PCCP PAPER. An optimized charge penetration model for use with the AMOEBA force field. 1. Introduction

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1 PCCP PAPER Cite this: Phys. Chem. Chem. Phys., 2017, 19, 276 An optimized chage penetation model fo use with the AMOEBA foce field Joshua A. Rackes, a Qiantao Wang, b Chengwen Liu, b Jean Philip Piquemal, c Pengyu Ren b and Jay W. Ponde* d Received 31st August 2016, Accepted 23d Novembe 2016 DOI: /c6cp06017j The pincipal challenge of using classical physics to model biomolecula inteactions is captuing the natue of shot-ange inteactions that dive biological pocesses fom nucleic acid base stacking to potein ligand binding. In paticula most classical foce fields suffe fom an eo in thei electostatic models that aises fom an ability to account fo the ovelap between chage distibutions occuing when molecules get close to each othe, known as chage penetation. In this wok we pesent a simple, physically motivated model fo including chage penetation in the AMOEBA (Atomic Multipole Optimized Enegetics fo Biomolecula Applications) foce field. With a function deived fom the chage distibution of a hydogen-like atom and a limited numbe of paametes, ou chage penetation model damatically impoves the desciption of electostatics at shot ange. On a database of 101 biomolecula dimes, the chage penetation model bings the eo in the electostatic inteaction enegy elative to the ab initio SAPT electostatic inteaction enegy fom 13.4 kcal mol 1 to 1.3 kcal mol 1. The model is shown not only to be obust and tansfeable fo the AMOEBA model, but also physically meaningful as it univesally impoves the desciption of the electostatic potential aound a given molecule. 1. Intoduction A gand challenge of molecula mechanics (MM) foce fields is modeling the physics of molecula inteactions with an accuacy and efficiency that allows ealistic, tactable simulations of lage systems. The goal is not only to coectly captue the physics of molecula inteactions, but also to be able to answe impotant pactical questions posed by biology, mateials science and a numbe of othe fields. To do this, MM models make classical appoximations to the 1st pinciples quantum mechanics diving the tue dynamics of a molecula system. Typically, this is done via a set of classical hamonic potential tems descibing the intamolecula inteactions of bonded atoms in the system and a sepaate set of non-bonded tems to descibe intemolecula inteactions. In paticula, the electostatic nonbonded tems ae especially impotant fo accuately modeling both shot and long ange molecula inteactions. 1 a Pogam in Computational & Molecula Biophysics, Washington Univesity, School of Medicine, Saint Louis, Missoui 63110, USA b Depatment of Biomedical Engineeing, The Univesity of Texas at Austin, Austin, Texas 78712, USA c Laboatoie de Chimie Théoique, Sobonne Univesités, UPMC Pais 06, UMR 7616, case couie 137, 4 place Jussieu, F 75005, Pais, Fance d Depatment of Chemisty, Washington Univesity in Saint Louis, Saint Louis, Missoui 63130, USA. E mail: ponde@dashe.wustl.edu Electonic supplementay infomation (ESI) available. See DOI: /c6cp06017j The AMOEBA foce field is unique in its teatment of these impotant intemolecula electostatic inteactions. Most MM foce fields use point chages to appoximate the chage distibution aound atoms in a system and paameteize these point chages based on themodynamic measuements. AMOEBA takes a moe physically ealistic appoach. The AMOEBA model appoximates the chage distibution aound atoms as a point multipole expansion of the chage distibution obtained fom ab initio quantum mechanics (QM) calculations. 2,3 Using a multipole expansion deived fom ab initio QM calculations povides a much moe accuate desciption of electostatic inteactions at medium-ange (B2 to 4 times the vdw adius), and has been shown to yield satisfactoy esults fo simulations of wate, poteins, nucleic acids and small molecules. 1,2,4,5 The multipole appoximation of electostatics, howeve, stats to beak down at shot-ange. While the multipole expansion is igoously coect fo inteactions of atoms at sufficient distance, it is no longe stictly valid once the electon clouds of inteacting atoms stat to ovelap. This phenomenon is known as chage penetation. Chage penetation is simply the change in the electostatic inteaction between two atoms due to thei electon cloud ovelap and the associated loss of nuclea sceening. It is a simple accounting fo the fact that atoms in a system ae not points; they epesent finite chage distibutions. Accuately modeling electostatics has been a pioity with AMOEBA since its inception. The impotance of these inteactions was a key motivation fo the oiginal AMOEBA multipole model. 276 Phys. Chem. Chem. Phys., 2017, 19, This jounal is the Owne Societies 2017

2 Pape PCCP Fig. 1 Electostatic potential as a function of distance. An inceasing level of theoy is needed as the adial distance fom an atom of inteest deceases. Qualitatively, accounting fo chage penetation is the logical next step in impoving this model. As depicted in Fig. 1, the cuent model coves the accuacy of long- and medium-ange electostatic inteactions. What is needed is a desciption of chage penetation to accuately model shot-ange inteactions. In addition to being physically elevant, chage penetation has been shown to be an impotant facto in many intemolecula inteactions. A paticulaly instuctive set of examples lies with what ae commonly called pi pi stacking inteactions. 6 The benzene sandwich dime, as illustated in Fig. 2, should classically be consideed electostatically epulsive since like chages ae lined up acoss fom one anothe. High level ab initio quantum mechanical calculations, howeve, show the counteintuitive esult that the benzene sandwich dime is electostatically attactive. 7 This is almost entiely due to chage penetation. Fig. 2 shows that the ovelap of electon clouds causes the electostatic enegy of the inteaction to become moe negative as the two monomes get close togethe. This same phenomenon Fig. 2 Electostatic enegy of the benzene sandwich dime. AMOEBA oveestimates the electostatic enegy of the inteaction compaed with the benchmak QM calculations. The eo gets pogessively wose at shot ange. is obseved with stacking inteactions between nucleobases. Pake and Sheill have ecently shown that without chage penetation, it is difficult, if not impossible to accuately captue the electostatics of inteacting nucleobases. 8 These consideations show that if AMOEBA is to be successful in accuately modeling biologically elevant inteactions such as nucleic acid folding o ligand binding, we must account fo the shot-ange electostatics of chage penetation. A numbe of studies have suggested functions fo incopoating chage penetation into existing molecula mechanics foce fields The deivation of most of these functions has followed the same basic stategy. The electostatic desciption of each atom in the system is split into two pats. The fist is the coe chage (often, but not necessaily simply the nuclea chage), teated as a point and second a smeaed electon cloud chage epesenting the emaining chage of the atom. This splits what was a single inteaction into fou inteactions, as illustated in Fig. 3. The functions listed in Table 1 ae fou methods suggested fo how best to handle this fou-pat inteaction between atoms. Tafipolsky and Engels took a moe diect appoach and calculated a numeical integal between spheical po-molecule chage densities. 17 This is simila in spiit to the appoach of the GEM (Gaussian Electostatic ) foce field, whee hemite gaussians ae used to epoduce the ab initio electon density. 9,21,22 While being physically staightfowad, these methods cuently lack the efficiency needed fo simulating lage systems. The othe thee methods use damping functions to appoximate how the electostatic potential of an atom changes in its electon cloud and use those damping functions to appoximate the value of the ovelap integal fo U 4. In a pevious poof-of-pinciple study, we implemented the fom of Piquemal and co-wokes in the AMOEBA foce field. 23 The study showed that accounting fo chage penetation can stat to ecove the tue natue of shot-ange electostatic inteactions between molecules. A follow-up study extended the model fo use with smooth paticle mesh Ewald. 24 In the pesent wok we seek to develop a compehensive model based This jounal is the Owne Societies 2017 Phys. Chem. Chem. Phys., 2017, 19,

3 PCCP Pape Fig. 3 Electostatic enegy of chage penetation coected, smeaed chage atomic inteactions. The total electostatic enegy is split into fou pats. The fist tem is the enegy of the coe coe, point point inteaction. The second and thid tems ae the enegies of each coe in the electostatic potential of the opposing smeaed chage. The fouth tem is the enegy of the ovelap between smeaed chage distibutions. Table 1 Poposed methods fo incopoating chage penetation into molecula mechanics electostatic enegy. Fo consistency, Z is the nuclea chage, is the total chage density of the electons, q is the total chage of the electon cloud, V is the numbe of valence electons, c is the patial chage, n is the numbe of sceening electons, and is the intenuclea distance. In the fist ow, the chage density is eithe a pomolecula chage density (Engels) o a density fom hemite gaussians in the GEM model (Cisneos) Coe A coe B Coe A smeaed chage B Smeaed chage A coe B Smeaed chage A smeaed chage B Engels; Cisneos Z A Z B ð 1 1 Z A B ð 2 Þ jr A 2 j d 2 ð 1 1 Z B A ð 1 Þ jr B 1 j d 1 ðð 1 1 A ð 1 Þ B ð 2 Þ d 1 d 2 j 1 2 j Godon Piquemal Tuhla Z A Z B V A V B ðc A þ n A Þðc B þ n B Þ Z A q B f damp ðþ V A ðc B V B Þ f damp ðþ ðc A þ n A Þn B f damp ðþ Z B q A f damp ðþ V B ðc A V A Þ f damp ðþ ðc B þ n B Þn A f damp ðþ q A q B f ovelap damp ðc A V A Þðc B V B Þ n A n B f ovelap damp f ovelap damp ðþ on the pevious wok that best captues the physics of electostatic intemolecula inteactions and the aims of the AMOEBA foce field. Given the potential impovement ou pevious wok has shown possible in such a model, the question becomes: what featues would we like the AMOEBA chage penetation model to have? In the wok pesented hee we aim to implement a chage penetation function that best meets the following citeia: (1) The model should be physically deived. (2) The model should be computationally efficient to compute. (3) The model should be numeically stable. (4) The model should accuately epoduce ab initio QM measuements fo elevant molecula inteactions. (5) The model should be consistent with the AMOEBA multipole model. In Section 2, we pesent the physical deivation of the models that wee consideed and deive coesponding damping tems fo highe-ode multipoles. In Section 3, the scheme fo paameteizing the models is pesented. Section 4 lays out esults compaing the pefomance of the models. Section 5 shows validation that the chage penetation model is captuing physical eality. And lastly, Section 6 daws ou conclusions. 2. Theoy Stone illustated the phenomenon of chage penetation with a simple example. 25 Conside the inteaction of a poton with a hydogen-like atom with nuclea chage Z. Fom quantum mechanics we know that the wave function of a hydogen-like atom is s cðþ ¼ Z3 This gives us the electon density of the atom, ðþ ¼ e Z : (1) p Z3 p e Z : (2) 278 Phys. Chem. Chem. Phys., 2017, 19, This jounal is the Owne Societies 2017

4 Pape This tells us how dense the electon distibution of the atom is as a function of the adial distance () fom its nucleus. To get the potential this density geneates, we must apply Poisson s equation, 2 V ¼ e 0 ; (3) whee e 0 is the pemittivity of fee space. Applying eqn (3) to eqn (2) we obtain VðÞ ¼ 1 þ Z þ 1 e 2Z ; (4) the familia potential due to the electon density of a hydogenlike atom. At lage distances fom the atom, the fist tem in eqn (4) dominates the second tem due to the second s exponential decay and we have the classical point chage coulomb appoximation of the potential. At close distances, howeve, as shown in Fig. 4, the second tem becomes non-negligible. This second tem epesents the chage penetation. We can exploit the fact that V() conveges to 1/ at lage distances and ewite eqn (4) as whee, VðÞ ¼ ð1 þ ZÞe 2Z ¼ f dampðþ (5) f damp () =1 (1 + Z)e 2Z. (6) The potential in this fom is epesented simply as the point chage coulomb potential multiplied by a damping function. This is convenient because the damping function has the following staightfowad popeties: (1) It appoaches a value of one as becomes lage. (2) It appoaches a value of zeo as appoaches zeo. (3) It is a diect multiplication of the classical point-chage coulomb potential. (4) It descibes chage penetation as a deviation fom the classical potential. To this point thee ae no appoximations made in ou deivation. Cucially, howeve, most atoms in systems of inteest fo molecula simulation ae not stictly hydogen-like. This means that f damp fo non-hydogen-like atoms is not exactly given by eqn (6). The popeties and fom of eqn (6) ae instuctive, howeve. To captue the physics moe geneally, we intoduce a paamete, a, in place of the 2Z and emove the pefacto in font of the exponential to obtain f damp () =1 e a. (7) This moe geneal constuction of f damp etains all of the elevant damping function popeties listed above and allows us to tune the paamete, a, to epoduce ab initio electostatic enegies. This is identical to the damping function poposed sepaately by both Godon and co-wokes 11 and Piquemal and co-wokes. 10 Using the damping fomulation of eqn (7), we have now effectively changed the potential due to evey atom in a given system. The potential at any point in the system is descibed by, VðÞ ¼ Z þ f dampðþv classical ¼ Z þ ð 1 e a Þ V classical (8) whee the potential due to the nucleus is unchanged, but the potential due to the electons now accounts fo the chage penetation effect. This, howeve, is not quite enough to get the inteaction enegy between two atoms. Recall fom Fig. 3 that although the second and thid tems of the chage penetation coected electostatic inteaction enegy involve simple point chages inteacting with the potential due to smeaed chage distibutions, the fouth tem has two smeaed chage distibutions inteacting with each othe. In this unique case, we must deive a second ovelap damping function to account fo this inteaction. Fo the fouth, ovelap tem we ae attempting to appoximate the ovelap integal between the two chage distibutions, U 4 ¼ ð A B dv A dv B ¼ 1 2 ð A V B ðaþdv A þ ð B V A ðbþdv B PCCP ; (9) whee V A and V B ae the chage penetation coected potentials due to atoms A and B espectively. Godon and co-wokes appoximate this integal using the one-cente method given by Coulson 26 to yield Fig. 4 Classical coulomb potential vs. hydogen like atom potential. Plotted is the electostatic potential of a point electon vs. the hydogen like electon (Z = 2 to emphasize the distinction). The classical potential diveges fom the hydogen like esult at shot ange. U 4 ¼ q Aq B ¼ q Aq B a B a ða 2 B a 2 A Þ e a A A ða 2 A a 2 B f ovelap1 damp ðþ Þ e a B (10a) whee q A and q B ae the total electon chages of atoms A and B, fo the chage chage potion of the inteaction. Piquemal and This jounal is the Owne Societies 2017 Phys. Chem. Chem. Phys., 2017, 19,

5 PCCP co-wokes take a two-cente appoach to appoximating the integal, U 4 ¼ q Aq B 1 e b A 1 e b B q A q B ¼ f ovelap2 damp ðþ (10b) whee, as laid out in ou pevious wok (ef. 20), a second paamete is intoduced to descibe the ovelap. While the deivations of these fomulae ae slightly diffeent, mathematically these U 4 ovelap damping functions constitute the only functional diffeence between the models of Godon and co-wokes and Piquemal and co-wokes. Fo simplicity s sake, the appoach of eqn (10a) will be efeed to as model 1 and eqn (10b) as model 2. They can be implemented, howeve, in an identical manne. These ovelap damping functions allow us to calculate the chage penetation coected chage chage electostatic inteaction between any two sites: U chage chage electostatic ¼ Z AZ B þ Z Aq B f damp ðþþ Z Bq A f damp ðþ þ q Aq B f ovelap damp ðþ: (11) The AMOEBA model, howeve, has moe than just chages on evey atom. It uses a multipole expansion epesenting the chage distibution at evey site. The enegy between two AMOEBA multipole sites, i and j, is given by, U multipole = M t it classical ij M j (12) whee M i and M j epesent the multipole moments on atoms i and j espectively, and T classical ij @ @ i j j (13) is the classical point multipole inteaction matix. We can see in eqn (13) that the inteaction matix, T ij, fo AMOEBA without chage penetation is obtained simply by taking epeated deivatives of the classical coulomb potential, 1/. To account fo chage penetation, not just in chage chage inteactions, but in all multipole inteactions up to abitay ode, we simply inset the chage penetation damped potential in place of the classical potential. This yields the chage penetation coected multipole inteaction j T ij ¼ 1 f j (14) whee f damp is eithe 1 (fo nuclea nuclea inteactions), the damping function fom eqn (7) (fo the second and thid tems of the inteaction enegy), o the ovelap damping function fom eqn (10a) o (10b) (fo the fouth tem of the inteaction enegy). Using the chage penetation coected multipole inteaction matices, we can expess the new AMOEBA multipole inteaction enegy of any two sites as: U CP electostatic ¼ Z iz j þ Z i T damp ij M j þ Z j T damp ji M i þ Mi t T ovelap ij M j : (15) Eqn (15) allows us to account fo the effects of chage penetation up to abitay ode multipole expansion. Fo AMOEBA, which has multipole inteactions up to quadupole quadupole, this means that the chage penetation model can be made fully consistent with the multipole model. See ESI fo explicit damping functions fo all AMOEBA multipole inteaction components. 3. Paameteization Pape The goal of including chage penetation in the AMOEBA model is to moe accuately epoduce the enegies of electostatic inteactions between molecules at shot ange. Because both models 1 and 2 contain empiical paametes, we will seek to optimize them by fitting to a database of elevant intemolecula electostatic enegies. In ou pevious wok, the S101 and S101x7 databases whee constucted fo this pupose. 23 The S101 database contains 101 unique pais of both homodimes and heteodimes of common oganic molecules. It contains the widely used S66 database 27 along with some additional elevant biomolecula inteactions. The S101x7 database is constucted by placing each dime pai fom the S101 database at 0.70, 0.80, 0.90, 0.95, 1.00, 1.05 and 1.10 times thei equilibium intemolecula distance. A schematic epesentation of all the dime pais included in the S101 database is shown in Fig. 5. In all of the paameteization that follows, the entie S101x7 database was used with the exception of inteactions 280 Phys. Chem. Chem. Phys., 2017, 19, This jounal is the Owne Societies 2017

6 Pape Fig. 5 Dime pais in the S101 database. Aows connect monomes that fom dimes. A /2 designation indicates a homodime. A / + designation indicates both neutal and positively chaged foms. Repoduced fom ef. 20. involving ethyne. The omission of ethyne allows diect compaison with the esults fom ou pevious wok. To paameteize the chage penetation models against the S101x7 database, accuate intemolecula electostatic enegies ae needed fo all dime pais. In the pevious wok, Symmety Adapted Petubation Theoy (SAPT) 28 calculations whee pefomed to obtain these enegies. SAPT calculations decompose intemolecula enegies into physically meaningful components; the intemolecula enegy between two monomes is boken down into electostatic, induction, exchange-epulsion and dispesion enegies. Fo the S101x7 database, SAPT2+ calculations, 29,30 estimated at the complete basis set (CBS) limit as descibed in ef. 22, wee caied out to etun the ab initio electostatic inteaction enegy of each dime pai. The paametes of model 1 and model 2 wee optimized by pefoming a nonlinea least squaes fit to minimize the diffeence between the AMOEBA electostatic enegy (with chage penetation), U AMOEBA electostatic, and the SAPT electostatic enegy, U SAPT electostatic, fo each dime pai. Fo models 1 and 2, two methods of paameteizing ae poposed. In the fist method one paamete, a, is assigned pe element. In the second, one a is assigned pe chage penetation class. These classes, as listed in Table 2, ae simply chosen to allow fo diffeent desciptions of atoms of the same element but diffeent physiochemical classifications. The choice of classes is based on the knowledge that the electonic stuctue of an sp 2 hybidized cabon, fo example, will be geneally diffeent than that of an aomatic cabon. While it is cetainly tue that diffeences in electon distibution exist even amongst atoms of the same chage penetation class (the electonic stuctue of evey sp 2 hybidized cabon is not exactly the same), the guiding pinciple is to include only the minimal level of atomic classification to allow the model to be easily tansfeable. Fo model 2, the paamete, b, is fixed as a faction of a, b = ga. PCCP whee the paamete, g, is taken to be univesal to avoid ovefitting. Allowing b to float fo evey chage penetation class has the potential, of couse, to impove the oveall fit, but at the cost of losing physical meaningfulness. Recall fom eqn (10b) that although the b paamete is specific to the ovelap function in model 2, the two electon clouds that ae ovelapping ae supposed to aleady be descibed by the paamete a. Allowing both a and b to float in the fit would allow two diffeent paametes to descibe essentially the same physics. Instead fitting one univesal paamete g simply descibes how b should Table 2 Atom classes and fitted paametes fo chage penetation models Element Chage chage damping 1 a (Å 1 ) 2 a (Å 1 ) 2 g 3 z (Å 1 ) Chage penetation class Chage chage damping 1 a (Å 1 ) 2 a (Å 1 ) 2 g Highe ode damping 1 a (Å 1 ) 2 a (Å 1 ) Hydogen (H) Non pola (H C) Aomatic (H C) Pola, wate (H X) Cabon (C) sp 3, tetahedal sp 2, aomatic sp 2, cabonyl, etc Nitogen (N) sp 3, tetahedal sp 2, aomatic sp 2, othe Oxygen (O) sp 3, hydoxyl, wate sp 2, aomatic sp 2, cabonyl Phosphoous (P) Phosphate Sulfu (S) Sulfide, thiol Sulfu(IV) Fluoine (F) Oganofluoide Chloine (Cl) Oganochloide Bomine (B) Oganobomide g This jounal is the Owne Societies 2017 Phys. Chem. Chem. Phys., 2017, 19,

7 PCCP be geneally elated to a in appoximating the ovelap between molecules. It should be noted that the paameteization stategy hee fo model 2 diffes slightly fom pevious wok. It is chosen in this way to best fit the AMOEBA multipole model and povide fo a diect compaison with model 1 on the same test set. The esults of fitting model 1 and model 2 ae shown in Table 2. Thee fits wee pefomed fo each model. Fist the S101x7 database of intemolecula electostatic enegies was fit using only chage chage damping with paametes assigned by element. Next, the same chage chage damping fit was pefomed with paametes assigned by class. Then the database was fit using highe-ode damping with damping of all AMOEBA multipole inteactions (up to and including quadupole quadupole). In addition to paameteizing models 1 and 2, a thid model, due to Wang and Tuhla has been paameteized as well. This model, developed fo application in QM/MM calculations is included as a point of compaison. Howeve, it is not developed any futhe than chage chage damping using paametes assigned by element as it has seveal popeties that make it unsuitable fo implementation in AMOEBA. Fist, the model can be unstable with espect to the paametes of inteacting atoms. If two closely inteacting atoms have paametes that ae close, but not identical, the ovelap damping functions of the model beaks down. Second, expanding the model to include highe-ode damping to make it fully consistent with the AMOEBA multipole model is computationally intactable with this model. The expessions that fom the ovelap damping functions, as seen in eqn (8) and (9) in ef. 19 ae much moe complex functions of the adial distance between atoms,. Taking the successive deivatives necessay fo highe-ode damping tems would poduce expessions too expensive to calculate fo ou puposes. Thid, even if such deivatives wee deemed necessay, the model s famewok is incompatible with highe-ode damping. The damping functions used in Wang and Tuhla s model ae meant to simulate the oute Slate-type obitals of atoms. With this being the case, athe than teat all of an atom s electons as damped, the model only teats a maximum of 2 as damped. This teatment is acceptable fo chage chage damping since chage is spheically symmetic and one simply teats the emaining electons as pat of the coe. This is, howeve, poblematic fo highe-ode damping because thee is no such simple patitioning of the electons that make up an atom s dipole and quadupole moment. It would be nonsensical to apply the model s damping tems meantfotwoelectons,toanatom sdipoleandquadupole inteactions. In the following section the fits poduced by the paameteization of all thee models is pesented. The fits of each model to the S101x7 database will be used along with some impotant validation tests and theoetical aguments to detemine which model and which paameteization stategy to implement in AMOEBA. 4. Results Pape To undestand how chage penetation impoves the electostatic model of AMOEBA, we must undestand how the cuent AMOEBA model without a chage penetation coection pefoms.fig.6showshowamoeba spedictionofintemolecula electostatic enegies compaes to the SAPT ab initio electostatic enegy values on the S101x7 database. Fig. 6 eveals that using only a multipole expansion to descibe the electostatic inteactions between molecules systematically oveestimates the electostatic enegy at shot ange. The pevasive gap illustated Fig. 6 AMOEBA, multipole only intemolecula electostatic enegy of dimes in S101x7 database. The multipole only electostatic enegy fo each dime is plotted against the benchmak SAPT electostatic enegy. The diagonal, y = x line indicates what would be pefect ageement. Compaed to the benchmak calculations, the multipole only model systematically oveestimates the electostatic enegy. 282 Phys. Chem. Chem. Phys., 2017, 19, This jounal is the Owne Societies 2017

8 Pape in Fig. 6 illustates the need fo including chage penetation in the electostatic model of the AMOEBA foce field. The most naïve method of applying a chage penetation coection is to assign one paamete pe element and damp only the chage chage electostatic inteactions. As a fist test of the theoy, this stategy was implemented fo models 1, 2 and 3. Each model was then paameteized by fitting to the S101x7 database. The oveall esults of assigning paametes by element and damping only the chage chage electostatic inteactions ae illustated in the fist cluste of columns in Fig. 7. It is clea that all thee models pefom much bette than the cuent AMOEBA multipole only model. The RMS eo of the multipole-only model fo electostatic enegies on the S101x7 database is 13.4 kcal mol 1. s 1, 2 and 3 bing that eo down to 2.1 kcal mol 1,2.1kcalmol 1 and 4.5 kcal mol 1 espectively, showing that even a naïve damping stategy stats to captue the missing physics. It is also appaent that models 1 and 2 pefom much bette, even at this low level of implementation, than model 3. Additionally, note that despite having fewe paametes, model 1 pefoms nealy identically to model 2 fo this implementation. Complete statistics fo each of these fits, including a beakdown by intemolecula distance, ae available in ESI. While assigning paametes by element poduces an impovement ove the multipole-only AMOEBA model, it ignoes some key physiochemical popeties of elements in diffeent bonding envionments elevant to intepeting the a paamete. The a paamete with units, Å 1, can be undestood as the invese of the physical extent of the electon cloud of an atom. Fom ab initio electonic stuctue calculations we know that in geneal this popety can change substantially based on the bonding envionment of an atom. Fo this eason we fit models 1 and 2 with paametes assigned by class to the S101x7 as descibed in the peceding section. The oveall esults of assigning paametes by class and still damping only the chage chage electostatic inteactions ae illustated in the second cluste of PCCP columns in Fig. 7. The fist thing to note is the absence of a fit fo model 3. Once the paamete set is expanded to include classes, model 3 becomes highly unstable. As noted befoe this is due to numeical instability when paametes in the model become close. This is pactically unavoidable fo class-based paametes, so model 3 is excluded fom this point fowad. Moe impotantly, howeve, we notice also that splitting out diffeent paamete classes impoves the oveall fit to the S101x7 database fo models 1 and 2. Assigning paametes by class impoves the pefomance on the RMS eo. Again despite having fewe paametes, model 1 outpefoms model 2 in this case. This impovement is lagely due to allowing diffeent classes fo the same element. Fo example, Table 2 shows that fo model 1 the paamete fo hydogen in the element based fit splits quite significantly when one allows diffeent classes to vay. The element paamete, 4.0 Å 1 splits into paametes of 3.4 Å 1,3.9Å 1 and 5.0 Å 1 fo non-pola, aomatic and pola hydogen espectively. This exta flexibility in the paameteization, ooted in basic physiochemical popeties impoves ou oveall desciption of the electostatics. Again specific statistics fo classbasedfitscanbefoundintheesi. Splitting out sepaate chemical classes fo paametes impoves the pefomance of ou chage chage damping chage penetation model, but it unfotunately does not meet the citeia of being fully consistent with the AMOEBA multipole electostatic model. To test the fully integated model we implemented chage penetation damping fo all multipole inteaction tems (up to and including quadupole quadupole) fo both models 1 and 2. We will efe to this model as highe-ode damping. The oveall esults, illustated in the thid and final cluste of columns in Fig. 7, show the impovement that this model bings. Implementing a fully integated highe-ode damping model with class-based paametes bings the RMS eo on the entie S101x7 database fo models 1 and 2 down to 1.31 kcal mol 1 and 1.52 kcal mol 1 espectively. Full statistical analysis can be found in ESI. These numbes epesent a damatic impovement ove the cuent Fig. 7 Root mean squae eo of AMOEBA electostatic enegy with chage penetation on S101x7 database. Multiple chage penetation models wee tested. The fist cluste of columns epesents the esults of paametes fit by element with chage chage damping only. The second cluste is the esults of having paametes assigned by class and chage chage damping. The thid cluste is the esults fo including highe ode damping in addition to having paametes assigned by class. (RMS eo of AMOEBA with multipoles only is 13.4 kcal mol 1 ). This jounal is the Owne Societies 2017 Phys. Chem. Chem. Phys., 2017, 19,

9 PCCP AMOEBA multipole-only RMS eo of kcal mol 1. Moe impotantly they also impove on the eos fom ou chage chage damping implementations. A significant potion of the impovement is due to impovement in the pefomance on the closest dime pais in the S101x7 database. Among dimes that ae sepaated by 0.70 and 0.80 of thei equilibium distance, model 1 with highe-ode damping educed that eo fom 2.75 kcal mol 1 to 2.27 kcal mol 1, and model 2 educed it fom 4.36 kcal mol 1 to 2.64 kcal mol 1. Impotantly, this impovement does not sacifice the fit at moe accessible distances. Fo model 1 the RMS eo on dimes with intemolecula sepaations of 0.90 to 1.10 times thei equilibium distance dopped to unde 1 kcal mol 1 compaed with an eo of ove 4 kcal mol 1 fo the cuent multipole-only model. Lastly, these fits give a slight edge to the simple model 1 ove model 2. 1 pefoms 16% bette than model 2 on oveall RMS eos in the S101x7 database when highe-ode damping is included. The absolute pecent eo of model 2 on the electostatic enegies of the S101x7 database is 10%, while model 1 gives 7%. Fig. 7 lays out the oveall pefomance of each of the implementations descibed above. It is clea fom this data that model 1 with highe-ode damping and paametes assigned by class gives the best fit to the electostatics of the S101x7 database. The impovement this model gives on each individual dime pai is shown in Fig. 8. Fig. 8 shows that acoss the boad model 1 with highe-ode damping is supeio to simple chage chage damping, and epesents a damatic impovement ove the cuent multipole-only model. This is boun out in a handful of impotant and instuctive examples. Fig. 9 lays out the esults fo fitting the wate dime, Fig. 10 shows two Pape Fig. 9 Wate dime electostatics. AMOEBA dime electostatic enegies without (multipoles only) and with (model 1 with chage chage and highe ode damping) chage penetation ae plotted against benchmak SAPT electostatic enegies. impotant oientations of the benzene dime and Fig. 11 shows the model s pefomance on phosphate ions. These thee examples epesent impotant elevant biomolecula inteactions that the cuent multipole-only model fails to accuately captue. Moeove, all thee also show that an integated highe-ode damping model is needed to achieve the highest level of ageement with SAPT electostatic data. These examples show that not only does the model geneally impove the quality of electostatics acoss a wide dataset, but it also pefoms well on individual examples, such as the benzene sandwich dime, that inspied ou investigation of the chage penetation phenomenon. Fig. 8 AMOEBA intemolecula electostatic enegy with and without chage penetation of S101x7 database dimes. The AMOEBA electostatic enegy both without (mulitpole only) and with (model 1 with chage chage o highe ode damping) chage penetation is plotted against benchmak SAPT electostatic enegy calculations. The diagonal, y = x line indicates what would be pefect ageement. Including highe ode damping in the chage penetation model yields the best ageement with ab initio electostatic enegies. 284 Phys. Chem. Chem. Phys., 2017, 19, This jounal is the Owne Societies 2017

10 Pape PCCP Fig. 10 Benzene (a) sandwich and (b) T shape dime electostatics. AMOEBA dime electostatic enegies without (multipoles only) and with (model 1 with chage chage and highe ode damping) chage penetation ae plotted against benchmak SAPT electostatic enegies. 5. Validation The fit to the S101x7 database with model 1 highe-ode damping is a welcome esult. The model damatically impoves the quality of the electostatic fit fo those electostatic inteactions ove AMOEBA s cuent multipole-only model and it outpefoms all of the othe elevant damping models poposed. Thee ae, howeve, some consideations that need to be addessed to validate model 1 with highe-ode damping as the best option fo captuing the physics of chage penetation. Fist, we would like to show that in addition to giving the best fit, model 1 is also the most obust option. Second, we need to know to what extent this chage penetation model is independent of the AMOEBA multipole model. And most impotantly, we must validate that this model is captuing a eal physical phenomenon. It is impotant ou chage penetation model not only povides a good fit to ab initio electostatic data, but also that the model is obust. To evaluate obustness we must evaluate the sensitivity of the model to small changes in the paametes. 3 does not pass this paamete sensitivity equiement. Fig. 12 shows the behavio of the oxygen sulfu electostatic inteaction in the DMSO wate dime as the diffeence between oxygen and sulfu paametes gets smalle. Clealy model 3 beaks down as the two paametes get close to one anothe. Moeove, the poblem is compounded as the intemolecula distance deceases. Since the zeta paamete multiplies the inteatomic distance,, eveywhee in the damping function, the poblem gets wose as monomes get close togethe. 2 does not suffe fom any such numeical instability, but it is sensitive to the paamete, g, that detemines the ovelap This jounal is the Owne Societies 2017 Fig. 11 Phosphate wate dime electostatics. AMOEBA dime electo static enegies without (multipoles only) and with (model 1 with chage chage and highe ode damping) chage penetation ae plotted against benchmak SAPT electostatic enegies. Results ae shown fo PO4H (a), PO4H2 (b) and PO4H3 (c). damping function. Table 3 shows that if the closest dimes ae left out of ou fit to the electostatic data, g changes fom 0.88 to Moeove, if we use the g that comes out of the fit whee we leave out the closest points, the RMS eo fo the full S101x7 database jumps fom 1.52 kcal mol 1 to 1.83 kcal mol 1. 1 on the othe hand does not suffe fom any such sensitivity. If we leave out the closest dime pais and fit paametes to ou model, Table 3 shows that those paametes do almost as well as the paametes fit to the full S101x7 database. The RMS eo fo model 1 in this case goes up by less than 0.1 kcal mol 1. Phys. Chem. Chem. Phys., 2017, 19,

11 PCCP Pape Fig. 12 Chage penetation model stability. The oxygen sulfu electostatic inteaction enegy fo the wate DMSO dime is plotted as a function of the diffeence between the oxygen and sulfu chage penetation paametes. As the atio of the paametes appoaches unity, model 3 becomes unstable. Table 3 Chage penetation model paamete sensitivity. s 1 and 2 wee fit to the S101x7 database excluding the closest points (all dimes except those at 0.7 times the equilibium distance). The paametes geneated fom that fit ae then tested on the full database. 2, paticulaly the g paamete, poves to be the moe sensitive to this change 1 2 Paametes fom fit 1.31 kcal mol kcal mol 1 (g 0.88) to full S101x7 database Paametes fom fit to S101x7 database excluding the closest points ( ) 1.40 kcal mol kcal mol 1 (g 0.90) By these tests model 1 shows the stength with espect to numeical stability and paamete tansfeability we expect a obust chage penetation model to have. In addition to being the most obust option, model 1 also shows good model independence fom the AMOEBA multipole model. AMOEBA follows a defined potocol fo detemining chage, dipole and quadupole paametes fo each monome 2 and we should expect that ou model should, fo the most pat, be independent of that specific potocol. In othe wods the multipole model and the chage penetation model should not depend on each othe. To test this we use the toy example, benzene. When detemining the electostatic paametes fo benzene, multiple values fo the opposing chages of the cabons and hydogens will give nealy identical fits to the electostatic potential on a gid of points aound the molecule. Although the AMOEBA multipole potocol fixes those chage values semiabitaily, we wanted to see if choosing othewise would beak ou model 1 chage penetation model. Fig. 13 demonstates that model 1 accuately epoduces the electostatic potential egadless Fig. 13 Chage penetation model independence. Thee diffeent benzene multipole models wee chosen with chages fixed at e, 0.15 e, and 0 e that give oughly equivalent electostatic potential fits. The chage penetation model was then applied to all thee models. RMS eos of the electostatic potential on a gid of points aound benzene fo each model ae plotted. The chage penetation significantly lowes the eo egadless of multipole model. of which potential-fitted chage dipole quadupole model one chooses. This validates an impotant featue of the model: that it is independent of the specifics of potential fitting potocol fo the AMOEBA multipole model. Lastly, but most impotantly, fo ou model to be valid, we must pove that it is captuing a eal physical effect. At the heat of the chage penetation phenomenon is the fact that the electostatic potential aound an atom at shot ange cannot be epoduced by a simple point multipole appoximation without accounting fo the extent of the atom s chage density. To validate that the model is descibing this physics we tested to see if ou chage penetation model, model 1 with highe-ode damping, could accuately epoduce the ab initio electostatic potential aound a molecule at shot ange. Fig. 14 shows that without exception the chage penetation model damatically impoves the electostatic potential fit aound evey monome in the S101 database. This is the validation we ae looking fo. Not only does ou model coect the pactical poblem of bad intemolecula electostatic enegies at close ange, but it does so by accuately captuing the physical eality of molecules finite chage distibutions. 6. Test case: nucleic acid base stacking As stated in the intoduction, chage penetation effects ae impotant in a boad ange of close-contact biomolecula inteactions. One essential example is the stacking inteactions of nucleobases in DNA and RNA sequences. Pake and Sheill ecently showed that without an explicit accounting fo chage penetation, foce fields stuggle to accuately epoduce the ab initio electostatic enegies of these inteactions. 8 Fo instance in an AC:GT base step, the mean absolute eos (MAE) of the AMBER 31,32 and CHARMM 33 foce fields elative to the SAPT electostatic enegy wee ove 20 kcal mol Phys. Chem. Chem. Phys., 2017, 19, This jounal is the Owne Societies 2017

12 Pape PCCP Fig. 14 Chage penetation model pefomance on electostatic potential of monomes in S101 database. The RMS eo of the electostatic potential on a gid of points aound each monome is plotted. Including chage penetation impoves the fit to the electostatic potential fo evey monome. Likewise, we find that AMOEBA without chage penetation gives an electostatic enegy MAE ove 20 kcal mol 1 as well. Howeve, when we apply ou chage penetation function with paametes fixed to thei values fom the S101x7 fit, the MAE dops damatically to nealy 2 kcal mol 1. This impovement is not unique to the AC:GT base step. As shown in Fig. 15, Fig. 15 Mean absolute electostatic inteaction enegy eo elative to SAPT0 fo ten stacked base steps. Including chage penetation lowes the MAE in the electostatic inteaction enegy fo evey base step combination. Fig. 16 Mean absolute electostatic inteaction enegy eo elative to SAPT fo six stuctual paametes. Including chage penetation lowes the MAE fo vaiation along evey degee of feedom in the nucleobase stacking inteaction. Inset epoduced fom ef. 7. This jounal is the Owne Societies 2017 Phys. Chem. Chem. Phys., 2017, 19,

13 PCCP Fig. 17 Electostatic enegy of a stacked TA:TA inteaction vs. ise. Including chage penetation epoduces the ab initio SAPT electostatic enegy ove the ange of ise paametes. The behavio is consistent with that of the benzene dime inteaction (see Fig. 10). Fig. 18 Electostatic enegy of a stacked TA:TA inteaction vs. Tilt. Includ ing chage penetation epoduces the ab initio SAPT electostatic enegy ove the ange of tilt paametes. Tilt like inteactions ae not pat of the S101x7 database, so this behavio shows a level of tansfeability fo the model. the MAE of ou AMOEBA model with chage penetation is significantly lowe fo evey base step combination. Moeove, this impovement in the electostatic desciption of nucleobase stacking holds even fo non-equilibium stacking aangements. Fig. 16 shows that fo the six stuctual paametes that define the stacking inteaction, 34 the AMOEBA + chage penetation model does fa bette than AMBER, CHARMM o the cuent AMOEBA foce field. These data confim, as asseted by Pake and Sheill, that including chage penetation is an absolute necessity fo a obust nucleic acid foce field model. This impeative is highlighted in two standout cases of the TA:TA base step. Fig. 17 shows the pefomance of foce field models against SAPT electostatics vesus the nucleobase ise. It is immediately clea that the AMOEBA + chage penetation model put fowad hee is the only model that accuately epoduces the electostatic natue of this inteaction. The same is seen in Fig. 18 wheeweexaminetheelectostaticenegyasafunctionofthetilt paamete. Again, the model including chage penetation is the only model that agees with the quantum mechanics. This same impovement pesists acoss all stuctual paametes of the TA:TA base step. Figues fo the othe fou paametes can be found in the ESI. It is woth noting that not only is this an impotant test case because of its diect elation to biomolecula applications fo the foce field. It is also impotant because it shows that the model, paameteized against a paticula test set (S101x7) pefoms well on inteactions well outside of that set. These esults give us confidence in the tansfeability of ou chage penetation model. 7. Conclusions Pape The goal of the AMOEBA foce field is to model the physics of biomolecula inteactions using appoximations that make calculations on lage systems tactable. Ou wok hee shows that to accuately captue the physics of shot-ange intemolecula Fig. 19 Chage penetation model ageement with AMOEBA potential fit multipole model. s 1 and 2 ae fit to the S101x7 database using eithe DMA o potential fit multipoles. RMS electostatic enegy eo is plotted. 2 pefoms slightly bette when DMA multipoles ae used, but model 1 with potential fit multipoles gives the best oveall fit. 288 Phys. Chem. Chem. Phys., 2017, 19, This jounal is the Owne Societies 2017

14 Pape inteactions, a chage penetation tem is absolutely necessay. Without accounting fo chage penetation, even an advanced point multipole model cannot accuately epoduce electostatic inteactions at shot ange. These discepancies in intemolecula inteactions cucial to biomolecula systems ae lage enough that they cannot be ignoed. Fotunately, we have also shown that chage penetation can be coected fo with the implementation of a simple set of damping functions.thisisnotnecessailyanew conclusion. Pevious wok on AMOEBA as well othe classical foce field models have demonstated the efficacy of using damping functions to captue chage penetation. We have demonstated hee that the highe-ode damping functions we have developed fo model 1 epesent the best, most integated method fo implementing chage penetation in the AMOEBA foce field. Thee ae some key easons why using model 1 with higheode damping makes the most sense fo AMOEBA. The fist eason is the most obvious. On an extensive test set of elevant molecula dimes, model 1 with highe-ode damping poduced the most accuate esults. We have shown that including highe-ode damping povides a substantial incease in model accuacy and model 1 pefoms well at this pupose. The pactical pupose of including chage penetation in the foce field is to accuately descibe intemolecula inteactions and by this diect measue model 1 with highe-ode damping does the best. The model does moe than simply give good numbes, howeve. 1 is deived fom the fundamental physics of atomic chage distibutions. The damping function that descibes the electostatic potential aound an atom in this model comes diectly fom the chage distibution of a hydogenlike atom. The ovelap damping function comes diectly fom an appoximation of the ovelap integal between two hydogen-like chage densities. The model does contain empiical paametes, but those paametes ae given physical meaning by the deived functions they sit in. A natual question is why the simila model 2 with one exta paamete does not give bette esults than model 1. The simple answe is that it appeas the two models ae intinsically aligned with diffeent mulitpole models. AMOEBA takes a two step appoach to assigning multipole paametes. Fist distibuted multipole analysis (DMA) is pefomed to obtain initial chage, dipole and quaduople paametes. Then, those paametes ae optimized by fitting to the electostatic potential on a gid of points aound the molecule. Because the ovelap function in model 1 is constucted stating fom a simple one-electon potential, model 1 seems to align nicely with the electostatic potential fit method fo detemining AMOEBA multipoles. In contast it seems that the two-cente integal method used by model 2 might pefom bette with multipoles that ae not potential-fitted. This theoy is bone out by the esults of Fig. 19. Fig. 19 illustates that model 2 with its exta fee paamete, does pefom bette on the S101x7 database when simple DMA multipoles ae used instead of potential fitted ones. Using the AMOEBA potential fitted multipoles howeve does bette oveall and much bette when paied with PCCP model 1. The oigin of this diffeence between models 1 and 2 in instuctive. It shows that despite its elative simplicity, model 1 seems to povide a bette intinsic fit fo the AMOEBA foce field. Not only is the model conceptually aligned with the AMOEBA multipole model, but it is fully integated with it as well. Pio chage penetation models have damped chage chage inteactions o a handful of highe ode inteactions, 13,14 but hee we have deived damping functions fo multipole inteactions up to abitay ode. This does two impotant things. Fist, it impoves the oveall accuacy of ou intemolecula electostatic enegies. And second, it gives us a fully integated multipole electostatic chage penetation model. The chage, dipole, quadupole moments of a multipole expansion ae all functions of the undelying chage density distibution. Thus evey inteaction of these moments should be damped by the function that descibes that chage density. Ou highe-ode chage penetation model satisfies this equiement and does so in a simple, staightfowad way. Impotantly, the chage penetation model doesn t just fit one set of data. We have demonstated that it passes multiple validation tests. Fist, the model poved to be obust. Thee is no numeical instability and the paametes ae not ovely sensitive. Second, the model is independent of the multipole model. This means that even if a slightly diffeent set of multipole moments that fit the electostatic potential ae chosen fo a given molecule, ou chage penetation model will still give the same impovement in the fit. These validation tests indicate not only that ou model is viable, but that it is not beholden to the test set o the multipole model. In addition we have shown that ou chage penetation model has some measue of pedictive powe. On the biologically significant test of electostatics in nucleic acid base stacking, ou chage penetation model accuately pedicted the electostatic enegies of base stacking ove a wide ange of non-equilibium stuctual paametes. This esult displays the pomise this model shows in its application to simulations of eal biological systems. Finally, ou highe-ode chage penetation model captues a eal physical effect. The chage penetation phenomenon is a diect esult of the fact that atoms have chage distibutions epesenting thei electon densities. We have shown that ou chage penetation function captues exactly this physics. When we use ou model to fit the electostatic potential on a gid of point suounding a molecule, the eo in the electostatic fit fom the simple point multipole appoximation goes down fo evey tested case. This gives us the highest degee of cetainty that we ae doing moe than just adding in anothe degee of feedom to ou electostatic function. The damping functions deived fo ou highe-ode damping model accuately descibe the electostatic envionment aound molecules, and since the effect is necessaily shot-ange, the computational cost of accounting fo chage penetation in this way is minimal. The damping tems can be implemented utilizing a shot-ange cutoff, o can be computed fo evey paiwise inteaction in the eal-space potion of an Ewald summation appoach. In eithe case, the additional cost beyond that of the This jounal is the Owne Societies 2017 Phys. Chem. Chem. Phys., 2017, 19,