Mapping saddles and minima on free energy surfaces using multiple climbing strings

Abstract

Locating saddle points on free energy surfaces is key in characterizing multistate transition events in complicated molecular-scale systems. Because these saddle points represent transition states, determining minimum free energy pathways to these saddles and measuring their free energies relative to their connected minima are further necessary, for instance, to estimate transition rates. In this work, we propose a new multistring version of the climbing string method in collective variables to locate all saddles and corresponding pathways on free energy surfaces. The method uses dynamic strings to locate saddles and static strings to keep a history of prior strings converged to saddles. Interaction of the dynamic strings with the static strings is used to avoid the convergence to already-identified saddles. Additionally, because the strings approximate curves in collective-variable space, and we can measure free energy along each curve, identification of any saddle’s two connected minima is guaranteed. We demonstrate this method to map the network of stationary points in the 2D and 4D free energy surfaces of alanine dipeptide and alanine tripeptide, respectively.

I. INTRODUCTION

Stationary points on multidimensional potential energy surfaces (PES’s) and/or free energy surfaces (FES’s) can be important in gaining insights into the thermodynamics and kinetics of complex systems of many interacting degrees of freedom. These zero-gradient points correspond to stable or metastable minima and saddle points on their respective energy surface. Although minima can be located relatively easily using standard minimization techniques, the task of locating saddles and measuring their energies relative to their connected minima remains challenging. These saddle points correspond to transition states along the minimum energy path connecting two stable/metastable states and thus act as dynamic bottlenecks for the given transition. Therefore, comprehension of these stationary points is pivotal to compute reaction rates and relative free energies and to understand the underlying transition mechanisms. Unfortunately, these high energy states are highly unlikely to appear in classical molecular dynamics (MD) or Monte Carlo simulations.

To track down saddle points, one needs to find an energy maximum only along one of the directions while ensuring that energy is locally minimal among orthogonal directions. Depending on the initial input information, the methods finding saddle points can be broadly categorized into two classes: (1) the closed or double-ended search methods that require prior knowledge about the location of two minima, for example, the elastic band (EB), the nudged elastic band (NEB),¹ the minimum Hamiltonian path (MHP), step and slide method,² and the string method,³ and (2) the open or single-ended search methods requiring the initial state as the only input, for example, the eigenvector following method,^4,5 the activation-relaxation technique (ART),^6–8 the dimer method,⁹ the one-side growing string (OGS) method,¹⁰ the shrinking dimer method,^11,12 gentlest-ascent dynamics (GAD),^13,14 etc. (Multiple-image methods such as the climbing image NEB¹⁵ and the climbing string¹⁶ are also available which only require information about the local gradient to climb uphill.) Open methods are relatively more challenging as well as interesting, as they may lead to saddles connecting to unidentified stable/metastable states. In these methods, the system is made to climb uphill on the energy surface using full or partial knowledge of the Hessian matrix to find a direction of smallest curvature and stop when the saddle is found. A good comparison of these methods can be found in the review by Henkelman et al.¹⁷

Most of the methods listed above are used to locate the minima and saddle points on potential energy surfaces. However, these methods become less practical for investigating phenomena in complex systems at finite temperatures described by collective variables (CV’s). In such cases, entropic contributions are relevant, as in conformational distributions, protein folding, and protein-ligand binding. In these complex systems described using CV’s, free energy governs thermodynamics and kinetics, and therefore it is often desirable to characterize the transition events on the associated FES. Here, the choice of CVs is crucial and, in general, they should represent the slow modes of the system which can suitably distinguish between different macroscopic configurations and whose saddles represent actual bottlenecks through which a system must pass in order to execute a transition. In recent decades, many enhanced sampling methods have been proposed to explore the FES in CV spaces, including metadynamics, adaptive biased force sampling, umbrella sampling, and others.^18–28 Although such methods work efficiently in low-dimensional CV-spaces, methods to locate saddle points in high-dimensional spaces are relatively rare. In a recent example, Tuckerman and co-workers have recently developed a Hessian-based stochastic activation-relaxation technique (START) by combining the machine learning optimization method with the ART approach to map minima and saddle points on a high-dimensional FES.²⁹ Although the START is a powerful method of navigation on an FES, it is not designed to compute free-energy differences between stationary points concurrently with FES exploration.

In this work, we propose a method for navigating CV spaces that provides free-energy differences by design, based on the string method in CV’s³⁰ and the climbing string method on potential energy surfaces.¹⁶ Our approach introduces “climbing multistrings” in which more than one string tracks connected stationary points. The climbing multistring method takes an initial guess of a path in CV space and optimizes its shape by two evolution and reparameterization operations while climbing uphill to converge one end at a saddle point. In contrast to other methods which evolve a single image, the climbing multistring method uncovers details of the minimum free-energy pathway (MFEP) between a minimum and a connecting saddle. Provided the good set of CVs, the transition/reaction events follow the well-defined channels or thin tube around the MFEP, representing the projection of swaths of trajectories in configuration space. Like a regular string method in CV’s, the climbing multistring method may employ high-dimensional CVs since the efficiency of the method primarily depends on the number of images, not on the dimensionality of the FES. In the climbing multistring method, two kinds of strings are utilized: dynamic strings and static strings. Dynamic strings evolve on the high dimensional FES to locate saddles connected to minima. Static strings store the positions of the images on the strings which have already converged to saddles. Dynamic strings that converge to a saddle point are made static, and new dynamic strings are made to avoid existing static strings. Therefore, the efficiency of the method also depends on the number of static strings. The proposed method only uses the local force gradient and, thus, does not need computation of the Hessian matrix. Despite using the local force gradient, the climbing multistring method converges to first order saddles as the higher order saddles are unstable.

The remainder of this paper is organized as follows: Sec. II introduces the climbing multistring method to locate saddle points and minima. Section III demonstrates the applicability of the multistring method for alanine dipeptide and alanine tripeptide. Concluding remarks are made in Sec. IV.

II. METHODS AND COMPUTATIONAL DETAILS

A. Climbing multistrings

The climbing multistring method is a modification of the simplified string method in CVs³⁰ to locate the saddle points connected to a given minimum through a MFEP. The MFEP can be imagined as a curvilinear path z(α) in CV space that is tangential to the gradient of free energy (∇F) and thus satisfies^30,31

$${\left(M\left(z\left(\alpha \right)\right)\nabla F\left(z\left(\alpha \right)\right)\right)}^{\perp }=0,$$ (1)

where M(z(α)) and ∇F(z(α)) are the metric tensor and negative gradient of free energy, respectively. In order to acquire the MFEP between two locations in the CV space, an initial guess of the curve z(α) is required which comprises a string of N discrete images of the system with a particular parameterization; here, we situate the image at the first position along the string at α = 0 and the final image at α = 1. Provided that the endpoints are fixed or evolving toward the nearest minimum, the position of each intermediate image 0 < α < N on the string is then updated according to

$$\gamma ż\left(\alpha ,t\right)=-M\left(z\left(\alpha ,t\right)\right)\nabla F\left(z\left(\alpha ,t\right)\right)+\lambda \left(\alpha ,t\right)z^{\prime }\left(\alpha ,t\right)$$ (2)

to optimize the pathway. Here, γ is the friction parameter, ż and z′ are the derivatives of z with respect to t and α, respectively, and λ is a Lagrange multiplier which renders the combined effect of the tangential force of the gradient and reparameterization of the string.

The M(z(α, t)) and ∇F(z(α), t) in Eq. (2) are obtained as the conditional expectations by performing restrained simulations at the assigned CV values; each is found from an independent simulation, requiring two MD systems per image. Then, Eq. (2) is solved iteratively to obtain the steady-state solution as the MFEP satisfying Eq. (1). The full details of the string method can be found in Refs. 3, 30, 31, and 32.

In climbing mode, only one end of the string is attracted to (or fixed at) a minimum, while the other end of the string climbs uphill on the FES to locate saddles. The climbing is achieved by modifying the forces on the final image after evolving the string according to Eq. (2) which requires boundary conditions at the end points of the string. The α = 0 end evolves according to

$$\gamma ż\left(\alpha =0,t\right)=-M\left(z\left(\alpha =0,t\right)\right)\nabla F\left(z\left(\alpha =0,t\right)\right).$$ (3)

The climbing end at α = 1 climbs in the direction tangent to the string. This is achieved via the following update:

$$\begin{array}{ccc}\hfill \gamma ż\left(\alpha =1,t\right)& =\hfill & -M\left(z\left(\alpha =1,t\right)\right)\nabla F\left(z\left(\alpha =1,t\right)\right)+\nu \tau \left(t\right)\left(\right.\tau \left(t\right)\hfill \\ \hfill & \hfill & \cdot M\left(z\left(\alpha =1,t\right)\right)\nabla F\left(z\left(\alpha =1,t\right)\right)\left)\right.,\hfill \end{array}$$ (4)

where ν > 1 is a parameter that controls ascent speed and τ = z′(α = 1, t)/|z′(α = 1, t)| is the unit tangent along the string. That is, in Eq. (4), the tangential force along the string is reversed at the end. The unit tangent is approximated between the images N − 1 and N (where image N lies at position α = 1) via backward-difference,

$$\tau =\frac{z\left(N,t-1\right)-z\left(N-1,t-1\right)}{|z\left(N,t-1\right)-z\left(N-1,t-1\right)|}.$$ (5)

To generalize the multistring method, let us adopt the notation that z_i denotes string i and that there are N_d evolving (termed “dynamic”) strings. The multistring method proposed here is designed to launch and evolve a set of such strings $\left\{z_1,z_2,\dots ,z_{N_d}\right\}$ in one go. Hence, unlike conventional climbing image methods which use only one string to locate a single saddle at a time, the climbing multistring method is designed to locate multiple saddles. To prevent two dynamic strings from locating an already-identified saddle, the first time a saddle is located, the last N_f images of the string that located it are stored as a “static” or nonevolving string. As more saddles are found, the number of saddles N_s grows and there is one static string per saddle [Fig. 1(d)]. Each time a static string is created, the dynamic string from which is created is reinitialized along the next trial direction to locate another possible saddle point [Fig. 1(d)]. Strings are tracked with indices, and at any one time instant, there are N_d (constant) dynamic strings and N_s static strings.

FIG. 1.

Snapshots from the climbing multistring simulation in the 2D FES for alanine dipeptide in water at 300 K. (a) The initial pair of strings with string indices 1 (black open circles) and 2 (black solid circles). (b) String 2 initializes along the next trial direction (gray solid circles) as it encounters string 1. (c) Strings converge to a minimum free energy pathway connecting the minimum to saddles. (d) The final N_f images of converged strings are stored as static strings (white solid circles) followed by reinitialization of dynamic strings in next trial directions (black open and solid circles). The units of the free energy are kcal/mol.

Although each dynamic string evolves independently, the advantage of using more than one string, including static strings, is that now these strings can be made to communicate with each other. As a result, strings “know” about the presence of other strings in CV space via interstring communication that can be used to avoid the convergence of more than one string to a given saddle. For each dynamic string z_i in a multistring simulation, we compute an effective “energy” V_i via

$$V_i=\sum _{j>i}^{N_d}g\left(\left|z_i\left(N\right)-z_j\left(N\right)\right|\right)+\sum _{k=1}^{N_s}\sum _{l=N-N_f+1}^{N}g\left(\left|z_i\left(N\right)-z_k\left(l\right)\right|\right),$$ (6)

where

$$g\left(r\right)={\left(\frac{\sigma }{r}\right)}^9.$$ (7)

If V_i exceeds some user-defined tolerance value, string z_i is then reinitialized along a new random trial direction [Figs. 1(a) and 1(b)]. Note that the first summation runs over all dynamic strings whose index j is greater than i. This asymmetry is such that the string with the lower index will always have a higher V_i and will thus be subject to reinitialization first. In addition, the first summation only considers string ends z_i(N) and z_j(N) of dynamic strings, whereas the second term sums over all the fixed images (N_f) corresponding to each static string (N_s).

For a given order of repulsive potential and tolerance, the choice of σ determines the sensitivity of one string toward another string. If the σ is low, one string will sense another string only at very close separations, while if the σ value is too high, any unidentified saddles near an identified saddle might be missed. In general, the value of sigma depends on the number of strings and length of initial string, and should be less than the distances among final images on different initial strings.

B. Simulation details

In order to validate the method, two test systems were considered: (1) alanine dipeptide in explicit water and (2) alanine tripeptide in the vacuum. MD simulations were carried out by employing the CHARMM22 force-field³³ and NAMD 2.11, a software package for MD simulations.³⁴ The multistring method was implemented by using the tclforces interface of NAMD. The equations of motion were integrated using the velocity-Verlet scheme with a 1 fs time step. Periodic boundary conditions were used, and long-range electrostatics were computed using particle-mesh Ewald summation with a grid spacing of 1 Å. The van der Waals interactions were cutoff beyond a distance of 10 Å. All simulations were performed in the canonical ensemble. The temperature was maintained at 300 K using the Langevin thermostat with a damping constant of 5 ps⁻¹.

For alanine dipeptide (acetyl-ALA-methylamide), a single molecule solvated by 274 TIP3P water molecules was considered. Here, ALA represents a single alanine residue. The covalent bonds in water were kept rigid via the RATTLE algorithm. Initially, the system was equilibrated for 10 ns in the isothermal-isobaric ensemble at 300 K and 1 bar. The temperature and pressure were maintained using a Langevin Nosé-Hoover piston with a damping constant of 5 and 10 ps⁻¹ for thermostat and barostat, respectively. For alanine tripeptide (acetyl-ALA-ALA-methylamide), a single molecule was considered in a large cubic box of 30 Å. The system was initially equilibrated for 10 ns in the canonical ensemble. For both systems, the CMAP corrections for torsional angles were not included in the CHARMM force-field. The CMAP term is a cross-term for the backbone dihedral angle ($\varphi$, $\psi$), introduced in CHARMM force-field to improve conformation properties of protein backbone.

C. Procedure to locate saddles and minima

In order to locate the saddles and minima, a combination of climbing and regular string method calculations was employed. For both systems, the Ramachandran (ϕ, ψ) angles of the ALA units were used as CVs. Starting from any location on the FES, the nearest minimum was located by using a regular string method calculation. Subsequently, the pair of strings was employed to locate the saddles connected to the given minimum. The initial length of each string in the CV space was 0.3, and the following procedure was used for evolving the multistrings:

• 1.Evolve each string using the mean force ∇F(z) and the metric tensor M(z).
• 2.Reparameterize each string using piecewise linear interpolation.
• 3.Update the position of the final image on each string using climbing forces.

For each update of the string, an inverse friction of 0.00 005 ps⁻¹ was used. The approximate mean force to update the string is obtained by the 20 restrained MD steps using a coupling constant of 100 kcal/(mol⁻¹/rad⁻²). However, the value of inverse friction parameter is chosen to be low enough such that z values evolve on a much lower time scale. The climbing forces were updated using the ascent speed ν = 2. For computing the interstring repulsive forces, σ = 0.1 was employed with a tolerance value of 0.01. The V_i’s on each dynamic string were calculated after every 200 climbing steps and only when the string length is greater than 0.4. In addition, to prevent the convergence of the string on the saddles which were indirectly connected to a given minimum, an additional constraint on the free energy was also imposed to ensure a monotonic increase along the path. In this regard, the mean forces were accumulated for 2000 climbing steps to get an approximate free energy profile along the string by employing thermodynamic integration. In the case of the violation of monotonic increase in free energy, the location of the climbing image was reset to the location of the image at the first maximum followed by reparameterization. The convergence of the string was monitored by computing the root mean square distance (RMSD) of the string from a reference string, according to

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{D}\left(t\right)=\sqrt{\frac{1}{N}\sum _{j=1}^N|z\left(j,t\right)-z_{○}\left(j,t\right){|}^2},$$ (8)

where z( j, t) is the instantaneous location of image j in CV space at time t, z_○( j, t) is the reference position of image j (which here is the position after the latest update), and N is the number of images in the string. A string was considered to be converged when RMSD represents thermal fluctuations. Once a string converged, the positions of the last few images along the converged string were stored as a static string [Fig. 1(d)]. The number of images to be stored was obtained by rounding off [N × (interimage spacing/0.3)]. Subsequently, the penultimate image on the string was made to climb for 1000 steps to obtain the initial guess direction toward the connecting minimum. The string was then reinitialized along a new trial direction to locate another saddle.

From the identified saddles, the connecting minimum on the other side of the saddle was located by performing the regular string method calculations with an initial end fixed on the saddle. The position of the other end of the string was set to the location which was obtained after the climbing of the penultimate image. In case if the length of the string was below 0.05 (length in CV space), the string was extrapolated to a length of 0.2 along the same direction.

III. RESULTS AND DISCUSSION

A. Alanine dipeptide in aqueous phase

Alanine dipeptide’s (AD) Ramachandran dihedral angle space is one of the most studied systems to verify the validity of methods for exploring free-energy surfaces. Here, we demonstrate using the climbing multistrings approach to map stationary points on the 2D FES for AD in explicit water. All dynamic strings consisted of 12 images. For each identified minimum, 15 uniformly distributed initial trial directions are proposed to define launch directions for the strings to locate the possible saddles. The complete procedure for locating saddles and minima is discussed in Sec. II A. A total of 11 stationary points are identified which include 4 minima (termed M1–M4) and 7 saddles (termed S1–S7). The locations of these stationary points are given in Table I. A full network map of minima and saddles connected through their MFEPs is shown in Fig. 2. Figure 2 shows the association of four saddles (S1–S4) with M1 ${\left(-79.3,-65.2\right)}_{{\alpha }_R}$, three saddles (S3, S4, and S7) with M2 (−82.0, 166.1)_β, four saddles (S1, S2, S5, and S6) with M3 ${\left(58.67,-117.21\right)}_{C{7}_{ax}}$, and three saddles (S5, S6, and S7) with M4 ${\left(57.0,59.0\right)}_{{\alpha }_L}$. Figure 2 also shows the free-energy difference of each saddle from its respective minima, which was calculated by employing the thermodynamic integration along the MFEP. Based on the obtained free energies, M1 is found to be the global minimum followed by M2, M3, and M4 with the relative free energies of 0.12, 3.95, and 6.45 kcal/mol, respectively. Since M1 and M2 are separated by lower free energy barriers (along the transition pathways through S4 and S3), the regions in M1 and M2 would be visited more frequently in classical MD simulations. On the other hand, the transition pathways to M3 and M4 exhibit significantly higher free-energy barriers, and therefore, these configurations are expected to less frequently visited. The simulation time to construct a network of MFEPs connecting saddles and minima is ≈350 ns.

TABLE I.

Locations of the minima (M) and saddle points (S) on the 2D FES of alanine dipeptide in water at 300 K.

Index	(ϕ, ψ) (deg)
M1	(−79, −65)
M2	(−82, 166)
M3	(58, −117)
M4	(57, 59)
S1	(9, −96)
S2	(126, −69)
S3	(−94, 25)
S4	(−89, −144)
S5	(73, −7)
S6	(68, 111)
S7	(14, 87)

FIG. 2.

A network of minima and saddles connected through MFEP’s on the two dimensional FES of alanine dipeptide in water at 300 K. White circles and rectangles show the locations of minima (M1–M4) and saddles (S1–S7), respectively. The free-energy difference from each minimum to each saddle on the minimum-to-saddle direction is shown along the transition paths in kcal/mol. The 2D FES for alanine dipeptide is generated using the single-sweep method for which the details are given in the supplementary material.

All the identified locations of minima and their corresponding free energies were in good agreement with those reported in Ref. 35. However, it was noted that the free energy surface reported in other simulation studies also exhibits additional minima.^36,37 The absence of such minima in our work is likely due to our choice not to use of CMAP corrections in the torsional potentials, in order to maintain consistency with the results of the START method.²⁹ To demonstrate the difference, Fig. S1 in the supplementary material shows the distribution of Ramachandran angles for alanine dipeptide in water obtained with and without CMAP corrections from a single MD simulation. The different distribution of dihedral angles with the CMAP corrections shows modified locations of minima as well as the presence of additional minima. On a FES including CMAP corrections, by using the procedure described in Sec. II, the total number of minima and saddles is found to be 13 and 17, respectively. The locations of these minima and saddle points on the 2D FES are shown in Fig. S2.

B. Alanine tripeptide in vacuum

To further illustrate the applicability of the method in higher dimensions, the network of stationary points is mapped on the 4D (ϕ₁, ψ₁, ϕ₂, ψ₂) FES of alanine tripeptide in the vacuum. Saddles from a given minimum are located using a pair of strings consisting 16 images each and 40 initial trial directions (20 uniformly distributed directions for each pair of Ramachandran angles). A total of 61 stationary points are located which include 17 minima and 44 saddles. A network of all the identified minima connected through their respective saddles is shown in Fig. 3. Figure 3 also shows the free energy of each saddle relative to its connecting minimum, which is calculated using thermodynamic integration along the associated MFEP. The relative free energy of each minimum is then calculated using the path with the least number of saddles. For the minima which are connected through more than one path having the same number of saddles, the free energy is obtained as an average over these paths. Based on the calculated relative free energies of each minimum, M7 (−81, 78, −80, and 70) is found to be the global minimum, while M14 (67, −97, −154, and 155) is found to be the minimum exhibiting the highest free energy of 7.0 kcal/mol. The locations and free energy of the minima are given in Table II. The locations of saddles are given in Table III.

FIG. 3.

A network of minima (M) and saddle points (S) on the 4D FES of alanine tripeptide in a vacuum. The relative free energies of the saddles (gray circles) relative to their connecting minimum are shown along the edges. The single-ended saddles S5, S19, S23, S29, and S37 are periodic. The color bar represents the free energies of the minima with respect to global minimum M7 (red). All the free energy barriers reported here are in kcal/mol. The network is generated using the Gephi package.³⁸

TABLE II.

Locations of the minima (M) on the 4D FES of alanine tripeptide in a vacuum at 300 K. The nomenclature for the minima and saddles is adopted from Ref. 29.

	(ϕ1, ψ1, ϕ2, ψ2)	Relative free energy
Index	(deg)	(kcal/mol)
M1	(−146, 165, 70, −70)	3.80
M2	(−147, 165, −79, −35)	5.28
M3	(−146, 170, −145, 168)	1.21
M4	(69, −66, −81, 68)	2.57
M5	(67, −72, 67, −67)	4.08
M6	(63, −84, −86, −26)	2.14
M7	(−81, 78, −80, 70)	0.00
M8	(−143, 165, −83, 76)	1.25
M9	(72, −58, −83, 147)	4.46
M10	(−66, −16, −84, −8)	3.73
M11	(−82, 68, 69, −69)	1.83
M12	(−81, 64, −87, −41)	2.52
M13	(−72, −44, −86, 64)	3.25
M14	(66, −97, −154, 155)	7.00
M16	(−79, 73, −143, −65)	2.56
M17	(−79, −33, −148, 164)	3.86
M18	(−150, −73, −93, 160)	4.52

TABLE III.

Locations of the saddle points (S) on the 4D FES of alanine tripeptide in the gaseous phase at 300 K. The nomenclature for the minima and saddles is adopted from Ref. 29. S9 and S16 are not shown as these saddles are not located by the climbing multistring method.

Index	(ϕ1, ψ1, ϕ2, ψ2) (deg)	Index	(ϕ1, ψ1, ϕ2, ψ2) (deg)
S1	(−96, −75, 69, −72)	S25	(−81, 76, 16, 60)
S2	(−144, 163, −93, −69)	S26	(−84, 6, −85, 65)
S3	(70, −69, 132, −67)	S27	(17, 58, −82, 76)
S4	(−9, −66, 68, −72)	S28	(−152, 171, −5, −70)
S5	(75, 110, 69, −70)	S29	(−81, 70, 78, 104)
S6	(−82, 68, 5, −74)	S30	(−85, 106, −131, 165)
S7	(−111, −88, −82, 77)	S31	(−151, 172, −83, −2)
S8	(125, −59, −87, 158)	S32	(−92, 26, −154, 158)
S10	(120, −128, −144, 163)	S33	(132, −114, −78, −28)
S11	(76, 115, −151, 166)	S34	(−143, 166, −92, 130)
S12	(63, −88, −93, 29)	S35	(−146, 164, 129, −115)
S13	(−8, −60, −79, −13)	S36	(−117, 146, 70, −68)
S14	(−65, −33, −93, 24)	S37	(69, −67, 77, 105)
S15	(−78, −41, −109, 126)	S38	(128, −77, −81, 82)
S17	(−82, 75, 130, −68)	S39	(−121, 148, −79, −35)
S18	(−80, 80, −140, −119)	S40	(69, −85, −151, 153)
S19	(−147, 161, 73, 104)	S41	(68, −70, −5, −69)
S20	(−81, 68, −87, 8)	S42	(72, −58, −82, 138)
S21	(71, −60, −108, −122)	S43	(128, −106, 70, −70)
S22	(70, −62, 9, 65)	S44	(−75, 13, −86, −22)
S23	(72, 104, −82, 77)	S45	(1, −78, −148, 151)
S24	(−141, 164, 17, 60)	S46	(−124, 158, −82, 76)

The notations used for minima and saddles in Table II, Table III, and Fig. 3 are adopted from the work of Chen, Yu, and Tuckerman.²⁹ In that work, the authors demonstrate an efficient method called stochastic activation-relaxation (START) for identifying stationary points on high-dimensional free-energy surfaces but which does not compute free-energy differences. The set of stationary points identified using our climbing multistrings approach is essentially identical to those determined using the START, with a few minor exceptions. Two saddles, S9 and S16, which START shows to connect M18–M15 and M3–M9,²⁹ respectively, have not been identified. Climbing strings launched from M18 did not converge to the anticipated region of S9, suggesting a relatively flat region with negligible transverse forces on the climbing end. Consequently, M15 could not be located as the connecting saddle S9 remained unidentified. Moreover, in addition to 43 identified saddles, the multistring method realized one more saddle we term S46 connecting M7–M8. In contrast to what was found using the START,²⁹ the saddle points S7 and S34 are found to be nonperiodic. The locations of the identified saddles are given in Table III.

In comparison to the START,²⁹ the computational cost of climbing multistrings is substantially higher, primarily because it identifies MFEP’s and computes free-energy differences, rather than only locating stationary points on the FES. For alanine tripeptide, the average simulation time is ≈20 ns per image per minimum. The higher computational cost is straightforwardly due to the fact that strings have multiple images, and it takes minimally one string convergence to locate one saddle. In addition, the time required to converge the whole string in comparison to the single image also contributes. Furthermore, the minima which connect to saddle points through relatively broad and flat channels, such as M3, M8, and M18, required more computational time to achieve convergence. For such minima, an inverse friction of 0.00 008 ps⁻¹ with 40 restrained MD step is found to be more appropriate.

IV. CONCLUSION

We have illustrated a climbing string method in CV’s, modified to launch multiple strings in one go, to map the network of minima and saddles on a high dimensional FES. Using solvated alanine dipeptide and vacuum alanine tripeptide as test cases, the multistring method has been shown to capture all the significant minima and saddle points on the 2D and 4D FES, respectively. Because it is a collective-variable approach, it should be easy to adapt for high dimensional free energy surfaces provided the appropriate set of CVs with no hidden barriers on orthogonal variables as well as for other complex systems. However, the computational cost of the climbing multistring method is high in comparison to the methods which evolve only a single image of the simulation system. Hence, the START approach suggested by Tuckerman and co-workers can be used to locate stationary points on the FES more efficiently.²⁹ However, at the same time, the multistring method provides the relative free energies of each landmark as well as the complete details of the minimum free energy pathway connecting them. It is anticipated that such information about networks of minima and saddles connected via MFEPs can be further used to estimate the accurate rate constants.³⁹ It should also be mentioned that the knowledge of free energy along the path also helps to ensure the direct connectivity of the saddle to a given minimum. One of the reasons for high computational cost is that the current multistring method employs a full string to explore the FES. Therefore, for complex systems, it might be advantageous to approximate the climbing directions first using a single image method followed by full climbing strings to locate the saddle and corresponding MFEP.

SUPPLEMENTARY MATERIAL

See the supplementary material for the details on reconstruction of the free energy surface for alanine dipeptide under aqueous conditions and the effect of CMAP corrections.