Converging Ligand-Binding Free Energies Obtained with Free-Energy Perturbations at the Quantum Mechanical Level

Transcription

1 FULL PAPER Convergng Lgand-Bndng Free Energe Obtaned wth Free-Energy Perturbaton at the Quantum Mechancal Level Martn A. Olon, [a] P ar S oderhjelm, [b] and Ulf Ryde* [a] In th artcle, the convergence of quantum mechancal (QM) free-energy mulaton baed on molecular dynamc mulaton at the molecular mechanc (MM) level ha been nvetgated. We have etmated relatve free energe for the bndng of nne cyclc carboxylate lgand to the octa-acd deep-cavty hot, ncludng the hot, the lgand, and all water molecule wthn 4.5 Å of the lgand n the QM calculaton ( atom). We ue ngle-tep exponental averagng (EA) and the non-boltzmann Bennett acceptance rato (NBB) method to etmate QM/MM free energy wth the em-emprcal PM6-DH2X method, both baed on nteracton energe. We how that EA wth cumulant expanon gve a better convergence and ue half a many QM calculaton a NBB, although the two method gve content reult. Wth 720,000 QM calculaton per tranformaton, QM/MM free-energy etmate wth a precon of 1 kj/mol can be obtaned for all eght relatve energe wth EA, howng that th approach can be ued to calculate converged QM/ MM bndng free energe for realtc ytem and large QM partton. VC 2016 The Author. Journal of Computatonal Chemtry Publhed by Wley Perodcal, Inc. DOI: /jcc Introducton One of the larget challenge for computatonal chemtry to develop method to etmate bndng energe of mall molecule to bomacromolecule. If uch energe could be accurately etmated, mportant part of drug development could be performed computatonally. Conequently, many method have been developed wth th am, rangng from fat corng method, over end-pont method, to trct freeenergy mulaton (FES) method. [1 3] Owng to the ze of the macromolecule, uch calculaton have typcally been performed at the molecular-mechanc (MM) level of theory. However, t well-known that the MM force feld ued for bochemcal molecule nvolve evere approxmaton, for example, omttng polaraton, hgher-order multpole, charge tranfer, and charge penetraton. All thee effect are automatcally ncluded n quantum-mechancal (QM) calculaton. Therefore, there have lately been qute ome nteret to mprove bndng-affnty calculaton by QM method, [4 6] for example, a a potproceng of corng calculaton, mprovement of dockng calculaton, or a a component of end-pont calculaton. [7 15] Many dfferent QM method have been employed, rangng from ememprcal QM (SQM) method, [7,10,12] va dperon-corrected denty-functonal theory (DFT) method, [13,14] and many-body perturbaton theory, [11] to coupledcluter method. [13,15] Some calculaton nvolved only the lgand n the QM calculaton, [8,9] wherea other ncluded alo the near-by group, [11,13 15] or even the whole ytem. [7,10,12] It would be even better f QM calculaton could be combned wth the FES method, whch n prncple hould gve correct reult, f ued wth a perfect energy functon and complete amplng of all relevant part of the phae pace. Unfortunately, QM method are extremely demandng n term of computatonal tme and memory requrement. Currently, QM energy calculaton can be performed for a full proten at the SQM level, wherea more accurate DFT calculaton can be performed on one or a few thouand of atom, and very accurate hgh-level QM calculaton, uch a the gold-tandard CCSD(T) method can only be appled to a few ten of atom. Moreover, FES method are baed on extenve amplng of the phae pace, typcally nvolvng energy calculaton n a molecular dynamc or Monte Carlo mulaton. Therefore, ome ort of approxmaton needed to perform FES calculaton at the QM level. One approach to ue QM for only a mall, but nteretng, part of the ytem (e.g., the lgand) and MM for the remander, the QM/MM approach. A few full FES lgand-bndng tude have been publhed wth uch a parttonng, treatng only the lgand by QM and ung SQM calculaton. [16 18] Another approach to perform the amplng at the MM level and then evaluate QM/MM energe only for a retrcted number Th an open acce artcle under the term of the Creatve Common Attrbuton-NonCommercal-NoDerv Lcene, whch permt ue and dtrbuton n any medum, provded the orgnal work properly cted, the ue non-commercal and no modfcaton or adaptaton are made. [a] M. A. Olon, U. Ryde Department of Theoretcal Chemtry, Lund Unverty, Chemcal Centre, P. O. Box 124, Lund, SE , Sweden E-mal: Ulf.Ryde@teokem.lu.e [b] P ar S oderhjelm Department of Bophycal Chemtry, Lund Unverty, Chemcal Centre, P. O. Box 124, Lund, SE , Sweden Contract grant ponor: Swedh reearch councl (project ); Contract grant ponor: Knut and Alce Wallenberg Foundaton; Contract grant number: KAW VC 2016 The Author. Journal of Computatonal Chemtry Publhed by Wley Perodcal, Inc. Journal of Computatonal Chemtry 2016, 37,

2 FULL PAPER Fgure 1. The varou thermodynamc cycle employed n the EA, NBB4, and NBB13 method to calculate bndng free energe at the QM level. The cycle apply for the lgand mulated both wth and wthout the hot, gvng ether DG QM bound or DGQM free n eq. (2) (ndcated by DGQM n the fgure). [Color fgure can be vewed n the onlne ue, whch avalable at wleyonlnelbrary.com.] of naphot. Vald QM/MM free energe can be obtaned ether by a MM!QM/MM FES calculaton, employng the thermodynamc cycle n Fgure 1a, [19 21] or by reweghtng of the MM naphot toward the QM/MM energy functon (Fg. 1b and 1c). [22] Such approache have been ued for lgand bndng, [13,23 25] a well a for olvaton free energe [26 31] and qute extenvely for enzyme reacton. [19 21,32 34] The challenge wth th approach to obtan converged reult for the MM!QM/MM perturbaton, whch mut be performed n a ngle tep to avod the need of QM/MM amplng, that, to enure that the overlap of the MM and QM/MM potental large enough (a few approache nvolvng QM/MM amplng have been uggeted [24,26,34 36] ). For enzyme reacton, proper convergence ha been obtaned by keepng the QM ytem fxed; [19 21] wthout th approxmaton, very poor convergence ha been oberved, whch could only partly be decreaed by employng SQM/MM amplng. [34] For bndng affnte, uch an approxmaton eem napproprate, a the entropy and reorganaton of the lgand expected to be mportant for the bndng. Eex and coworker have addreed th problem by conderng only the electronc polaraton energy, whch eem to gve converged ngle-tep MM!QM/MM energe calculated by exponental averagng (EA;.e., ung the Zwanzg freeenergy perturbaton approach; [37] Fg. 1a) wth 24,000 QM calculaton for flexble lgand bound to cyclooxygenae-2, a well a for mall molecule n water oluton, n both cae wth only the lgand treated by QM. [23,28] However, they have alo obtaned converged QM/MM olvaton free energe for mall phenol analogue, ncludng 200 water molecule n the QM calculaton, conderng nteracton energe wth only 1080 QM calculaton. [27] By performng full QM mulaton, they have alo hown that nteracton energe (n contrat to total QM energe) gve converged and content free energe for the MM!QM perturbaton. [38] K ong et al. ntead reweghted the MM naphot wth QM energe, ung the non-boltzmann Bennett acceptance rato method (NBB; Fg. 1b). [22] Wth th approach, they have obtaned converged QM/MM hydraton free energe ung ,000 QM calculaton, treatng only the lgand by QM. [29,30] On the other hand, Mulholland and coworker ued full QM/ MM Monte Carlo mulaton, but employed the Metropol Hatng approach to reduce the number of QM calculaton requred. [26] They have tuded the relatve hydraton energy of water and methanol, a well a the bndng of water molecule to neuramndae, treatng only the lgand by QM. [24,39] Stll, the approach very demandng, requrng QM calculaton. However, recently Skylar and coworker have ued a mlar approach to calculate hydraton free energe wth full QM calculaton, ung QM/MM tructure obtaned by hybrd Monte Carlo mulaton from MD mulaton a an ntermedate 1590 Journal of Computatonal Chemtry 2016, 37,

3 FULL PAPER teppng tone. [31] They obtaned converged relatve olvaton energe by only 6000 QM calculaton for each tate. We have employed both the EA and NBB approache to calculate the relatve bndng affnte of nne cyclc carboxylc acd to the octa-acd deep-cavty hot molecule (Fg. 2 and 3a) and for two ynthetc daccharde bndng to galectn-3, ung the full hot, all proten group, and water molecule wthn 6 Å of the lgand n the QM calculaton ( atom for the hot guet ytem and atom for galectn-3) and dperon-corrected denty-functonal theory wth large ba et (quadruple or trple zeta qualty, repectvely). [13,25] Unfortunately, t wa not poble to obtan converged MM!QM/MM free energe for ether ytem ung 3600 QM calculaton for each tranformaton. The full advantage of ung QM calculaton not obtaned untl both the lgand and at leat the cloet group of the receptor (4.5 6 Å [40 42] ) are ncluded n the QM calculaton. So far, no converged QM/MM bndng affnte have been obtaned wth uch an approach, owng to the ue of too demandng QM method. [13,25] Therefore, we n th artcle turn to the cheaper (but more approxmate) SQM method and tudy what needed to obtan converged MM!QM/MM free energe for the octa-acd hot guet ytem. The empha on convergence and what method gve the bet convergence (we compare dfferent varant of the EA and NBB method), not on reproducng expermental data. We how that 720,000 QM calculaton per tranformaton are requred to converge the MM!QM free energe to wthn 1 kj/mol. Method Smulated ytem Fgure 2. Guet molecule for the etmaton of bndng free energe to a truncated octa-acd hot. [Color fgure can be vewed n the onlne ue, whch avalable at wleyonlnelbrary.com.] In th artcle, we tudy the bndng of nne cyclc carboxylate lgand to the octa-acd hot, ung expermental data from the SAMPL4 challenge. [43,44] The lgand are hown n Fgure 2 and the octa-acd hot n Fgure 3a. Startng tructure for the calculaton were taken from our prevou tudy of th ytem. [13] To reduce the ze and the large negatve charge of the hot and reduce t flexblty, we deleted the four proponate group and alo the four carboxylate group on the rm of the rng ytem, gvng re to a neutral cavtand (NOA) wth 144 atom, hown n Fgure 3b. We wll how below that th truncaton ha only a mnor effect on relatve bndng affnte etmated at the MM level, but t mprove the convergence of the FES calculaton. The general Amber force feld [45] wa ued for both the NOA hot and the lgand, [13] and the TIP3P force feld wa ued for water molecule. [46] Retraned electrotatc potental (RESP) charge [47] for the lgand were taken from our prevou tudy [13] and thoe of NOA were etmated n the ame way: The hot wa optmzed at the AM1 level [48] and the electrotatc potental wa calculated at the Hartree Fock/6-31G* level at pont ampled around the molecule accordng to the Merz Kollman cheme, [49] albet at a hgher-than-default denty (10 layer wth 17 pont per unt area, gvng 2000 pont per atom), ung the Gauan 09 oftware. [50] The charge were then ftted to thee potental ung the antechamber program n the Amber 14 ute. [51] It wa enured that all ymmetry-equvalent atom had the ame charge (gvng only 16 unque charge). The force feld ued for NOA ncluded n the Supportng Informaton, Table S1. FES calculaton at the MM level All molecular dynamc (MD) mulaton and FES calculaton were performed wth the Amber 13 (pre-releae) and 14 Fgure 3. Structure of the full octa-acd hot (a) and the neutralzed hot, NOA wthout the proponate and carboxylate group (b). Journal of Computatonal Chemtry 2016, 37,

4 FULL PAPER oftware. [51] NOA and the lgand were olvated n a truncated octahedral box of water molecule, extendng at leat 9 Å from the olute ung the leap program n the Amber ute, gvng 4100 and 1800 atom n total for the calculaton wth and wthout the hot, repectvely. Ffteen ndependent mulaton were run for each lgand by olvatng the ytem n 15 dfferent TIP3P water boxe of explct water molecule and employng dfferent random eed for the tartng velocte, to ncreae the dfference between the ndependent mulaton [52] ). No counter-on were ued n the calculaton (mplyng that a neutralng plama were added to the ytem n the mulaton), becaue we have prevouly hown that they only have a mnor nfluence on the calculated freeenergy dfference. [13] The relatve bndng free energy between two lgand, L 0 and L 1 (DDG bnd ), wa calculated for eght tranformaton: MeBz!Bz, EtBz!MeBz, pclbz!bz, mclbz!bz, Hx!Bz, MeHx!Hx, Hx!Pen, and Hep!Hx (the name of the lgand are defned n Fg. 2). The FES calculaton were run wth the pmemd module of Amber, [51,53] ung the dual topology cheme wth both lgand n the topology fle. We employed 13 tate wth k , 0.05, 0.1, 0.2,..., 0.8, 0.9, 0.95, and 1.00, ung a lnear tranformaton of the potental: V k 5 ð1 kv 0 1kV 1 ; (1) where V 0 the potental of the larger lgand and V 1 the potental of the maller lgand. Electrotatc and van der Waal nteracton were perturbed concomtantly, ung oft-core potental for both type of nteracton. [54,55] The oft-core potental were ued only for atom dfferng between the two guet molecule, that, for the tranformed CH 3!HorCl!H group for the MeBz!Bz, EtBz!MeBz, pclbz!bz, mclbz!bz, and MeHx!Hx tranformaton, but for all atom n the rng ytem for the Hx!Bz, Hx!Pen, and Hep!Hx tranformaton. Tet calculaton have hown that ung oft-core potental for the whole guet molecule alo for the maller tranformaton doe not change the reult gnfcantly. [13] To make the calculaton comparable between the two veron of Amber, we ued the keyword thake 5 1 for the Amber 14 calculaton. For each k value, we frt performed 100 tep of mnmaton, wth the heavy atom of the hot and guet molecule retraned toward the tartng tructure wth a force contant of kj/mol/å 2. Th wa followed by 20 p contant-volume equlbraton wth the ame retrant and 2 n contantpreure equlbraton wthout any retrant. Fnally, an 8 n producton mulaton wa run, durng whch tructure were ampled every 2 p. In the MD mulaton, bond nvolvng hydrogen atom were contraned wth the SHAKE algorthm, [56] allowng for a tme-tep of 2 f. In all mulaton, the temperature wa kept contant at 300 K ung Langevn dynamc [57] wth a collon frequency of 2 p 21, and the preure wa kept contant at 1 atm ung a weak-couplng otropc algorthm [58] wth a relaxaton tme of 1 p. Long-range electrotatc were handled by partcle-meh Ewald (PME) ummaton [59] wth a fourth-order B plne nterpolaton and a tolerance of The cut-off for Lennard Jone nteracton wa et to 8 Å. The relatve bndng free energe were etmated ung a thermodynamc cycle that relate DDG bnd to the free energy of alchemcally tranformng L 0 nto L 1 when they are ether bound to the hot, DG bound, or are free n oluton, DG free [60] DDG bnd 5 DG bnd ðl 1 2 DG bnd ðl 0 5 DG bound 2 DG free (2) DG bound and DG free can be etmated by the Bennett acceptance-rato method [61,62] (BAR). In th approach, an MD mulaton run for each k, wth the potental n eq. (1). For each par of neghborng k value, A and B, the free energy dfference between the two tate etmated from DG A!B 5RT ln hf V ð A2V B 1C B 1C (3) hfðv B 2V A 2C A where f(x) 5 (1 1 exp(x/rt)) 21 the Ferm functon, R the ga contant, T the temperature (whch wa 300 K throughout th artcle), and C a contant [f the number of ample are dfferent n the two mulaton, n A 6¼ n B, a correcton factor ln(n A /n B ) hould be added to the rght-hand de of eq. (3)]. An teratve procedure appled to fnd a value of C that make the frt term of the rght-hand de of eq. (3) vanh. Free energe were alo calculated by mult-tate BAR (MBAR), [63] thermodynamc ntegraton, [64] and exponental averagng, [37] ung the pymbar oftware. [63] Preented reult were obtaned wth MBAR. SQM calculaton SQM ngle-pont calculaton were run on each of the MM naphot, both for the mulaton wth and wthout NOA. For thee calculaton, water molecule were wrapped back nto the orgnal perodc box, centred on the lgand wth the ptraj module. In the SQM calculaton, the 48 water molecule cloet to the lgand were ncluded n the calculaton wthout NOA, wherea the 19 water molecule cloet to the C atom n the carboxylate group were ncluded for the calculaton wth the lgand n NOA (n total or atom, repectvely; Fg. 4). Th repreent all water molecule wthn 4.5 Å of the lgand and they were obtaned n the ame way a n our prevou tudy. [13] The PM6-DH2X method [65] wa employed for the SQM calculaton, that, ncludng dperon, hydrogen-bond, and halogen correcton, [66 68] ung the MOPAC oftware [69] (th wa the mot accurate SQM method n th oftware when th nvetgaton wa tarted). The calculaton employed the keyword Prece, to enhance the energy convergence crteron to kj/mol. For each naphot, nteracton energe were obtaned by eparate calculaton for the complex, the guet, and the remander (.e., water molecule wth or wthout NOA): [13,39] MMfQM free energe DE nteract 5E complex 2 E guet 2 E remander (4) Several dfferent method were teted to calculate the MM!QM free energe. Frt, the QM nteracton energe 1592 Journal of Computatonal Chemtry 2016, 37,

5 FULL PAPER Fgure 4. Example of tructure ued for the PM6-DH2X calculaton, ncludng 19 or 48 water molecule for the calculaton wth (a) and wthout (b) NOA, repectvely. [Color fgure can be vewed n the onlne ue, whch avalable at wleyonlnelbrary.com.] were ued drectly to calculate bndng free energe for all k value wth the MBAR approach, that, gnorng the fact that the MD mulaton were not performed at the QM level. Th wll be called the QM-MBAR approach. Second, we employed EA calculaton. [13,19,20,23,27,28] In thee, we employ the thermodynamc cycle n Fgure 1a, howng that DDG bnd frt etmated at the MM level and then two ngle-tep FEP calculaton are ued to calculate the effect of changng the energy functon from MM to QM, one for each of the two lgand: DG QM 5DG MM 1DG MM!QM ;L 1 2DG MM!QM ;L 0 (5) where DG MM the free energy of the tranformaton at the MM level for ether the bound or free tate [ubcrpt ;.e., DG bound or DG free n eq. (2)], obtaned by the tandard MBAR approach, and the other two term are correcton term for gong from the MM potental to the QM potental. The latter correcton need to be evaluated only at the endpont of the tranformaton, that, for L 0 n the k naphot and for L 1 n the k naphot (for both the bound and free mulaton). Each correcton wa evaluated ether ung exponental averagng (EA) [37] : D h E DG MM!QM ;L 52RTln exp 2 EL QM 2EL MM =RT ;L (6) or by ung the cumulant expanon to the econd order (DG5l2 2RT r2, where l the average and r the tandard devaton of the EL QM 2EL MM dtrbuton; EAc), [70,71] whch exact f the energy dfference follow a Gauan dtrbuton. Thrd, we employed the NBB approach to reweght the naphot. [22] Th method evaluate the free energy accordng to: 0D f DG A!B B 5RT@ D f EA QM 2EQM B E QM B 2EA QM E D 1 1CexpðEB ba =RT expðea ba=rt BD EA E C A1C 2CexpðEba A EA =RT expðeb ba =RT B (7) where E ba 5 E MM E QM.Thbaacorrectonforthefact that the mulaton are performed at the MM level, but the energe are calculated at the QM level. The advantage wth NBB that the free energe are calculated wth BAR, whch ha better convergence properte than EA, epecally when the overlap poor. [62] The dadvantage that at leat twce a many QM calculaton are needed, becaue BAR mprove the convergence by employng the nformaton from both a forward and backward calculaton. Two dfferent approache to obtan the net bndng free energe were ued, a llutrated n Fgure 1. In the frt, QM energe were calculated for all 13 k value n the perturbaton (Fg. 1c). Th approach wll be called NBB13 n the followng. In the econd approach, NBB wa ued only for the frt two and lat two k value n the perturbaton (Fg. 1b), a ha been uggeted by K ong and coworker. [29,30] Thu, the net bndng energy wa obtaned from DG QM 5DG QMðk50!MM ð k50:05 MM ð k50:95!qm ð k51 1DG 1DG and the MM!QM energe were obtaned from QMðk50!MM ð k50:05 DG 0D fðe QM B 5RT@ k50 2EMM k50:05 E;k50:05 1C MM ð k50:05!k50:95 D expðe ba D f ðek50:05 MM 2EQM k50 2CexpðEba k50 E;k50 =RT k50 =RT E;k50 (8) 1 C A1C; (9) becaue the MM(k )!QM(k 5 0) perturbaton baed on the MM(k ) mulaton, whch are not baed, wherea the revere tranformaton (QM(k 5 0)!MM(k )) baed on the MM(k 5 0) mulaton, rather than the correct QM(k 5 0) mulaton. It can be een that QM calculaton are needed only for L 0, but not for L 1. A mlar equaton apple for MM k50:95 DG ð!qm ð k51 (for whch QM calculaton are needed Journal of Computatonal Chemtry 2016, 37,

6 FULL PAPER for L 1, but not for L 0 ). Th approach wll be called NBB4. All potental energe (E QM, E MM, and E ba ) n eq. (6), (7), and (9) [and alo eq. (10) (12) below] were approxmated wth the correpondng nteracton energe, calculated by eq. (4). Moreover, the QM potental energe n the equaton were calculated ether for the olated QM ytem (x QM ;.e., the olated guet wth 49 water molecule or the hot guet complex wth 19 water molecule) or for the full ytem wth a QM/MM approach: E QM=MM 5E QM ðx QM 2E MM ðx QM 1E MM ðx all (10) For the EA method n eq. (5) and (6), the two approache gve the ame reult, becaue the energy dfference n the exponental n eq. (6) become n the QM/MM cae E QM=MM L ðx all 1EL MM ðx all 2E MM ð L 2E MM ð L x all x all 5EL QM ð 5EL QM ð x QM x QM 2E MM ð L 2E MM ð L x QM x QM (11) whch the ame a n eq. (6). However, for NBB4 Ek50 ba 5 Ek50 MM ð x QM2E QM k50ð x QM reman the ame accordng to eq. (11), but eq. (9) change to DG QM=MMðk50!MM ð k50:05 0D fðe QM=MM k50 2Ek50:05 E;k50:05 MM 1C B 5RT@ D fðe MM k50:05 2EQM=MM k50 D expðe ba 2CexpðE ba k50 =RT E;k50 k50 =RT E;k50 1 C A1C (12) wth E QM/MM calculated from eq. (10) and the MM k50:05!k50:95 DG ð energy calculated wth the full ytem, perodc boundary condton, and total energe. Uncertante, qualty etmate, and overlap meaure Reported uncertante are tandard error, that, tandard devaton dvded by the quare root of the number of ample, for example, the 15 et of ndependent mulaton. The uncertante of the free-energy etmate were obtaned by nonparametrc boottrap amplng (ung 1000 ample) of the potental-energy dfference n the BAR or NBB calculaton. The qualty of the bndng-affnty etmate compared to expermental data wa quantfed ung the mean abolute devaton (MAD), the root-mean-quared devaton (RMSD), the correlaton coeffcent (r 2 ), and the lope and ntercept of the bet correlaton lne. In addton, Kendall rank correlaton coeffcent wa calculated for the eght tranformaton explctly mulated ( r ). The uncertante of the qualty etmate were obtaned by a parametrc boottrap (ung 500 ample), aumng the etmate follow a Gauan dtrbuton wth the mean equal to the etmate and the tandard devaton equal to the reported uncertanty. To etmate the convergence of the varou perturbaton, x dfferent overlap meaure were employed. [72] We calculated the Bhattacharyya coeffcent for the energy dtrbuton overlap (X), [73] the Wu & Kofke overlap meaure of the energy probablty dtrbuton (K AB ) and ther ba metrc (P), [74,75] the weght of the maxmum term n the exponental average (w max ), [20] the dfference of the forward and backward exponental average etmate (DDG EA ), and the dfference between the BAR and TI etmate (DDG TI, although th dfference may alo reflect the ntegraton error n TI [76] ). [72] We ued w max alo to etmate the convergence of the EA and NBB4 calculaton. In the former cae, w max the weght of the maxmum term n the average n eq. (6). In the latter cae, w max wa calculated for each of the three average n eq. (9) after convergence of C and the larget of thee three value reported. However, calculated n th way and ung the ame data, w max for EA and NBB4 dentcal, becaue the latter alway domnated by the exp E ba term n eq. (9), whch the ame a n eq. (6). k50 =RT Reult and Dcuon Bndng affnte at the MM level In th artcle, we tudy the bndng of nne carboxylate lgand to the octa-acd (OA) hot molecule, hown n Fgure 2 and 3a. We calculate the relatve bndng energe of the lgand wth FES method and our goal to obtan converged relatve bndng energe at the QM/MM level, wthout performng amplng at the QM/MM level, but ncludng all group wthn 4.5 Å of the lgand n the QM calculaton (not only the lgand a n mot prevou tude [23,24,28 30,39] ). Our prevou nvetgaton of th ytem a well a the bndng of two lgand to galectn-3 faled to gve converged QM/MM bndng energe wth 3600 QM calculaton at the DFT level. [13,25] Therefore, we employ here the much fater SQM PM6-DH2X method, o that we can perform enough QM calculaton to enure converged reult. Moreover, we have removed the proponate and benzoate group of the octa-acd hot (yeldng NOA, hown n Fg. 3b), becaue our prevou tudy howed that t wa hard to obtan a proper amplng of the dhedral angle of the proponate group wthn a typcal mulaton tme (4 n). [13] Moreover, the large negatve charge (28) of the hot molecule ometme gave problem n the QM calculaton. To check that the truncaton of the hot doe not affect the reult gnfcantly, we frt calculated DDG bnd for the NOA hot at the MM level. From the reult n Table 1, t can be een that the calculaton wth NOA gave almot the ame reult a for the full octa-acd hot [13] : For fve of the tranformaton, the two hot gave reult that agreed wthn 1 kj/mol, wherea for the remanng three tranformaton (EtBz!MeBz, Hx!Bz, and Hep!Hx), the reult dffered by 2 3 kj/mol. However, owng to the hgh precon of both calculaton, the dfference tattcally gnfcant for all except two of the tranformaton (MeBz!Bz and Hx!Pen) at the 95% level. The reult of the NOA calculaton are apprecably more prece than the OA calculaton ( , compared to kj/mol). Th partly owng to the longer mulaton (8 n v. 4 n) and the larger number of ndependent mulaton (15 v. 10). However, there are alo clear ndcaton that the NOA calculaton are better converged than the prevou calculaton: The overlap meaure n Table 2 how a perfect overlap for all the eght tranformaton wth NOA wth all X , K AB 1.03, P 2.5, w max 0.03, DDG EA 0.08 kj/mol, and 1594 Journal of Computatonal Chemtry 2016, 37,

7 FULL PAPER Table 1. Reult for the eght perturbaton (DDG bnd n kj/mol) obtaned at the MM level (ung MBAR) for the NOA hot. Tranformaton OA Exp. [44] NOA MM calc. OA [13] MM calc. NOA SQM/MM MeBz!Bz EtBz!MeBz pclbz!bz mclbz!bz Hx!Bz MeHx!Hx Hx!Pen Hep!Hx MAD RMSD r lope nter r For comparon, calculated (BAR) [13] and expermental [44] reult obtaned wth the full octa-acd (OA) hot are alo ncluded. For both NOA and OA, the preented calculated reult are the average and tandard error over the 15 or 10 ndependent mulaton. In the lat column, the SQM/MM reult for the NOA hot, obtaned wth EAc and 15 ndependent calculaton are ncluded. The x lat row gve qualty meaure decrbng how well the calculaton reproduce the expermental data of OA n the frt column: The mean abolute devaton (MAD n kj/mol), the root-mean-quared devaton (RMSD n kj/mol), the correlaton coeffcent, the lope and ntercept of the bet correlaton lne, and Kendall rankng correlaton coeffcent for only the eght condered tranformaton ( r ). DDG TI 0.06 kj/mol (X goe from 0, no overlap to 1, perfect overlap [73] ; K AB goe from 0 no overlap, va 1 full overlap, to 2 the frt dtrbuton completely nde the econd dtrbuton [74,75] ; a negatve P ndcate poor overlap [74,75] ;1/w max ndcate how many naphot contrbute gnfcantly to the EA etmate; DDG EA the hytere n the forward and backward EA etmate; and DDG TI ndcate the dfference between the BAR and TI etmate). In fact, all free-energy meaure etmated by PYMBAR (TI, TI cubc EA forward, EA backward, BAR, and MBAR) agree wthn kj/mol for the eght tranformaton, and the mot accurate BAR and MBAR reult agree wthn 0.05 kj/mol, ndcatng extremely well-converged reult. For our prevou OA mulaton [13] (alo lted n Table 2), the convergence wa apprecably wore wth X down to 0.93, K AB down to 0.79, P down to 20.1, w max up to 0.95, and DDG EA up to 16 kj/mol, wherea DDG TI 0.3 kj/mol wa good. In partcular, Hx!Pen and Hep!Hx tranformaton gave negatve P value and w max > 0.38, whch ndcate that more k value or longer mulaton hould have been ued. Th alo reflected by the larger tandard error of thee two etmate (0.7 kj/mol). Moreover, the MeHx!Hx tranformaton gave w max and DDG EA 5 16 kj/ mol, whch ndcate that the overlap wa poor alo for th tranformaton. A large part of the mprovement for NOA can be attrbuted to the longer mulaton (60,000 naphot ntead of 4000). However, f we ntead conder the wort value n the 15 ndependent mulaton of NOA, each baed on 4000 naphot, NOA tll gve converged reult (X , K AB 0.93, P 1.7, w max 0.27, DDG EA 0.9 kj/mol, and DDG TI 0.06 kj/mol, although both w max and DDG EA have ncreaed by a factor of 6 17).Thhowthattheremovaloftheflexbleproponategroup ha trongly mproved the amplng for the NOA hot. Affnte at the SQM level Next, we tred to etmate bndng affnte alo at the SQM/ MM level ung 60,000 naphot for each k value (n practce, we frt dd the calculaton on 4000 naphot and baed on thoe reult, we decded how many ndependent mulaton were needed to converge the reult to a precon of 1 kj/ mol). A detaled n the Method ecton, we employed everal dfferent approache to calculate the MM!SQM free energe. Frt, we tred to ue the full NBB13 approach wth SQM Table 2. Overlap meaure for the eght perturbaton of NOA and OA, performed at the MM level, baed on 60,000 (NOA) or 4000 (OA) naphot. Each meaure the mnmum (X, K AB, and P) or maxmum (w max, DDG EA, and DDG TI ) value over the 26 k value for the mulaton wth and wthout the hot. X K AB P w max DDG EA DDG TI NOA MeBz!Bz EtBz!MeBz pclbz!bz mclbz!bz Hx!Bz MeHx!Hx Hx!Pen Hep!Hx OA MeBz!Bz EtBz!MeBz pclbz!bz mclbz!bz Hx!Bz MeHx!Hx Hx!Pen Hep!Hx The meaure are the Bhattacharyya coeffcent for the energy dtrbuton overlap (X), [73] the Wu & Kofke overlap meaure of the energy probablty dtrbuton (K AB ) and ther ba metrc (P) [74,75] the weght of the maxmum term n the EA (w max ), [20] the dfference of the forward and backward EA etmate (DDG EA n kj/mol), and the dfference between the BAR and TI etmate (DDG TI n kj/mol). Value ndcatng poor overlap or bad convergence are marked n bold face (X < 0.7, K AB < 0.7, P < 0.5, w max > 0.2, DDG EA > 4 kj/mol, or DDG TI > 1 kj/ mol). [72,74,75] Journal of Computatonal Chemtry 2016, 37,

8 FULL PAPER Table 3. NBB4, EA, EAc, and PA reult for the eght tranformaton (DDG bnd or DDG MM!QM n kj/mol). Quantty DDG bnd DDG MM!QM DDG bnd w max Method NBB4 EA EAc PA EAc EA Averagng all 15 ndep all all 15 ndep all 15 ndep all MeBz!Bz EtBz!MeBz pclbz!bz mclbz!bz Hx!Bz MeHx!Hx Hx!Pen Hep!Hx Reult are hown for ether all 60,000 naphot n a ngle calculaton wth tandard error obtaned wth boottrappng (all) or a the average p ffffffffff over 15 ndependent mulaton wth 4000 naphot each, obtanng tandard error from the tandard devaton over thee 15 reult, dvded by 15 (15 ndep). In the lat column, w max gven for the EA all calculaton. calculaton for all k value (Fg. 1c). However, th very demandng, requrng 60, ,800,000 QM calculaton for each tranformaton (60,000 naphot, 13 k value, two et of mulaton, that, wth or wthout the hot, calculaton wth both L 0 and L 1, and three calculaton for each geometry to get nteracton energe from eq. (4), but E remander the ame for the two lgand). Moreover, the calculaton gave many numercal problem and hghly uncertan reult. The reaon for th partly that the MM calculaton employ oft-core potental, whch ncreae the dfference between QM and MM and therefore deterorate the convergence of the MM!QM perturbaton. Therefore, th approach wa only attempted for the MeBz!Bz tranformaton and for 4000 naphot, gvng DDG NBB13 bnd kj/mol. We alo tred to calculate the bndng free energe drectly wth MBAR calculaton baed on the QM reult (QM-MBAR; ung the ame data a NBB13), gnorng the fact that the naphot were obtaned wth MD mulaton at the MM, rather than the QM level. [25] However, the reult baed on only 4000 naphot for the MeBz!Bz tranformaton wa poor ( kj/mol), wth large dfference between etmate obtaned wth dfferent method (BAR, TI, and EA) and many overlap etmate ndcatng poor overlap, for example, P down to 22.0, w max up to 1.0, and DDG EA up to 949 kj/ mol. Therefore, th approach wa not further purued. Intead, we teted the NBB4 approach uggeted by K ong and coworker. [29,30] In th approach, NBB ued to etmate the free energy of gong from k wth QM to k wth MM (and mlar between k and 1.00), wherea the dfference between k and 0.95 etmated at the MM level, a llutrated n Fgure 1b. NBB4 requre 60, ,440,000 QM calculaton (.e., only for k and 0.05 wth L 0, and for k and 1.00 wth L 1 ), whch 5.4 tme fewer than wth NBB13. The NBB4 reult are collected n Table 3. Two et of reult are preented: The frt for an NBB4 calculaton ncludng the concatenated reult of all 60,000 naphot wth the tandard error etmated by boottrappng. The econd the average over the 15 ndvdual ndependent calculaton wth 4000 naphot n each and the tandard error calculated from the tandard devaton over the 15 et. It can be een that the uncertanty of the former approach omewhat larger than for the latter, 2 7 v. 2 3 kj/mol. The oppote normally oberved, whch ndcate that the reult trongly depend on a few naphot, that, that the calculaton are tll poorly converged. In mot cae, the reult of the two et of calculaton agree wthn the etmated tattcal uncertanty, wth dfference of 1 8 kj/mol. However, for the mclbz!bz tranformaton, the dfference 20 kj/mol, howng that the NBB4 etmate do not fully how the expected tattcal behavor. Therefore, we ntead tred to etmate the MM!QM free energy by the EA approach. A drect applcaton of EA [.e., wth the full exponental averagng n eq. (6)], gave an uncertanty mlar to that for NBB4, 2 6 kj/mol (thrd column n Table 3). Moreover, w max (lat column n Table 3), howng that the exponental average domnated by one or a few term (naphot). More table reult could be obtaned by ung a cumulant expanon to the econd order [70,71] (whch equvalent of aumng a Gauan dtrbuton; EAc). Wth uch an approach, the uncertanty wa reduced to kj/mol for all eght tranformaton (baed on calculaton on all 60,000 naphot and boottrapped uncertante; column four n Table 3). Cloely mlar reult were obtaned by averagng the reult from the 15 ndependent mulaton (wth 4000 naphot n each; column fve n Table 3): The two approache gave reult that agreed wthn 0.2 kj/mol and the uncertante (etmated from the tandard devaton over the 15 mulaton) alo agreed wthn a factor of Moreover, the average tandard error for the ndvdual calculaton baed on 4000 naphot were tme larger than the tandard error for the pffffffffffffffffffffffffffffffffffffffffffffffffff calculaton baed on 60,000 naphot, p that, cloe to 60000=40005 ffffffffff , followng the pffffff expected n dependence of a normal dtrbuton (n fact, we elected the fnal number of naphot baed on uch an extrapolaton). Fgure 5 how how the predcton converge and the precon mprove wth the number of naphot. The reult of the EA and EAc method agree wthn 1 14 kj/mol, whch nde the 95% confdence nterval 1596 Journal of Computatonal Chemtry 2016, 37,

9 FULL PAPER Fgure 5. a) Convergence of the EAc predcton of DDG MM!QM wth repect to the number of condered naphot for the eght tranformaton. Pane b) how the correpondng tandard error of the calculaton, baed on 1000 boottrap. (domnated by the uncertanty of EA) for all tranformaton, except mclbz!bz and Hep!Hx, ndcatng that the EA reult are not fully well-behavng. In Fgure S1 n the Supportng Informaton, dtrbuton and normal-probablty plot are gven for three of the MM!QM perturbaton. It can be een that all E QM E MM dtrbuton are very cloe to normal, except n the low-probablty end. The two frt example how typcal reult for mulaton wth and wthout the hot, repectvely, for whch the dtrbuton Gauan beyond probablty, wherea the lat row how the pooret reult, for whch devaton from Gauan dtrbuton tart to emerge at 0.02 probablty. Three of the lgand are nvolved n more than one perturbaton (Bz, MeBz, and Hx). Therefore, we have 2 4 etmate of DG MM!QM for thee for the mulaton wth or wthout the hot, and thee etmate are collected n Supportng Informaton Table S2. In mot cae, the reult of thee calculaton agree, for example, to kj/mol for Bz wth the hot (average kj/mol), n agreement wth the etmated uncertanty of 0.4 kj/mol for the ndvdual etmate. However, n three cae, one of the mulaton gve devatng reult by 5 9 kj/mol. Th ndcate that the EAc reult tll are omewhat entve to rare event n the mulaton and occaonally the etmated precon too hgh. Fnally, we alo tred to etmate the MM!QM free energy wth the pure average (PA), ntead of the exponental average n the EA approach. Th gave well-converged reult wth a tandard error of 0.2 kj/mol, reflectng that a pure average ha much better convergence properte than the exponental average (xth column n Table 3). However, the pure average an approxmaton to the true exponental average, an approxmaton that vald only f the varaton n the E QM E MM energy dfference mall, whch not the cae. Therefore, the pure average converged to reult that were dfferent from thoe obtaned wth EAc. For the DDG MM!QM correcton n Table 3, the dfference between the PA and EAc reult for the varou tranformaton wa up to 14 kj/mol (5 kj/mol on average), a dfference that tattcally gnfcant for fve of the tranformaton. However, DDG MM!QM obtaned a the dfference of the reult for the calculaton wth and wthout the NOA hot, whch each are the dfference of the reult obtaned wth L 0 and L 1 [eq. (2) and (5)]. For thee four contrbuton to DDG MM!QM, the dfference between PA and EAc wa much larger, kj/mol. Th clearly how that pure average cannot be ued f you am at an accuracy better than 10 kj/mol, epecally a there no ueful etmate of the true uncertanty of the approach. The NBB and EA reult dcued o far are not comparable, becaue the former are full DG bnd free energe, wherea the latter only nclude the MM!QM free-energy correcton (DDG MM!QM n Table 3). Thu, to compare the reult, we need to add DDG MM bnd, obtaned for the olated QM ytem at the MM level. Th done n the penultmate column n Table 3. It can be een that the NBB4 and EAc reult agree wthn 1 9 kj/mol (baed on average of the 15 ndependent mulaton), whch reaonable, conderng the qute large uncertanty of the NBB4 reult. All SQM reult dcued up to th pont have been baed on calculaton of the olated QM ytem. More realtc energe can be obtaned by a QM/MM approach. A dcued n the Method ecton, QM and QM/MM energe gve the ame reult for EA, o we ealy reach a fnal reult by addng the MM!QM EAc free-energy correcton n Table 3 (column fve) to the DDG bnd free energe, obtaned at the MM level for the full perodc ytem from the econd column n Table 1, gvng the reult n lat column of Table 1. Thee reult dffer lghtly from the reult n the penultmate column n Table 3, becaue the latter employ MM DDG MM bnd bndng free energe obtaned for only the QM ytem and ung nteracton energe ntead of total potental energe (to make the reult drectly comparable wth the NBB4 reult n Table 3 whch ue the ame MM DDG MM bnd free energe). The two reult dffer by up to 3.5 kj/ mol (1.5 kj/mol on average), howng that already the mall QM ytem gve reaonable DDG MM bnd free-energy etmate. From Fgure 6, t can be een that all MM!QM correcton are n the correct drecton (aumng that NOA hould gve the ame reult a OA), that, reducng the relatve affnte, except n the EtBz!MeBz cae, for whch the MM!QM correcton cloe to zero. Unfortunately, the correcton are too large n x of the cae, gvng too mall relatve bndng affnte. Conequently, the SQM/MM reult reproduce the expermental OA reult apprecably wore than the MM reult, a hown n Table 1. For example, MAD ncreae to kj/mol and r 2 vanhe. Of coure, th omewhat Journal of Computatonal Chemtry 2016, 37,

10 FULL PAPER Fgure 6. Comparon of the MM and SQM/MM (EAc) reult for NOA, compared to the expermental relatve affnte [44] for the eght condered tranformaton. The black lne how the perfect correlaton. [Color fgure can be vewed n the onlne ue, whch avalable at wleyonlnelbrary. com.] dappontng after performng almot 6 mllon SQM calculaton. However, th reult much better than prevou attempt to obtan QM/MM FES bndng free energe for the ame ytem, whch gave MAD of kj/mol and no convergence for the MM! QM perturbaton. [13] It alo much better than approache baed on optmzed tructure and energe calculated by dperon-corrected DFT method or even CCSD(T) calculaton, gvng MAD of 8 37 kj/mol. [13 15] Moreover, our am ha been to fnd out what requred to converge the MM!QM/MM, not to reproduce the expermental reult. Therefore, we have elected a rather cheap method, PM6-D2HX, whch among the bet avalable SQM method, although t apprecably wore than dperoncorrected DFT method. [77] Thu, at leat for th data, EAc wth the cumulant expanon gave a lower uncertanty than NBB4. A poble reaon for th agan the ue of oft-core potental n the MM calculaton, whch ncreae the dfference between QM and MM: In the EA approach, only the k and 1.00 tate are condered, for whch the oft-core potental are not actve. However, NBB4 conder alo the k and 0.95 tate, for whch the oft-core actve. The EA approach alo ha the advantage of ung only 60, ,000 QM calculaton (.e., only for L 0 at k and L 1 at k ), that, half a many a for NBB4. Moreover, the reweghtng n NBB often badly condtoned, eentally pckng out a ngle energy (naphot) n the econd average n the nomnator of eq. (9). In fact, w max calculated for th method exactly the ame a for EA (hown n the lat column n Table 3), that, , ndcatng that the etmated free energe are completely domnated by one or a few QM energe (and that the ue of numerou QM calculaton only a way of fndng thee value). Th explan the rather poor convergence of both thee method. On the other hand, wth the cumulant expanon, all QM value are ued to etmate the average and tandard devaton of the Gauan dtrbuton (but both value are tll omewhat domnated by a few value). All reult n Table 3 were obtaned ung nteracton energe [eq. (4)] rather than total energe. Th ha the advantage of makng the E QM E MM energy dfference maller and le varyng by gnorng the dfference n the two energy functon for the nternal nteracton wthn the lgand or the hot. On the other hand, th an approxmaton. At the MM level, t a good approxmaton: For even of the perturbaton, DDG bnd calculated wth nteracton energe (and wthout perodcty and Ewald ummaton) reproduce the reult n Table 1 (baed on total energe) wthn 0.6 kj/mol (MAD 0.3 kj/mol). However, for the Hx!Bz perturbaton, the dfference lghtly larger, 2.7 kj/mol. On the other hand, ung total energe for the MM!QM perturbaton ncreae the varaton n E QM E MM by a factor of 2, makng the convergence much wore. A a conequence, DDG MM!QM etmated by EAc change by 1 12 kj/mol (5 kj/mol on average; pure average change by 1 4 kj/mol) and the boottrapped precon etmate become very large, llutratng that thee reult are far from converged. Th how that nteracton energe trongly mprove the convergence of the MM!QM perturbaton, n agreement wth other tude. [38,78] Concluon In th artcle, we have tuded what needed to obtan converged QM/MM relatve bndng free energe, performng amplng only at the MM level and ncludng a gnfcant urroundng of the lgand n the QM calculaton ( atom). Prevou tude wth uch an approach have gven poorly converged reult both for a hot guet ytem and a full proten, probably owng to the ue of too few QM calculaton (3600 per tranformaton). [13,25] Therefore, we have here employed a ytem for whch we can perform many more QM calculaton: Frt, we ued the octa-acd hot guet ytem, whch maller and mpler than a proten. Second, we removed all eght carboxylate group on the hot molecule to further reduce the ze of the ytem, to remove poble problem of the extenve net charge of the hot, and to reduce the flexblty of the hot and therefore mprove the amplng of the phae pace. Thrd, we employed the SQM PM6-DH2X method, whch computatonally much cheaper than the DFT method we have ued n our prevou tude. On the other hand, th mean that we cannot trctly compare to any expermental reult and that we ue a QM method wth an apprecably lower accuracy than dperon-corrected DFT method. [77] We frt howed that the truncaton of the hot gave rather retrcted change n relatve bndng free energe, a etmated at the MM level, up to 3 kj/mol. Moreover, the new calculaton were much more prece than thoe baed on the full OA hot, wth a precon of kj/mol. Th wa partly an effect of the longer mulaton (120 n per k value), but there were alo clear ndcaton that the amplng ha been mproved. In partcular, our overlap meaure clearly howed that the mulaton were perfectly converged, n varance to the orgnal mulaton Journal of Computatonal Chemtry 2016, 37,

11 FULL PAPER Next, we teted x dfferent method to calculate relatve bndng free energe at the QM level. We howed that approache baed on QM calculaton for all 13 k value (both NBB13 and QM-MBAR) were very expenve and gave poorly converged reult. On the other hand, NBB4 and EA gave promng reult, although a full convergence could not be obtaned even wth 1,440,000 or 720,000 QM calculaton per tranformaton, repectvely. Intead, the reult ndcated that 3 50 tme more QM calculaton are needed for convergence and th mot lkely an underetmate owng to the bad condtonng of thee method (they trongly depend on one or very few of the calculated value). However, the EAc approach wth the cumulant expanon gave ncely converged reult wth a tandard error of 1 kj/ mol ung 720,000 QM calculaton per k tranformaton for all eght tuded relatve free energe. Moreover, t howed the expected quare-root dependence of the tandard error wth repect to the number of calculaton. It alo requred half a many QM calculaton a the NBB4 approach. Pure average for the MM!QM perturbaton alo gave converged free energe, but the reult dffered from thoe obtaned by EAc by up to 14 kj/mol, becaue th only an approxmate method that trctly hould not work when the varaton n the MM QM energy dfference large. The requred number of QM calculaton of a comparable magntude to what ha been ued n prevou QM/MM-FES tude by Mulholland and coworker, [24,26] epecally a they ncluded only a ngle rgd water molecule n the QM ytem. K ong et al. alo had to perform 20,000 60,000 QM calculaton to obtan an uncertanty of up to 2 kj/mol, agan ncludng only a mall olute n the QM ytem. [30] However, Skylar et al. have ncluded the olute and 200 water molecule n the QM calculaton, calculatng the free energe by EA, baed on nteracton energe. [27] Stll, they clam to obtan converged relatve olvaton free energe (wthn 4 kj/mol) wth only 1080 QM calculaton per tranformaton. The reaon for th eem to be a maller dfference between the QM and MM potental (although they ue the ame GAFF/TIP3P MM method a we do): They report a range for the E QM E MM energy dfference of 55 kj/mol, wherea t almot four tme larger n our tudy, kj/mol. The convergence of FES trongly depend on th range n the well-converged FES calculaton at the MM level (wth 13 k value), the range typcally 10 kj/mol wth a maxmum of 38 kj/mol. The reaon for the lower range n the tude by Skylar et al. probably that they tudy only mple and rgd phenol dervatve. In a recent tudy wth more flexble (but maller) molecule, they employed more QM calculaton and ncluded alo an ntermedate QM/MM tep to mprove the convergence wth only the olute n the QM, performng QM/MM MD mulaton. [31] Fnally, t hould alo be noted that K ong et al. have n two tude uggeted that NBB4 gve better convergence than EA (for abolute olvaton free energe). [29,30] However, they dd not employ the cumulant expanon for EA, whch trongly mprove the convergence n th tudy. In fact, very recently they publhed a tudy of hydraton free energe of 20 organc molecule, n whch they come to concluon very mlar to our: [78] Wth a cumulant expanon, they obtan a lghtly better convergence of the QM/MM free energe wth EAc than wth NBB4, both analytcally and n practce. Th how that our concluon apply alo to other type of ytem. Conequently, we recommend the EAc method to obtan converged QM/MM bndng free energe. Stll, our reult how that a very large number of QM calculaton are needed to obtan trct QM/MM FES bndng free energe, 720,000 per perturbaton. Th provde a ueful gude for future tude of QM/ MM bndng free energe: The QM method and the ze of the QM ytem have to be elected to allow for uch an amount of QM calculaton. Moreover, t provde a frm ba for comparon wth alternatve method. It ha recently been hown that reaonable bndng free energe can be obtaned wth ngle tructure optmzed wth dperon-corrected DFT method for hot guet ytem; [77,79] n fact, relatve energe for lgand bndng to the ame hot were reproduced wth a MAD of 5 kj/ mol. Such an approach requre only a few hundred QM energy calculaton. Unfortunately, the approach worked apprecably wore for the OA ytem, wth MAD of 5 10 kj/mol, probably owng to the larger flexblty and the hgh charge of th hot. [13,14] Alternatvely, full QM/MM MD mulaton could be performed; 720,000 QM calculaton correpond to n mulaton, dependng on the tme tep, whch may gve converged FES reult. Thu, t mght be better to pend the QM calculaton on true FES calculaton wth amplng at the QM/MM level ntead. Th currently nvetgated n our group. Acknowledgment The computaton were performed on computer reource provded by the Swedh Natonal Infratructure for Computng (SNIC) at Lunarc at Lund Unverty and HPC2N at Umeå Unverty. 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