asymmetric cluster structures are involved.10,32–34 Such effects are best

described using a QM approach that explicitly treats valence electrons responsible for

polarization and charge transfer. The quasichemical theoretical approach described below

makes this possible.

1.2 Division into Local Configurations Using Quasichemical Theory

A thermodynamic approach for incorporating QM appears in quasichemical theory (QCT).

QCT shows the connection between ion-coordinated and unrestrained local configuration

points of view by exactly defining the requirement for formation of a solvation structure and

its thermodynamic consequences. This connection is formally incorporated into Gibbsian

statistical mechanics by introducing constraints on the allowed system phase space:

Cn: The subset of allowed system configurations is limited to those in which a

molecular complex A·Xn is formed.

The ambiguity of “the complex is formed” must be resolved in practice with an indicator

function, I(x;Cn), which is one whenever a particular point in phase space satisfies specific

local structural requirements, and zero otherwise.

This structural division bears a close resemblance to the Stillinger-Weber inherent structure

model for liquids.35,36 In that model, a similar constraint was formulated for each global

minimum of the liquid. Thermodynamic properties of the entire system could then be

written as sums over these structural states and approximations used to estimate partition

functions near energy minima. Here instead we define a constraint for each coordination

complex of interest. The present definition does not require the presence of a potential

energy minimum. Applying this division to flat potential energy surfaces gives a

conceptually simple method for calculating the entropic consequences of choosing explicit

ligand positions. This generalization is also appropriate for discussing chemical reactions

occurring in arbitrary, inhomogeneous environments due to the natural appearance of the

environmental potential of mean force when calculating partition functions with

coordination constraints.

The local, structural division of QCT permits a discussion of solute properties in terms of

alternative coordination states. Each coordination state has an intrinsic set of ion selectivities

and spectroscopic properties given by a constrained average. External constraints (for

example protein conformational preferences, bulk system composition, applied membrane

electric field or surface tension, pressure, etc.) determine the overall equilibrium between

alternate states in the form of a probability for formation of each state. For an example, we

may choose a definition of C4, call it C4(2.3), selecting only states where exactly two

chlorides and two water oxygens are within 2.3Å of a pre-defined center. This condition is

appropriate for a distinguishing states of a platinum coordination complex in solution.37

Alternately, we may choose a definition appropriate for a sodium ion coordination complex,

say C4(tet), which is satisfied when four water oxygens are arranged tetrahedrally (with

some RMSD tolerance). Or we may choose a constraint appropriate for the S2 site of the

celebrated potassium-selective ion channel protein, KcsA,38 where 8 carbonyl oxygens are

located within 3.1Å of a K+ ion. In any case, the observed system properties are a function

of the averages conditional on the formation of each Cn and the set of probabilities ℘ (Cn) in

an obvious way.

At this point, the non-structural thermodynamic and local configuration viewpoints may

diverge for the following reason. Some ionic binding sites display a strong tendency toward

local order (for example a solid crystal), whereas others are not subject to strong

environmental constraints and become disordered. Associating the former with a single

minimum energy structure seems natural but need not be the only recognized type of

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structural order, while the latter disordered state may be, but is not necessarily, characterized

by a conformational space containing many minima. Therefore a given definition of a

coordination state may contain multiple local minima and be realized with high probability

without contradiction. Practical investigations usually involve calculating the relative

energetics and properties of several coordination states. These need not contain only one

minimum energy structure, but instead represent a partitioning of phase space into

alternative regions of ion-complexed and uncomplexed configurations as appropriate to the

experiment at hand.

1.3 Approximations to Quasichemical Theory

Arbitrary divisions of configuration space are allowed through the definition of Cn. A

consequence is that some region of configuration space can always be found that dominates

the thermodynamic average by containing essentially the whole region of structures sampled

during a given process (that is, any set of connected states in a thermodynamic cycle).

Although not necessary in the present framework, the rigid rotor, harmonic oscillator

(RRHO) approximation is often employed in QM thermodynamic calculations. However,

using this approximation requires making a rigorous connection between coordination states

and coordination structures by verifying that the specified constraint contains a single

minimum energy structure and is well described by a harmonic expansion of the potential

energy surface. A primitive approximation to quasichemical theory (QCA)21–23,39 assumes

that a stiff coordination structure is formed with high probability, and accordingly makes

both the RRHO and thermodynamically dominant, or maximum term, approximations.

Despite the many advantages of the structural view outlined above, there exist serious

criticisms of the idea that constrained averages could yield a correct free energy.40 The

argument is that any given coordination state, Cn, may have multiple minimum-energy

structures for its solvation shell. The existence of multiple minima invalidates free energy

calculations based on a single structure (that is, the RRHO approximation), and therefore

minimum energy local solvation structures cannot play any role in modern statistical

mechanical theories. As an example, Ref. 40 considered the definition Cn(3.5), where the

ion-water complex is taken as formed and the condition satisfied whenever n water centers

are within 3.5Å of an ion fixed at the origin. This condition omits consideration of ligand

structure inside this constraint (that is, whether or not the waters directly coordinate the ion

in a chemical sense). They proceeded to test the validity of a simplistic approximation to

solvation free energy that neglects thermal and entropic contributions, ΔEmin ≈ ΔG, and

found a deviation of 1–2.5 kcal/mol. Related problems were noted when considering

minimum energy structures in complicated biological environments, the worst of which is

the apparent unavailability of a simple answer as to which bound ion should be used during

the minimization process for studying competitive Na+ / K+ binding in the KcsA selectivity

filter (where most authors have chosen the endogenous ligand).

In view of these recent studies, confusion clearly exists about the thermodynamic

consequences of placing constraints on configuration space. The results of this report will

show conclusively that the complex formation free energy can be calculated based on any

desired constrained state when the free energy for binding site rearrangement is included.

Until this point the free energy cost for placing “microdroplet” half-harmonic constraints

(chosen, for example, to match the upper bound of binding site fluctuations) has been

ignored by some authors. The reason this omission usually leads to correct qualitative results

is that there is a negligible cost for constraining a system to the most probable biological

state. However, because the cost for constraining the binding site conformation is incurred

by the environment in order to produce a selective chelation complex, it is a critical element

in understanding solutions nature has developed to solve the design problem.

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By recognizing this constraint energy, QCT additionally produces meaningful conclusions

about individual coordination (that is, minimum energy) structures. The free energy

requirement for binding site rearrangement also answers the question of energy

minimization protocol for simple Na+/K+ selectivity questions – the coordination geometry

arrived at must closely resemble the native environment produced by the channel selectivity

filter. This requirement is, in fact, achieved by four glycine dipeptide strands when energy

minimized in the presence of K+, but not Na+.18

This report begins by introducing a statistical mechanical notation for thermodynamic cycles

that facilitate discussion of ion binding thermodynamics. We then give two thermodynamic

cycles for calculating solvation free energies based on QCT ideas that permit use of gas-

phase QM calculations for cluster formation energies. The first is applicable to

homogeneous solutions, and a small adaptation gives the second, which is applicable to non-

homogeneous environments such as enzyme binding sites and spatially constrained potential

of mean force calculations. We will use chemical equilibria to show the physical meaning of

each term that appears. The resulting theory is applied to the problem of calculating binding

site formation free energies. From our decomposition of the free energy of ionic selectivity,

we find that 1) Individual coordination structures (that is, containing only one energy

minimum) are natural chemical objects for carrying out inner-shell reactions; 2) Each

structure reproduces the full cluster formation free energy when the probability of

coordination structure formation is taken into account; 3) The approximation error of the

maximum term method used in QCA is rigorously bounded by the disorder of the ion

+binding site complex; and 4) Conditions can be found under which several useful

approximations will hold, making the theory amenable to computation in ion-biomolecule

binding sites.

The discussion will combine these ideas of chemical reaction constants into an insightful

picture of the competing roles of intrinsic binding site structure and solute binding

energetics in the determination of solvation free energies. An important conclusion is that

local solvation structures are both intuitively and computationally simple physical objects of

modern thermodynamic theory, appropriate for analysis of competitive ion binding in

enzyme active sites.

2 Theory

Useful, intuitively simple, derivations of QCT can be expressed in terms of thermodynamic

cycles.26,34,41 To form a compact notation for such expressions, we need only a language for

specifying the form of the partition function, Z, for each state in the cycle. Free energy

changes between states A and B are then given by the conventional expression

. A natural alphabet for describing a particular partition function is the set of

its constrained averages, such as energy (〈H〉), volume (〈V〉), or particle number (〈N〉), as

well as any constraints on the state space, for example the formation of a coordination state,

Cn. For ease of exposition, the location and conformation of the solute, X, is taken as fixed

throughout this report2.

As an example, a coordination complex in a solution at constant temperature and pressure

(state E of Fig. 1 and 2) has a partition function given by constrained average energy, 〈HN +

HX + ΔH〉, volume, 〈V〉, and coordination, Cn, yielding

2This condition should accordingly be included in all the partition function specifications, but is implicitly understood here.

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(1)

Here, HN and HX are the Hamiltonians of the isolated N-solvent molecule system and the

isolated solute, X, at a given point in phase space while ΔH represents the interaction

energy, ΔH ≡ HN+X − HN − HX. The prefactor removes degenerate configurations to

account for the indistinguishability of solvent molecules other than the ligand molecules

specifically labeled by Cn. The last term in the integral is the indicator function taking on the

value one or zero, depending on whether or not the coordination state is occupied.

For solute transfer reactions between environments, it becomes useful to invoke several of

these systems containing differing chemical species and constraints. In this case, a slight

abuse of the notation in Eq. 1 to center it around the identity and location of individual

molecules re-casts it in familiar chemical terms. This is done by writing each molecule

present in the calculation along with its physical location (that is, the “box” in a Grand

Canonical Monte Carlo simulation), implying an average energy constraint in each box.

Other constraints such as an average (V̄, constant pressure) or fixed volume (V), constraints

on local solvation structures, Cn, dipole moments, chemical potential, N̄, etc. should be

explicitly specified for each box. Thus the partition function of Eq. 1 could also be specified

by

or, equivalently,

This notation provides a consistent and useful way to state free energy cycles involving

coordination constraints as long as boxes and molecules are not created or destroyed along

the way. These are understood as present in all steps of the process, though they may not

contain any molecules or interact.

Using our notation, the reaction defining solvation free energy is,26

Here the starting position of the solute is formally a vacuum3, and has been given a

superscript to distinguish it as contained in a separate volume from (and thus non-interacting

with) the constant pressure solution, V̄.

2.1 Homogeneous QCT Process

We can proceed to write down a thermodynamic cycle mirroring the original formulation of

QCT for solvation in homogeneous solutions.42 The result is shown in Fig. 1. A direct, brute

force, thermodynamic integration (TI) or free-energy perturbation (FEP) can usually be

carried out to get the solvation free energy. It is usually not recognized that the results of

3Because the solute location is fixed, the partition function is just exp[−βEX].

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FEP calculations can be separated into local contributions, as shown in Fig. 1, to make

conclusions about specific coordination states. An added benefit of the formulation is

isolation of a chemical association step in gas phase, permitting efficient application and

comparison with quantum methods that account for non-pairwise-additive effects such as

polarization and charge transfer.32 The intermediate stages are shown schematically by

enclosing non-interacting molecules with dotted lines and indicating coordination and

excluded volume conditions by shading. Each step has a useful chemical interpretation.

Approximations used in practical calculations may also be expected to cancel partially

between left (downward) and right (upward) arrows, which correspond to similar processes.

2.1.1 Ligand Extraction, (A→B)—Starting from the non-interacting system in the upper

left, ligand molecules (Ln) forming the inner coordination shell are first removed from

solution into separate, labeled, non-interacting volumes, Vbn. The free energy cost is

determined by their solvation free energies (from a standard state at density Vb−1),

(2)

These quantities are usually available experimentally and have been tabulated for many

important solvent force field models. Note that if the ligands were not individually

distinguished in B, this step could be done in

useful to label each ligand/solvent molecule in both state B and wherever the coordination

possible ways. In practice, it is more

condition Cn is applied, leading to a degeneracy factor change of

reaction A→B and corresponding addition of lnn! in step E→F (Eqns. 2 and 3). Taking the

in

thermodynamic limit in the above, generates the solvent density, ρH2O.

2.1.2 Binding Site Formation, (B→C)—The next contribution (B→C) comes from the

likelihood that the given coordination geometry is formed in the absence of interactions with

the solute ion (denoted by the zero subscript). Arranging the (now non-interacting) ligands

into their inner-shell (IS) binding geometry, Cn, gives rise to an entropic contribution,

−nlnVIS/Vb. This is because of the change of volume to

translational constraints.

for purely

In addition, we can choose the inner-shell condition to restrict the surrounding solvent to lie

outside the specified coordination complex. Logically, the indicator function I(Cn) factors

into a constraint on the ligand geometry times a “non-occupancy” constraint C0 on the

surrounding solvent. The corresponding free energy comes from

the probability that the rest of the solution obeys the coordination condition. This term is the

origin of packing contributions to the hard-sphere solvation free energy,41 and depends on

the exact constraint geometry.

Note that the above choice to include the solvent cavity is made throughout this report in

order to make a consistent comparison to the equations presented in Ref. 26 (corresponding

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to the thermodynamic cycle in section2.2). However, if the constraint Cn ignores the

configurations of the surrounding fluid, then the −ln℘0 (C0) packing term can be moved to

step D→E to make a more naturally defined

of a definition for the solvation free energy of the cluster that excludes the packing term,

. In this way, figures 1 and 2 make use

2.1.3 Chemical Association / Complex Formation (C→D)—Because the ligands are

not interacting with the rest of solution at this stage, the chemical binding step (C→D) may

be carried out in vacuum. The chemical interpretation of the free energy for this step

remains well-defined because of the constraint on the locations of the ligands. In addition,

this step usually produces the dominant contribution to the total thermodynamic cycle and

thus carries the most relevance to selectivity of like-charged ions in the given coordination

state.18 Importantly, the rigorous separation of the solvation process has made this term

amenable to high-level ab-initio calculations.43

Since state B contains n non-interacting ligand molecules, each in a standard state volume

Vb, we may alternately envision forming a cluster from these molecules by immediately

combining their volumes into a new container, Cn. The path directly from B to D thus

represents a gas-phase equilibrium constant,

It should be noted that this equation explicitly shows how the magnitude of the inner-shell

volume VIS vanishes by construction in the complete cycle and is not essential to solvation

free energies.

2.1.4 Solvation of the Complex (D→E) and Spontaneous Complex Formation

(F→E)—After the complex is formed in D, it is re-introduced into the rest of solution,

paralleling the first step in the cycle. Because the solute location and coordination structure

are fixed during D→E, only small changes in the ligand distribution within the coordination

state Cn are expected. Finally, the constraint on the solute binding geometry is removed in

the presence of solution in step E→F, with free energy corresponding to

(3)

Here we have defined the coordination condition

any of the n! ligand orderings, and noted that its probability is n! times larger.

corresponding to Cn, but satisfied for

These probabilities are observables from an equilibrium system at state F. They indicate the

relative propensity for chemical association and formation of each type of complex in that

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solution. For example, suppose a metal ion is introduced between states A and F, and the

solvent for both states is composed of an aqueous mixture of two different chelation

compounds. The metal excess chemical potential, A→F, will be composed of contributions

from both possible chelation reactions. We label each coordination complex by separate a

geometric constraint, giving rise to two cycles (because of the two different bound states E).

Since the individual paths A→B→C→D→E→F must give the same free energy as the

overall reaction A→F, the proportion of metal bound to each complex (E→F) is completely

explained by variations in the binding energetics for each coordination complex, A→E.

For comparison with prior literature, the traditional gas-phase partition coefficient appearing

in step B→D is written in terms of indistinguishable ligands (C’n),43–46 so that it includes a

degeneracy term for the coupled state.

2.2 Non-homogeneous QCT

Figure 2 illustrates a recently developed variation of QCT applicable to non-homogeneous

solutions.26 Similar in spirit to Fig. 1, Fig. 2 contains at its base a cluster formation free

energy. Formation of the cluster can be carried out in the presence of the surrounding

medium, as described by previous authors26

– analogous to ΔG(Cn, QCT) of Fig. 1.

, or in an isolated environment

The principle difference from previous versions of QCT is the change in the ordering of

ligand removal and pre-arrangement of the coordination structure, Cn, so that only steps B

and C are altered. In A→D of Fig. 1, the ligands are individually removed from a

homogeneous fluid, grouped together in the (non-interacting) gas phase, and then all

intermolecular interactions are turned on simultaneously. Steps A→D of Fig. 2 first form

the coordination structure, Cn, then remove its interaction with the surrounding medium to

bring it into the gas phase before turning on solute-to-ligand interactions. The present cycle

therefore separates out thermodynamic contributions from ligand-ligand interactions

implicitly in step A→B, while Fig. 1 counts both ion and ligand interactions in C→D.

An important thermodynamic component is included in the first step of both cycles; namely,

a concentration contribution arising from the probability of finding the ligands in a given

neighborhood around the solute. Thus the solvent density dependence of cycle 2 contributes

to this probability, ℘0 (Cn), which will be identical to the concentration term of cycle 1

in the limit of negligible ligand-ligand strain force.

3 Methods

In order to achieve converged Na+ →K+ ion mutation free energy estimates, free energy

perturbation (FEP) calculations were carried out between states A→F (Fig. 3) using the

AMOEBA force field by linearly interpolating the ionic Lennard-Jones radius, well depth,

and polarizability – as was done in Ref. 30. We used 4 steps in-between Na+ and K+

parameters (for a total of 6 states). Hamiltonian replica exchange was utilized to improve

sampling for these calculations. Each 6-state simulation was run for 1.5 hours using a

version of Tinker47 modified for parallel replica exchange and was able to collect at least

2000 samples spaced 1 ps in time.

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Samples from all FEP intermediates were tested for satisfying each coordination constraint.

This provided a direct estimate of the coordination probabilities (D→F), as well as a set of

structures used to calculate conditional free energy differences (DNa+ →DK+ of Fig. 3)

between (and expectation values at) constrained states. Data sampled from all FEP

intermediates were efficiently combined using weights provided by the multi-state Bennett

acceptance ratio (MBAR) technique.48 In addition, one set of duplicate simulations were

carried out in order to verify the convergence of free energy estimates to within the stated

error calculated by MBAR. In all cases, the difference in free energy estimates came in very

close to or below the stated uncertainties, and the average between both final estimates is

reported in Fig. 6. The error-bars shown are one standard deviation, as estimated from the

first simulation.

3.1 Structural Classification

All ion-water cluster structures were classified according to a set of minimum energy

structures for each n in the following way. The ion position was fixed at the origin, and

hydrogen positions for each water were averaged to yield 2n comparison points. Root mean

square deviations (RMSD-s) were computed between the test and reference structures for all

possible water permutations of the test structure. The total number of points used for

normalizing the RMSD was 2n + 1. Test structures were clustered to the reference structure

with the lowest RMSD, and labeled by that RMSD value. The coordination condition thus

arrived at is,

Cn(x;ci,RMSD;{cj}): The structure x has lowest RMSD to ci among all cj and that

RMSD is less than the cutoff value.

This coordination condition depends on the set of reference structures used for comparison.

We arrived at a set of minima for each ion by energy minimizing and superimposing each

frame from our initial 1 ns dynamics calculations. Minima with an RMSD of greater than

0.3Å were added as new reference structures, while only the lowest energy structure was

retained if two fell within 0.3Å RMSD. The introduction details the number of minima

found within 3.1Å. The population of structures closest to each minimum was then

determined from the samples taken at thermal equilibrium and the minima with thermally

averaged oxygen-ion distance larger than 3.1Å removed from further analysis. The set of

sodium minima were mapped to the set of potassium minima to find corresponding

structures with reasonable RMSD and which had occupancy above 5% for at least one ion.

The set of potassium minima emerging from this process were used for the final

characterization presented in Fig. 4 and (a), left, and (a,b), right, of Fig. 5. There was a

single minimum for n=1–4, while n=5–8 had two (with three Na+ minima mapped non-

uniquely to two K+ structures for n=6).

4 Results and Discussion

Although they participate in much more complicated thermodynamic cycles, the primitive

objects of our theory are simple minimum energy structures, along with the region of

conformational space in their immediate vicinity. If the subset of allowed system

configurations is limited to those in the immediate vicinity of a minimum energy structure

(containing n inner-shell ligands with defined geometry), the resulting free energy change

for replacing a Na+ ion with K+ can be interpreted in a physically meaningful way. In this

section, we use Cn to denote this neighborhood and ΔF(Cn) to represent the constrained Na+

→K+ free energy change. Comparisons between ΔF(Cn) for different choices of Cn allow

mechanistic arguments based on a spatial partitioning of the free energy.

To illustrate these ideas, we have chosen to break out the minimum energy structure

contributions from the formation free energy of an ion-water cluster used in typical

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selectivity arguments.12,17,18 We will show step by step how a total selectivity (A→F)

arises from consideration of the occupancy (D→F) and conditional free energy (C→D) of

any coordination constraint through the appropriate thermodynamic cycle. Naturally,

coordination structures that occur with higher probability prove to be the most chemically

meaningful and computationally accessible, while separating out the conditional free energy

under each constraint can show the structural design principles responsible for selectivity.

The diversity of selectivities obtained in this way extends our insight well beyond a

thermodynamic integration approach that ignores structural information.

4.1 Selectivity of an Isolated Binding Site

To show the essential features of the proposed calculation, we have computed the Na+ →K+

mutation free energy for isolated ion-water clusters with n = 1–8 waters. The binding site

consists of n waters whose oxygen atoms (at distances rO from the origin) are restrained to

within a radius of Rmax = 3.1Å from the ion. This radius is an appropriate upper-bound for

the inner-shell of both Na+ and K+. The ion is fixed at the origin with a flat-bottomed,

harmonic potential

(4)

with force constant k=200 kcal/mol·Å2. The indicator function, I(·) is zero or one as defined

in the introduction. This system has the physical interpretation of ion selectivity in a water

cage embedded in a nonpolar environment.

The thermodynamic cycle analogous to Fig. 1 for this system is shown in Fig. 3. Note that

this cycle shares the gas phase coordination complex step,

common with the cycles illustrated in Figs. 1 and 2. It is universally arrived at via turning on

or mutating ion-ligand interactions in the presence of a coordination constraint and is the

hallmark of quasichemical theory. The results from this system are therefore useful as an

indication of the local contributions to selectivity in diverse ion-binding environments –

with the external potential and ligand density deciding the weighting between minima for

each system.

, in

Fig. 3 also shows the intrinsic connection between QCT and classical equilibrium theory in

that step [B] forms the basis for arbitrary gas-phase cluster formation reactions. The

constants KX·(H2O)n and KnH2O are the familiar gas-phase equilibrium constants for the

formation of ion-water and pure water clusters from an ideal gas standard state. The new

constant, KX·(H2O)n(Cn) is identical to the classical cluster formation constant except for the

explicit specification of the integration region via a constraint that the ligands must satisfy.

This step is essential for creating a local theory.

Since the choice of the coordination constraint, Cn, is arbitrary in the thermodynamic cycles

above, we should expect any partitioning of configuration space to give the same total

answer (A→F). Variations in the probabilities (℘X (Cn)) for forming each structure Cn

indicate the intrinsic conformational preferences of the ion-water cluster (X=Na+ or K+),

while variations in the conditional free energy, ΔF(Cn), give the ion selectivity belonging to

a specific coordination sub-state. For our particular criteria of RMSD from a set of reference

structures, labeled by c, ℘X (Cn(c,RMSD)) is the occupancy of a region of conformation

space around each reference structure. These occupancies indicate their relative importance

in determining the overall free energy of cluster formation.

Fig. 4 shows the components of the free energy differences corresponding to D→F and

between D(Na+) and D(K+) of Fig. 3. The constraint, fully explained in the Methods section,

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requires that the structure be closest in RMSD to a reference structure, and have an RMSD

within the cutoff. Successively larger cutoff values scale the allowed region about the

minimum from a single, minimum energy structure on the left (determined in the K+

Hamiltonian), to the whole set of structures closest to the reference on the right. For n = 1–4,

only one minimum energy structure was used, so that the right side of the plot is only

constrained by ligand number according to Eq. 4. For any value of the cutoff, it can clearly

be seen that when the probability for coordination state occupancy is properly taken into

account, the overall ionic selectivity is recovered exactly. This provides a validation of the

thermodynamic cycle (Fig. 3). Some regions of the plot are not shown, however, due to

insufficient sampling. For example, no structures closer than 0.2 RMSD to the minimum

were visited during 2 ns of sampling for the n = 6 structure. Because the free energy for

placing a constraint is the negative log of its probability of occurrence, this is expected for

constraints costing over 5 kcal/mol.

The AMOEBA force field model calculates the free energy cost for replacing an aqueous

Na+ ion by K+ at 17.3 kcal/mol,30 in agreement with experiment.49 As observed

previously,15 the lower cost of replacing Na+ by K+ in water clusters of size n = 1–8 implies

all of these water clusters are barely selective for K+ (vs. bulk water). Because of this weak

selectivity for n=6–8, relatively small conformational constraints imposed by the

environment on the metal coordination structure have the potential to shift the equilibrium in

favor of either ion. This system can thus succinctly illustrate the major features of the

selectivity debate.

4.1.1 Selectivity of a Minimum Energy Structure—First, for a single, fixed, ligand

configuration isolated from the surrounding medium, it can easily be seen that ion selectivity

(C→D) is determined solely by the simple, chemically intuitive ion-ligand interaction

strength. This energy change on replacing Na+ by K+ appears on the far left of each plot as

ΔF(Cn) at zero RMSD. In a biological environment, we must also add the response of the

protein/membrane medium surrounding the binding site

contribution can be easily approximated by a mean-field, Poisson-Boltzmann type model

and fits into the picture of ion-external environment interaction strength.18 However, if a

large-scale conformational change in the external medium occurs on binding, then the

environmental contribution to the free energy change between Na+ and K+ occupied states

would include this transition and become an interesting object for computation in its own

right.

. In most cases this

4.1.2 Structure, Energy, and Entropy—Second, allowing the binding site to explore a

small region around the reference structure moves us to the right on the ΔF(Cn) line. Here,

we begin to see entropic contributions arising from the shape of the potential energy surface

near the reference structure. This entropic contribution effectively penalizes ions requiring

more rigid coordination states. For n=1, it is apparent that the free energy cost, ΔF(Cn), is

increasing with the cutoff because the K+-water energy surface is more rigid around its own

minimum than the Na+-water surface. This same feature (ΔF(Cn) < ΔF) requires the free

energy cost for imposing the constraint C1(RMSD) on K+ to be less than that for imposing it

on Na+. The situation persists until the constraint is large enough to envelop the Na+-water

minimum (around 0.3Å RMSD) – at which point the constraint energy becomes effectively

zero. We should therefore expect this trend for all constraints based on K+-water minima.

Unexpectedly, this trend appears to be violated for n=4. Here, at R=0.7Å, the constraint

envelope is actually more easily satisfied by Na+’s 4 waters than K+, resulting in a higher

free energy when the ligand configuration is constrained to this region. The explanation is

that the K+-4 water structure at this RMSD is more diffuse than for Na+.

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4.1.3 Absence of Structural Information—Finally, at the far right of the plot, minimal

constraints are present, allowing the binding site to respond naturally to a change in its

bound ligand. At this point minimal structural information is available (that is, only the

“closest” matching reference structure), and it is no longer informative to explain the

selectivity differences in terms of structural preference. Instead, as it has been argued, the

selectivity (not considering variations in Rmax or n16,19) results from all contributions, and

has been described as a combination of ion-ligand attraction and ligand-ligand

repulsion.12,13 However, we should not ignore the cogent information that these interactions

give rise to preferences among compositional and structural states, Cn. Because each state

participates in the full equilibrium at [F] (where no constraints are present), any

decomposition is equally valid. However, each state we may define will have a specific

propensity for appearing and a specific ionic selectivity that determines its relevance.

If we choose to define our states using the number of waters placed in contact with the ion

(and further separating these into configurations near a set of minima), then we can

immediately gain structural insight into the selectivity question. Above n = 7, the AMOEBA

force field does not find any minimum energy structures for Na+ retaining all waters within

the cutoff. Furthermore, the plot for n = 8 shows the stabilizing effect of a minimum for K+,

which is itself K+ selective (vs. bulk liquid water). The present data lends support to the

current consensus view that for clusters constrained to have n = 8 inner-shell ligands with

small dipole moments, the Na+ →K+ mutation free energy is less than bulk and the binding

site selective for K+ because Na+ prefers a more ordered binding site with closer ligand

oxygens not afforded by the high ligand density at large n. This conclusion is difficult to

arrive at without structural reasoning since ligand-ligand repulsion must act via altering the

binding site conformational preferences. Indeed, interpretation of selectivity differences

through the mechanism of ligand field-strength, commonly tested by scaling/removing some

set of ligand-ligand interactions, has ultimately relied on this mechanism.50

Further evidence is provided by examining the distribution of the water oxygens within the

3.1Å constraint radius. At n = 8, the most probable configuration for Na+ is 5 waters inside

2.7Å and 3 outside. Within the set of such 5+3 configurations, the Na+ →K+ mutation free

energy is 20.9±0.1 kcal/mol – selective for Na+ over bulk water by 3.3 kcal/mol. For K+, the

most probable configuration is 1 inside and 7 outside 2.7Å, where (for 1+7 configurations)

the mutation free energy is only 11.3±0.1 kcal/mol – selective for K+ over bulk by 6 kcal/

mol. Thus, because the solvation shell of K+ is pushed out relative to Na+, the cost for

mutating Na+ →K+ decreases when internal structure is not imposed in the n=8 case. By

reasoning in the absence of structural information, it would have not be possible to quantify

these consequences of binding site architecture or understand them in the context of design.

4.2 Environmental Constraints Tuning Ion Selectivities

Because the set of coordination states were chosen based on distance from a K+-water

minimum in Fig. 4, it is not surprising to find a relatively lower cost for inserting K+ near

these special ligand configurations. This merely illustrates the structural, “snug-fit”51

mechanism for ligand selectivity (combined with over-coordination at n=8). Accordingly,

we should also expect that the ionic selectivity could be tuned in the other direction by an

environment imposing the appropriate Na+-centered constraint. Figure 5 shows the energy

decomposition for n = 4 clusters (left) and n = 6 clusters (right). The upper panels give

results for the K+-centered constraints as described previously, while the lower panel for

each (b,left and c,right) are re-calculated based only on distance from the most populated

Na+-water minimum. As expected, selectivity can also be achieved by structurally

constraining the binding site near the appropriate Na+-favoring geometry, even increasing

the Na+ →K+ cost above that of bulk (thus switching the binding site to Na+-selective).

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Direct structural control of the binding site conformation has been invoked to explain the K+

selectivity of valinomycin20,50 and the Na+ selectivity of the LeuT NA2 site.52

Although it is possible to design a metal chelator by enforcing rigidity, this is not always the

route taken by nature. As discussed in Ref. 15, constraining the environment around a

binding site to alter the probability of forming a coordination structure ℘0 (Cn) requires the

protein to incur a free energy cost during synthesis or folding because it is limiting its

conformational freedom. Taking Csite to be the region of ligand conformational space

allowed by the natural protein binding site, the free energy cost is dictated by the probability

of forming Csite in a higher entropy mutant, −ln℘unconstr. (Csite)/℘constr. (Csite). Now assume

the protein conformational space is further reduced to Cn ⊆Csite when bound to an ion. To

the extent that Csite excludes ligand conformational space required by an alternate ion, the

A→B cost may be directly translated into a selectivity gain by constraining the native state.

The appearance of −ln℘0 (Cn) in A→B of Fig. 2 thus implies that the synthesis cost must be

greater than or equal to the gain in selectivity. As a consequence of this energy balance, a

thermodynamic cost is paid at the cellular level for maintaining ion binding sites with

altered selectivities relative to the cheapest protein the cell could produce.

The discussion gives a quantitative basis to the general topological control viewpoint of a

system designed to employ specific environmental constraints to achieve selectivity.

Combined with the results above, this perspective shows that enforcing an 8-coordinate

structure is highly desirable for excluding Na+ from cation binding sites. Since Na+ does not

have stable 7 or 8-coordinate structures, the binding site may retain favorable flexibility

while biasing the conformation well away from Na+-selective regions. However, at lower

coordination, the binding site must use a rigid/mechanical mechanism to constrain its size or

shape in order to get away from structures favored by another ion.

The intrinsic conformational preferences of an isolated reference binding site including only

an ion-ligand distance restraint determine the “base case,” and can be rationalized as arising

from a combination of ion-ligand and ligand-ligand interactions. However, for definiteness,

this idea must also be accompanied by a calculation of those conformational preferences.

Ideally this would result in listing out individual reference structures, their relative

populations, and how they may be influenced by the environment. Without such information

it becomes difficult to apply structural reasoning consistently, as can be gleaned from the

discussion in Ref. 50. By failing to address the thermodynamic consequences of altering

structural constraints, recent discussions53 have not improved this situation. If, for example,

a hypothetical constraint is introduced on the positional fluctuations of the ligand atoms, its

effect is to shift the binding site equilibrium toward more rigid structures at a free energy

cost to the environment. The thermodynamic cycles introduced here place environmental

constraint and selectivity contributions on equal footing, and unambiguously show the

dilemma faced by nature of using all available constraints on binding site composition and

structure for optimizing selectivity.

Ion binding site conformations may be restrained by a variety of interactions with the

environment. As discussed at length in Refs. 20 and 17, these include conformational

restraints on ligand number, position, and water accessibility. The net function of these

restraints is to shift the ion-unoccupied and occupied coordination state probabilities ℘0 (Cn)

and ℘X (Cn) from their reference state, taken in this report as an isolated gas-phase cluster.

If ligand exchange is possible, the reference states A and F should include variability among

ligand numbers, n, at equilibrium with bulk solution.16,20 In this case, the equilibrium

between binding site compositions present at state F can be analyzed by considering steps

A→D for each composition. These must combine (in the same way shown in the last

section) to give the equilibrium propensities at state F for each coordination, ℘X (Cn).

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Longer-range interactions also play an important role in determining the distribution of ion-

binding conformations by altering the penalty for ligand extraction and complex

solvation.18,19 For example, around 4 inner-shell waters (near the distance defined by the

first maximum of the radial distribution functions) are preferred by both Na+ and K+ in

higher, water-like, dielectric environments. However, at lower dielectrics, Na+ prefers an

inner-shell coordination of 6 waters, whereas the larger K+ is most stable at n = 8. Other

features of protein environments will also play a role in shifting the stability of each

coordination structure. As an example, nearby charges that produce a net electric field will

shift the energy levels (and Boltzmann weighting) of polarized cluster structures.

These important effects are not usually probed because of the computational difficulty of

employing thermodynamic integration or FEP for changing binding site composition.

However, these effects can become amenable to study with properly validated

approximations.

4.3 Maximum Term Approximation

In certain circumstances, the strong ionic charge can give rise to highly rigid local

coordination geometries. Because our thermodynamic cycles are invariant to the choice of

Cn, it may make sense to limit investigation to the region around the most probable structure

in this case.

As has been noted previously42,43 it is possible to obtain an absolute solvation free energy

using a single coordination state because only the coupled probability (℘X (Cn)) is required

in the thermodynamic cycle (Fig. 1). Therefore, if the contributions A→E are obtained for

each of a set of mutually exclusive and exhaustive states Cn, then the missing contribution

from each is just the (necessarily negative) quantity βΔFE→F = ln℘X (C′n). The most

favorable ΔFA→E therefore comes from the C’n with highest probability, ℘X (C′n). As long

as this highest coordination probability is close to unity, it is appropriate to utilize the

maximum term approximation.

(5)

Comparing the maximum-term approximation of Eq. 5 with Fig. 1, we see that the

approximation error is given by the missing term, ln℘X (C′n). Since the minimum over all

states is taken in Eq.5, the approximation error is determined by the deviation of the most

probable Cn from ℘ (C′n) = 1.

In previous studies using the maximum term approximation for bulk solvation,21–23 the

maximum term was taken as the minimum energy structure with lowest RRHO free energy.

Relating this to our present results, it is appropriate to examine how the constrained free

energy calculated via FEP converges to the complete free energy with increasing RMSD.

From Fig. 4, it is evident that as the RMSD criterion is increased to the infinite value

appropriate to the RRHO, the free energy becomes essentially exact. This occurs even for n

= 6 and 8, where two low-lying, alternate minima are present. In the worst case for two

minima, both could have equal probability, making the maximum error kBT ln2 = 0.4 kcal/

mol. For rigid metal coordination complexes or environmentally constrained protein binding

sites, this difference should decrease according to the ordering of the ion+binding site

complex.

Finally, we show in Fig. 6 the consequence of applying the maximum term approximation to

neglect the ℘X (Cn) terms and the RRHO approximation for the free energy of complex

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formation ΔF(Cn). Our results show that these approximations give reasonable selectivities

(within 0.8 kcal/mol, achieved at n = 2) from a computationally trivial calculation up to n =

5, where several minima are present. Interestingly, the failure of the model at high n does

not arise from errors in the maximum term approximation (since the error there is limited to

0.4 kcal/mol), but instead from roughness of the potential energy surface not captured by the

harmonic approximation. Even very near a minimum, the approximate harmonic energy

surface (minimum energy plus quadratic term using second derivatives computed with

AMOEBA) shows noticeable deviation from the AMOEBA force field for clusters involving

two or more waters (data not shown). This is due to unphysical stretching of bond and angle

degrees of freedom by straightforward linear displacements along normal modes.

Practitioners who use harmonic approximations of the potential energy surface anticipate

this error.54 Our previous studies have thus considered small ion-ligand distances for QCT

analysis.18,19,21–23 If this is not done, the problem of multiple, anharmonic minima arises,

and less satisfactory results are obtained.39 Anharmonic corrections,55 internal rotors and

other approximations involving redundant internal coordinates56 may be used in the future

to extend the harmonic approximation to larger clusters. However, the general success of the

RRHO approximation seen in Fig. 6 may arise despite this issue if we assume a non-linear

mapping exists between harmonic displacements and coordinates. In this case, the

coordinate system will be slightly altered, destroying the linearity of harmonic

displacements while the harmonic integral still converges to a similar answer.

5 Conclusions

We have presented thermodynamic cycles (Figs. 1 and 2) for computational analysis of

competitive ion binding reactions from a chemical perspective that are appropriate for

evaluation of free energies in complex heterogeneous and biological environments. These

cycles give a clear interpretation to the physical forces contributing to ion selectivity as well

as a well-defined scheme for carrying out computations. They have allowed us to prove

rigorously that the ion mutation and conformational constraint free energies add to a

constant value, independent of the choice of coordination geometry or constraint distance. In

practice, several coordination constraints, corresponding to different pathways, B → E, can

be investigated simultaneously from unconstrained simulations at states A and F.

The physical picture gained is one of isolating individual coordination states and computing

equilibrium contributions from each. Figure 1 also contains the unification of the ligand

field strength, topological control, and phase activation viewpoints on general mechanisms

of selectivity. Ligand-ligand interactions contribute uniquely to the gas-phase coordination

entropy (B→C), while the external environment is primarily responsible for changes in

solvent extraction (A→B) and hydration of the complex (D→E). Chemically specific ion-

ligand interactions occur during complex formation (C→D), along with ligand-ligand

interactions that effectively shift this energy for each coordination state. Remembering

A→F is independent of the choice of coordination, we can mentally join this leg between all

such choices. The balance in occupancy between coordination states (E→F) is then

determined by the above considerations. Alternatively, ionic selectivities for a given

coordination state can be simplified further by considering only D→F, as was done in Sec.

4.1. The topological control perspective then says that the environmental factors controlling

complex solvation (D→E) and the topological constraints determining coordination state

occupancy (E→F) completely determine the selectivity. Recall that selectivity is defined as

the difference from mutation in solution (FA →FB) to mutation in the constrained binding

site (DA →DB). This picture leads naturally to the more symmetric Fig. 2.

This picture unifies the most prominent views on the selectivity question in a way that

would not be possible without consideration of individual structural states. Quasichemical

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theory does this by showing the trade-off between alternative combinations of ligand

chemistry, coordination number, cavity shape, rigidity constraints, and phase structure of the

external environment, which are individually capable of altering ion selectivity.

One of the primary benefits of re-stating previous quasichemical theoretical developments26

in thermodynamic cycle notation (Fig. 2) is that it immediately shows the physical

manipulations and logical progression inherent in formulas designed to break out structural

contributions to full solvation free energies. At the same time, it re-interprets the problem of

mathematically stating a local spatial partitioning of the free energy by requiring instead an

explicit statement for the intermediate states (see Sec. 2), eliminating the need for ancillary

quantities such as , the excess chemical potential of the ion under the constraint that no

inner-shell binding is possible. Because of this, the right-hand side of Fig. 1 (equivalent to

that of Fig. 2), succinctly states Eqns. 1–2 and 5–9 of Ref. 26. In addition, it becomes

simpler to conceptualize minor changes to the ligand density term of the theory required for

application to inhomogeneous environments. At the same time, it retains its original ability

to incorporate rigorous QM free energy calculations on gas-phase coordination structures.

Thermodynamic cycles based on QCT have made it possible to decompose the energetic and

structural contributions to ionic solvation free energies. Arguments based on this formalism

have already been applied to deconstruct the physical forces contributing to the propensities

for alternate coordination states in potassium channel selectivity filters.18 The additive gas

phase plus quasi-liquid plus concentration free energies of that work are here given by

(6)

As discussed in Sec. 4.3 the constant cavity formation contribution from ℘0 (C0) does not

change the relative free energies of alternate coordinations, Cn, while the variations of A→E

(Eq. 6) among alternate Cn imply their relative populations, ℘X (Cn). Therefore such studies

have not only predicted the relative propensities for specific binding site compositions and

structures, but have also shown the structural design mechanisms utilized by nature to

accomplish this tuning with minimal cost.

From Figs. 4 and 5 it is apparent that any choice of coordination state will lead to a correct

estimate of the ion binding free energy when the full cycle is taken into account. It therefore

stands to reason that if the energy landscape in the constrained region is well described by

an RRHO-like approximation, then the constrained free energy can be calculated using an

analytical integral. Since it is known that the RRHO approximation encounters increasing

difficulty at larger cluster sizes, alternate empirical procedures may be constructed to

estimate the amount of error in the harmonic approximation57 as a function of ligand cluster

size. In either case, performing the stratification of configurations into their closest

minimum energy basin shows a clear path forward for applying further approximations for

calculation of the chemical binding free energies ΔG(Cn, QCT) or

applying information models58 to packing terms A→B and E→F of Fig. 2.

or for

The major differences between the original formulation of QCT for homogeneous solutions

and the newer variant appropriate for non-homogeneous solution has been discussed in Sec.

2. Both share the ligand-constrained state [D], which is a hallmark of QCT theories. Even

for n=0, this state has been usefully employed for calculations on intrinsically disordered

binding sites.34,41,57,59,60 Expanding the present formalism to include multiple ion

occupancy and even protein conformational changes involves a trivial modification of the

coordination constraint, Cn, to place conditions on the protein and all ions present. The same

conclusions apply to solutes other than ions – for example, small molecule solvation and

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biomolecular interaction. However, as more conformational transitions are constrained, the

number of states also increases.

The free energy components in cycle 2, when based on small regions around single energetic

minima, are key to resolving present debates on the mechanisms of ion binding and

stabilization by protein environments. The ability to interpolate between single, minimum

energy structures, and native-like unconstrained states for the binding site shows that the

utility of ΔEmin = ΔFNa+→K+(Cn,RMSD = 0) lies in quantifying the ion-ligand interaction

strength. This is a characteristic of individual coordination structures, and the primitive

object upon which overall selectivities are built. This is not a separation into confinement

and geometric contributions,53 but rather a synthesis of the “caress of the surroundings” with

the crowding of the ligands.61 In contrast to previous speculations that including both field

strength and topological control (among coordination states) contributions to ion selectivity

would make the argument cumbersome,62 this report has shown that both are indispensable

for simultaneously explaining rigid/mechanical and disordered/entropic selectivity

mechanisms. Similar hybrid viewpoints have also been reached by Refs. 50 and 16.

Acknowledgments

This work was supported, in part, by Sandia’s LDRD program, and, in part, by the National Institutes of Health

through the NIH Road Map for Medical Research. Sandia National Laboratories is a multi-program laboratory

operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S.

Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

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Figure 1.

Free energies of the homogeneous QCT process. The local excess chemical potential

determining solute activity at a point in solution can be calculated following any path

between states leading from top left to top right. The particular choice of water for the

coordinating ligands, Ln, is arbitrary, as similar pathways can be defined using any

combination of solution components.

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Figure 2.

Free energies of the non-homogeneous QCT process, applicable to enzyme active sites.

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Figure 3.

QCT process for an isolated binding site. Due to the absence of a surrounding environment,

ligand extraction (A→B) completely removes all interactions between waters, and insertion

of the X·(H2O)n complex is trivial, merging states D and E from Fig. 1.

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Figure 4.

Sodium to potassium mutation free energies computed for individual coordination states, Cn,

as a function of distance (RMSD) from a K+-n water minimum energy structure. The total

selectivity is calculated from the sum of free energies for placing a constraint on Na+ (FNa+

→DNa+, green x-s), mutating the ion given Cn (DNa+ →DK+, magenta boxes), and removing

the constraint from K+ (DK+ →FK+, blue stars). At zero RMSD (far left of each plot),

is shown. These contributions sum to a constant for each n, the un-constrained

Na+ →K+ mutation free energy (red). In bulk water, the mutation free energy is 17.3 kcal/

mol.30

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Figure 5.

Comparison of selectivity components for alternate choices of minimum energy geometry at

fixed number of water ligands, n. For n = 4 (left), panel a is constructed using RMSD from a

K+ minimum energy structure, while b is constructed from a Na+ minimum. For n = 6

(right), a and b are based on K+ structures, and c is from the most highly occupied Na+

minimum.

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Figure 6.

Ion selectivities for constant water number, n, and wall force (Eq. 4) computed with the

AMOEBA force field using TI and the RRHO approximation (left scale). The RRHO

approximation to the free energy difference is plotted for Na+ clusters with at least 5%

percent occupancy at 298.15 K, subtracted from the free energy of the K+ cluster with

nearest RMSD minimum energy structure. The number of minimum energy structures

within the 3.1Å constraint is plotted on the right scale.

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