Keywords

1 Introduction

Coarse-grained (CG) models have proven to be useful in many fields of chemical research [1–10], allowing molecular simulations to be performed on larger system sizes and access longer timescales than is possible with atomistic-level models, enabling complex phenomena such as hierarchical self-assembly to be described [11, 12]. In CG simulations of aqueous systems, especially ones with significant amounts of hydrophobic and/or hydrophilic interactions, the water model is important and can have a major impact on the resulting properties of the system [13].

While the assignment of atoms to CG beads (i.e., defining the CG mapping) is relatively straightforward for most chemical systems (e.g., aggregating four methyl groups bonded in sequence into a single CG bead), mapping an atomistic water trajectory to the CG level (i.e., grouping several water molecules into a single CG bead) is not as well-defined given the lack of permanent bonds between water molecules. Even if a mapping were chosen, water molecules will diffuse away from their initial clusters over time, such that the initial mapping is no longer representative of the local clustering of water. This ambiguity presents a problem for structure-based methods that require an atomistic configuration to be mapped to the corresponding CG configuration, e.g., to generate a target radial distribution function (RDF) against which the forcefield is optimized. As such, the majority of many-to-one CG models of water (i.e., where one CG bead represents multiple water molecules) have instead been derived by assuming a functional form of the forcefield and optimizing the associated parameters to match selected physical properties of water, such as density, vaporization enthalpy, surface tension, etc. [13–19]. For example, Chiu et al. developed a 4:1 CG water forcefield by optimizing the parameters of a Morse potential to accurately reproduce the surface tension and density of liquid water [18]. Despite capturing the interfacial properties and density, this potential overestimates structural correlations, as one might expect given that structural data was not used in its optimization.

Recently, Hadley and McCabe [20] proposed a method for mapping configurations of atomistic water to their CG representations using the k-means clustering algorithm. Subsequently in related work, van Hoof et al. [21] developed the CUMULUS method for mapping atoms to CG beads. Both methods enable dynamic mapping of multiple water molecules to a single CG bead, allowing structure-based schemes to be used. Here, dynamic refers to a CG mapping that changes over the course of the atomistic trajectory, i.e., different water molecules are assigned to different CG beads in each frame of the atomistic trajectory. Both works employed the iterative Boltzmann inversion (IBI) [22] method to derive the intermolecular interaction by optimizing a numerical, rather than analytical, potential to reproduce RDFs calculated from the atomistic-to-CG mapped configurations [20, 21]. The forcefields derived are similar and show good agreement with the structural properties and density of the atomistic water models studied. However, neither model is able to accurately reproduce interfacial properties, since they were derived solely from bulk fluid data. This failure to capture interfacial properties is a consequence of the single-state nature of the IBI approach and may alter the balance of hydrophobic and hydrophilic interactions when using these water models in multicomponent systems.

Recently, the multistate IBI (MS IBI) method [23] was developed as an extension of the original IBI approach, with the goal of reducing state dependence and structural artifacts often found in IBI-based potentials [24–26].  While IBI-based potentials have been derived that show some degree of transferability [26–28] a significant issue related to the IBI method is that a multitude of potentials can give rise to similar RDFs, and the method cannot necessarily differentiate which of the many potentials is most accurate, as only RDF matching is considered. MS IBI operates based on the idea that different thermodynamic states will occupy different regions of potential “phase space” (i.e., regions where potentials give rise to similar RDFs), and that the most transferable, and thus most accurate, potential lies in the overlap of phase space for the different states. That is, by optimizing a potential simultaneously against multiple thermodynamic states, MS IBI provides constraints to the optimization, forcing the method to derive potentials that exist in this overlap region, and thus are transferable among the states considered. The MS IBI approach has been shown to reduce state dependence and improve the quality of the derived potentials, as compared to the original IBI method [23].

In this work, multistate iterative Boltzmann inversion (MS IBI) is used to derive an intermolecular potential that captures both bulk and interfacial properties of water, improving upon the CG water model of Hadley and McCabe [20]. Again, optimizations are carried out using the MS IBI method, where both bulk and interfacial systems are used simultaneously as target conditions for the optimization. MS IBI is also used, for the first time, in a multi-ensemble context, enabling optimizations in both the canonical (NVT) and isothermal-isobaric (NPT) ensembles to be performed simultaneously to derive the density-pressure relationship of the system. To further constrain the optimization, a slightly modified version of the Chiu et al. CG water forcefield, originally optimized for surface tension, is used as a starting condition, allowing the MS IBI method to make specific modifications to the potential to improve structural properties. The remainder of the paper is organized as follows: In Methods, a brief overview of the k-means clustering and MS IBI algorithms is given and the models used are described. The potential derivation is then presented, validated, and compared to existing CG water models in the Results section and finally, conclusions are drawn about the applicability of the derived CG model and the broader applicability of the MS IBI method discussed.

2 Methods

2.1 k-Means Clustering Algorithm

Mapping a water trajectory to a many-to-one CG level is inherently different than mapping a larger molecule’s trajectory, since for water, atoms mapped into a single CG bead necessarily exist on different molecules. Furthermore, the water molecules mapped to a common bead are not likely to remain associated throughout the full simulation because of thermal diffusion. A dynamic mapping scheme is therefore required to generate CG structures from atomistic configurations for water. Following the work of Hadley and McCabe [20], the k-means algorithm has been used to map atomistic water trajectories to the CG level. In short, k-means is a clustering algorithm that is used to find clusters of data points in a large data set. The positions of the water molecules are here analogous to the points in the data set and waters mapped to a single bead are analogous to the clusters. Additional details on the algorithm can be found elsewhere [20, 29]. While the k-means algorithm can be used to group together any number of water molecules, a 4:1 mapping is chosen, as this was found in prior work to provide the best balance between accuracy and computational efficiency [20] and 4:1 models are common in the literature [17, 18, 20].

2.2 Multistate Iterative Boltzmann Inversion Method

MS IBI was used to derive the intermolecular potential between water beads. The goal of MS IBI is to derive a single potential that can be used over a range of thermodynamic states. As an extension of the original IBI method [22], the potential is updated based on the average differences in CG and target RDFs at multiple states (i.e., a single potential for each pair is updated based on RDFs from multiple states). The potential is adjusted according to

$$ V_{{i{ {+} 1}}} (r )= V_{i} (r) - \frac{1}{N}\mathop \sum \limits_{s} \alpha_{s} (r)k_{\text{B}} T_{s} \ln\left[ {\frac{{g_{s}^{ * } (r)}}{{g_{s}^{i} (r)}}} \right], $$
(1)

where V i (r) is the pair potential as a function of separation r at the ith iteration; N the number of target states; α s (r) an effective weighting factor for state s, allowing more or less emphasis to be put on a particular target state; k B the Boltzmann constant, T s the absolute temperature of state s; g i s (r) the RDF from the CG simulation at state s using V i (r); and g ∗ s (r) the target RDF from state s. α s (r) was chosen to be a linear function of the form

$$ \alpha_{s} (r) = \alpha_{0,s} \left( {1 - \frac{r}{{r_{\text{cut}} }}} \right), $$
(2)

such that α s (r cut) = 0 and the potential remains 0 for r ≥ r cut. This form of α s (r) also places more emphasis on the short-ranged part of the potential to suppress long-range structural artifacts.

An initial potential is assumed for each pair interaction. In theory, there are no restrictions on the initial potential, so it may take any form; however, in practice, the initial potential is often taken to be the potential of mean force (PMF) calculated from the Boltzmann inverted RDF. In this work, rather than taking an average of the PMFs over the states used, the initial potential used was chosen to be a slightly modified version of Chiu et al.’s water model, as discussed below. That is, rather than starting from an initial potential that is likely to do a poor job of predicting the behavior, we start from a robust starting point as the Chiu et al. potential is known to accurately reproduce several properties of water.

A CG simulation is then run with the initial potential. Based on the RDFs from the CG simulation, the potential is updated according to Eq. (1). The updated potential is used as input to the next cycle, and the process is repeated until some stopping criterion is met. Here, the stopping criterion is determined using the following fitness function

$$ f_{\text{fit}} = 1 - \frac{{\mathop {\int\limits_{0}^{{r_{\text{cut}} }} {{\text{d}}r\left| {g^{i} (r) - g^{*} (r)} \right|} }\nolimits_{{}}^{{}} }}{{\mathop {\int\limits_{0}^{{r_{\text{cut}} }} {{\text{d}}r\left| {g^{i} (r)} \right| + \left| {g^{*} (r)} \right|} }\nolimits_{{}}^{{}} }}, $$
(3)

where the optimization is stopped when the value of f fit exceeds a specified value (i.e., meets some tolerance), given below.

2.3 Models

Atomistic simulations of pure water were performed with the TIP3P model [30]. All atomistic systems contained 5,832 water molecules and were simulated in LAMMPS [31, 32] using a 1 fs timestep. A cutoff distance of 12 Å was used for the van der Waals interactions; long-range electrostatics were handled with the PPPM method with a 12 Å real space cutoff. Three distinct states were simulated: bulk, NVT at 1.0 g/mL and 305 K; bulk, NPT at 305 K and 1.0 atm; and an NVT droplet state at 305 K, where the box from the bulk NVT state was expanded by a factor of 3 in one direction. Each atomistic simulation was run for 7 ns. The atomistic trajectories were mapped to the CG level using the k-means algorithm. Target RDFs were calculated from the final 5 ns of the mapped trajectory from each state (bulk NVT, bulk NPT, and droplet NVT). MS IBI was performed using the target data from each of the three states. The initial guess of the potential is given as a Morse potential of the form

$$ V(r) = D_{e} \left( {e^{{{ - 2}\beta \left( {r{ - }r_{\text{eq}} } \right)}} { - 2}e^{{ - \beta \left( {r{ - }r_{\text{eq}} } \right)}} } \right), $$
(4)

where r eq is the location of the potential minimum, −D e is the value of the potential minimum, and β is related to the width of the potential well. Parameters are taken to be those from Chiu et al: D e  = 0.813 kcal/mol, β = 0.556 Å−1, and r eq = 6.29 Å, however, we note that the potential was adjusted so that β = 0.5 Å−1 for r < r eq. This change was made to increase sampling at small separations, because numerical issues arise in the potential update when the CG RDF is zero but the target RDF is nonzero. This modification of the potential will slightly alter the properties as compared to the original model, as discussed below. The potential update scaling factor α 0,s (see Eqs. 1 and 2) was set to 0.7 to avoid large updates to the potential. The optimizations were stopped when f fit ≥ 0.98 and f fit(i) − f fit(i − 1) < 0.001 for each state.

All optimizations were performed with the open-source MS IBI Python package we developed [33], which calls HOOMD-Blue [34–36] to run the CG simulations and uses MDTraj [37, 38] for RDF calculations and file-handling. CG simulations were run at the same states as the atomistic systems. Initial CG configurations were generated from the CG-mapped atomistic trajectories at each state. As a result of the 4:1 mapping, CG water simulations contained 1,458 water beads. All CG simulations were run with a 10 fs timestep. The derived CG potential was set to 0 beyond the cutoff of 12 Å.

The surface tension γ of the droplet state was calculated as

$$ \gamma = \frac{1}{2}L_{z} \left\langle {P_{zz} - \frac{{P_{xx} + P_{yy} }}{2}} \right\rangle , $$

where L z is the length of the box in the expanded direction, P zz is the pressure component in the direction normal to the liquid-vapor interfaces, P xx and P yy are the pressure components in the directions lateral to the interfaces, and the angle brackets denote a time average. The factor of ½ is included to account for the two interfaces that are present in the droplet simulation setup.

3 Results and Discussion

3.1 Modified Chiu Potential

Since the MS IBI optimization of water uses a modified version of the Chiu, et al. potential as an initial guess, we first consider the impact of modifying the potential to create a softer repulsion. Figure 1 plots the RDFs of the three target states for the original and modified potentials and the RDF of the 4:1 mapped state (i.e., the target data used later for the MS IBI optimization). The peak location of the NVT state is relatively unchanged; however, upon modification, there is a slight shift in the first peak for the NPT and interfacial states, allowing the model to access smaller separations, as was intended and required for the potential update scheme. The softer potential allows closer contact and thus allows the MS IBI algorithm to modify this region of the potential where the 4:1 mapped atomistic water has non-zero values of the RDF. The density predicted with both potentials is the same (0.991 ± 0.003 g/mL); however, due to softening the potential, the calculated surface tension of the droplet changes from 70.3 to 45 mN/m after the modification, although this value is still sufficient for the droplet to maintain a stable interface. These surface tension values agree favorably with that of TIP3P water, which is reported to have a surface tension of 52.3 mN/m at 300 K [39].

Fig. 1
figure 1

RDFs from simulations using the original and modified Chiu potentials. Top NVT; top-middle NPT; bottom-middle interface; bottom comparison of the two potentials

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3.2 Potential Derivation and Validation

Starting from the modified Morse potential of Chiu, et al., the new water forcefield is optimized using the bulk NVT and NPT states and the interfacial state. This potential is chosen as the initial starting guess, rather than an arbitrary starting point, as the unmodified version has been shown to accurately reproduce many properties of water (e.g., density and surface tension), but overestimates the structural correlations. The use of MS IBI should allow for modification of this potential, such that it is able to reproduce structural quantities. The results of the potential derivation are summarized in Fig. 2, where it is clear that the modified Chiu, et al. potential (i.e., step 0) overestimates the structural correlations, as was also seen in Fig. 1 for both the modified and original potentials. After only a few iterations, the RDFs match the targets with a high degree of accuracy. This trend is shown in Fig. 3, which plots the fitness value from Eq. (3) as a function of iteration. The value of f fit changes most rapidly in the first 3 steps of the optimization. After 10 iterations, the stopping criteria are met and the optimization stopped. While the changes to the potential are small, there is a noticeable shift in the location of the minimum to a slightly larger r value and the potential becomes slightly more attractive. Although the shape of the attractive well is mostly unchanged, the potential more rapidly decays to 0 than the original Morse potential at larger r values, while the shape of the repulsive regime at small r is changed slightly. These subtle changes to the potential are sufficient to create significant changes in the RDF and provide excellent convergence of the structural correlations. These changes are made possible by modifying a numerical potential rather than adjusting parameters for an analytical potential. Note that in Figs. 1 and 2 the RDFs from the interfacial state do not decay to 1 at large r. This is due to the fact that 2/3 of the box is essentially devoid of particles, but the RDF is normalized based on the volume of the whole simulation box. This has no effect on the potential update scheme, as both the target and CG RDFs are normalized by the same factor, which cancels out in Eq. (1).

Fig. 2
figure 2

RDFs and potentials from the MS IBI potential derivation. Top NVT; middle-top NPT; middle-bottom interface; bottom potentials. The initial potential shows significant structural correlations missing from the target data. The derived potential at ten iterations shows excellent structural agreement with the target

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Fig. 3
figure 3

f fit from Eq. (3 ) as a function of iteration in the potential derivation. Convergence with the criterion is found after 10 iterations

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In addition to accurately capturing the RDFs, the multi-ensemble approach provides an accurate estimate of the density at 305 K and 1 atm. NPT simulations performed using the optimized CG forcefield find a density of 1.027 ± 0.006 g/mL, compared to 1.037 ± 0.004 g/mL for TIP3P water which was used to generate target data. This approach is successful because the RDFs will not match if the pressure-density relationship is not satisfied, as the density is implicitly represented in Eq. (1) through the RDF terms (i.e., the RDFs at the NPT state will not match the target RDFs if the density is significantly different than the density of the target state). In contrast, the original IBI method proposed the use of a pressure correction term of the form ∆V(r) = A(1 − r/r cut) to account for the pressure [22]. This approach has been successful, but requires a somewhat arbitrary estimate of the parameter A. While a method exists for estimating A based on the virial expression [40], some degree of trial-and-error is still necessary. Furthermore, the multi-ensemble approach within MS IBI does not require direct calculation of the pressure, which often demonstrates considerable fluctuations, providing a simpler route to account for pressure in the CG model.

Calculation of the surface tension of the derived MS IBI potential yields a value of 42 mN/m, lower than the original Chiu, et al. potential (70.3 mN/m) which was optimized to match experiment, but only slightly perturbed from the modified potential (45 mN/m). This reduction in surface tension appears directly related to the softening of the potential, although, we note that this softening is required to provide an accurate match of the structure and that this value reasonably approximates the surface tension of the atomistic TIP3P model used as target data (52.3 mN/m at 300 K) [39].

3.3 Validation and Comparison to Other Models

To further explore the efficacy of the MS IBI-derived model, comparisons are made to other CG water models in the literature, namely, the k-means based potential of Hadley and McCabe [20] derived via the single state (SS) IBI procedure (here referred to as the SS IBI potential) and the MARTINI potential [17]. These models were chosen because they are short-ranged, non-polarizable, and 4:1 models. For reference, these potentials are plotted in Fig. 4. Note that the MS IBI and SS IBI potentials are numerical (as they were derived via IBI), while the MARTINI potential is represented by a 12-6 Lennard-Jones potential with a well depth of 1.195 kcal/mol located at a separation of 5.276 Å. Note that all of the potentials considered in this paper provide a close estimate of the density of water at 1 atm and 305 K, as reported in Table 1.

Fig. 4
figure 4

Interaction potentials from the CG water models compared in this work. The MS IBI and SS IBI potentials are numerical, derived with structure-based methods. MARTINI is a Lennard-Jones 12-6 potential

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Table 1 Density of water at 305 K, 1 atm calculated with different models
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First considering the SS IBI potential, it can be seen that the well depth is approximately 0.5 kcal/mol weaker than the MS IBI potential and shifted to larger separations. While this has little impact on the density or the structural correlations of the bulk states (not shown), simulations of droplets show that the interfacial properties are not sufficiently captured. Specifically, as shown in Fig. 5, simulations of atomistic TIP3P, SS IBI, and MS IBI water were performed with interfaces. From these it can be clearly seen that the SS IBI potential model fills the box, rather than maintaining an interface. In contrast, the MS IBI model maintains a stable interface in agreement with the atomistic model. Thus, while an exact match to the experimental surface tension is not found for the MS IBI potential, as discussed above, it is still sufficiently strong to maintain a clear interface, providing a significant improvement over the SS IBI potential. We note that the difference between the SS IBI and MS IBI potentials is likely related to the aforementioned issue whereby many potentials can give rise to matching RDFs, and SS IBI provides no means to determine which ones are most physical. This limitation is overcome by the use of the interfacial state during the MS IBI optimization.

Fig. 5
figure 5

Simulation snapshots of droplets using the various models discussed. Top all-atom TIP3P; middle SS IBI; bottom MS IBI. Atomistic and MS IBI models agree, producing a system with a stable interface, whereas SS IBI does not form a stable interface

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It is also important that the potential is not so strong that the system can solidify at physiological conditions. For example, the MARTINI water model is known to spontaneously crystallize at physiologically relevant temperatures [17]. This phenomenon is enhanced by the presence of interfaces (e.g., a lipid bilayer surface), and requires the addition of unphysical “antifreeze” particles to avoid crystallization. While we note that modifications to the MARTINI water model exist (e.g., adding charge polarization) [41, 42], only the original MARTINI model was tested, since it more closely resembles the model derived via MS IBI (i.e., both represent 4 water molecules as a single, spherically symmetric interaction site). To test the crystallization tendency, a nucleation site is generated with the following protocol. A crystalline state is generated by running a simulation with the MS IBI potential in the NVT ensemble. During this simulation, the temperature is decreased from 305 to 1 K over 10 ns. A subsequent CG simulation is run at 1000 K, where the middle-most 1/8th of the beads are kept fixed, resulting in a configuration that contains a crystal seed surrounded by a fluid of CG water beads. The beads in the crystal seed are kept fixed in the nucleation site simulations, with interactions identical to the fluid interactions. While neither model shows a tendency to freeze at 305 K in the absence of a nucleation site over a 100 ns simulation, the MARTINI model rapidly crystallizes in the presence of a nucleation site, while the MS IBI potential remains fluid (Fig. 6 ). Note, for a direct comparison with the MS IBI model derived here, antifreeze particles were not used with the MARTINI model. To ensure that the MS IBI system is not an amorphous solid structure, the ratio of the diffusion coefficients with and without a nucleation site were calculated for each model from the slope of the mean-squared displacement. As shown in Table 2, the diffusion coefficient of the MS IBI potential model remains relatively unchanged when a nucleation site is added, whereas a significant drop is seen for the MARTINI model resulting from crystallization. Additionally, Fig. 7 plots the RDF of the MARTINI model for the bulk NVT state as compared to the 4:1 mapped target data. Clearly, the MARTINI potential does not accurately capture the structural correlations of bulk water, further demonstrating the significant improvement of the MS IBI model in reproducing key properties of water.

Fig. 6
figure 6

Configurations from simulations in the presence of a nucleation site with the MARTINI (left) and MS IBI (right) models. CG water beads colored silver were kept fixed during the simulations, but were treated as the same type as the blue particles (i.e., the color is different to show the nucleation site)

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Table 2 Ratio of diffusion coefficients from simulations with (D nuc) and without (D bulk) a nucleation site with different potentials
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Fig. 7
figure 7

RDFs of the MARTINI model and the atomistic TIP3P model mapped to the CG level for the bulk NVT state

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We note that the self-diffusion coefficient of MS IBI water is calculated to be 16.07 × 10−9 m2/s at 305 K and 1 atm, as compared to 3.05 × 10−9 m2/s for the atomistic TIP3P water at the same conditions, both run for 5 ns. This factor of ~5 difference is not entirely unexpected, given the softening of the free energy landscape that often comes with CG models and the fact that kinetic data was not used in the optimization. However, we also note that the dynamics of the CG model does not bear a strong connection with the atomistic level behavior, given that each CG bead represents 4 water molecules, but not necessarily the same water molecules through time, due to the lack of permanent bonds between the waters being grouped together.

4 Conclusions

In this work, the MS IBI method was used to derive the interactions for a 4:1 mapped CG water model, using a modified version of the Chiu, et al. potential as an initial guess. An improvement over previous models is made by simultaneously matching the fluid structure to target data from bulk and interfacial states. It was shown that a model that reproduces the structure and density of water does not necessarily reproduce the interfacial properties and that the addition of a droplet target state constrains the potential to also capture the interfacial properties. The resulting potential is able to accurately predict the density of water at 305 K and 1 atm, interfacial properties, and structural correlations. Additionally, the model shows no tendency to spontaneously crystallize at physiological conditions. This is important, since inaccuracies in a water model can propagate as more potentials are derived against it when simulating mixed systems.

This work highlights a key advantage of deriving potentials via the MS IBI approach. For simulations that cover multiple states, it is important to have a forcefield that is accurate across the states of interest. MS IBI allows this to be achieved by including target data from states that represent structures present in the states of interest. This is realized here by including a multi-ensemble state to accurately model the pressure-density relationship, and a droplet state to capture the interfacial properties of water. Another case where this would be beneficial is studying systems over multiple phases, e.g., phase transitions in liquid crystals. While clever approaches are taken to capture behavior across multiple states [43], a more systematic approach would be useful. Based on the results presented here, we foresee this method being useful for deriving CG potentials for a wide range of applications.