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TReK: a program for Trajectory REfinement and Kinematics

Note

Current Version 2.10 , July 5-2003

Please report problems or send suggestions to :

Stefan Fischer
Tel. +(49)6221-548879

Check for application examples and bugfixes under: http://www.iwr.uni-heidelberg.de/groups/biocomp/fischer

Note

  • For the busy user: please read at least the sections marked ”!!!” before starting to use TReK and CPR.
  • This interface to TReK replaces the TRAVel module of CHARMM, which is no longer supported. An effort was made to reproduce the “look&feel” of TRAVEL, whose input-scripts should work with TReK.

TReK is a collection of tools to find and smooth the minimum-energy path between two known structures representing the reactant and the product states of a reaction (here, the word “reaction” is used to design both a chemical reaction and/or a conformational transition). The energy can be an empirical energy (MM), a pure quantum (QM) potential, or a combined QM/MM energy.

The path is described by a series of structures which form a chain, or trajectory in conformational space, whose two end-points are the reactant and product structures. Watched in sequence, these molecular structures constitute a “movie” of the essential motions along the reaction, hence the appellation “kinematics”.

The main tool is a new implementation of the CPR (Conjugate Peak Refinement) algorithm (S.Fischer and M.Karplus, Chem. Phys. Letters 194, p.252, 1992), which refines an initial guess of the path (for ex. the linear interpolation between end-points) and finds the trajectory that follows the valleys of the high-dimensional energy surface, as well as all the exact saddle-points along that path.

Other tools are the SCM (Synchronous Chain Minimization) algorithm or the SDP (Steepest Descent Path) method, which both allow to smooth the shape of a path whose saddle-points are already known.

Other tools allow to analyze or manipulate reaction-paths.

Syntax of the TReK program

Keywords in [...] are optional. Default values are given in (...). Choose one from list : {...|...|...} or [...|...|...]

Main command (entering and leaving TReK)

TREK  [MAXPoints int (100) ]   [ { XAXI | YAXI | ZAXI } [ ROTAtion ] ]
                               [ SCALe [{ COMP | WMAIn }] ]

{ END | QUIT }

Subcommands (within TReK program)

[VERBose int (2) ]
[DISPlay int (80)]
[CHROno { RESEt | PRINt } ]

TRAJectory READ [UNIT int (40)] [REFErence] [ori]
  file1.crd
  file2.crd
     .
     .
  fileN.crd
DONE

TRAJectory READ  NAME file.dcd [UNIT int (40)] [REFErence] [ori]
                               [BEGIn int (1)] [SKIP int (1)] [STOP int (0)]

TRAJect WRITe NAME file.dcd  RESTART file.cpr  [UNIT int (40)]

TRAJectory ANALyze [SCAN [STEP real]]

TRAJectory { INCRease | DECRease [PEAK] [MINI] [SADD] }
           [ NGRId int | STEPsize real ] [ori]

SADDle int [ int [ int ...]]

CPR  [SADDle] [cpr_exit] [saddledef] [restart]
     [scanning] [relaxspeed] [oscillation] [miscrefine] [linextrem] [ori]

SCM  [NCYCle int (1)]    [PROJtol real (.01)] [ANGLe real (90.)]
     [MINUpdate int (5)] [MAXUpdate int (30)] [linextrem] [ori]

CROS [MINDist real (e-4)] [MINStep int (50)] [FIRStep real] [ANGL real (20)]

SDP  [SAVDistance real (.05)] [MINGrad real (e-3)]
     [NREActant int (maxpoints/2)] [NPROduct int (maxpoints/2)]
     [MODE int (4)] [ANGLe real (30.)] [MINCycle int (50)] [linextrem]

COPY [COMP] { SADDle | ORDEr int | INDEx int }

Detailed keywords

cpr_exit    :== exit-criteria for CPR :

                [NCYCle int (1)]  [NECAlls int (999999)]  [HIGHsad]

saddledef   :== saddle-point criteria :

                [SADGrad real (.05)] [SADMini int (sqrt(N))]

restart     :== read a TReK restart-file :

                RESTart filename [SCAN] [UNIT int (40)]

scanning    :== interpolation step-size for path-scanning :

                [INTErpol int (3)] [NGRId int (5) | STEPsiz real]
                [STPLowest real (0)]  [SCAN]

relaxspeed  :== stringency of path-relaxation :

                [TOL1proj real (1.)] [TOL2proj real (3.)] [LINMax int (2000)]

oscillation :== oscillation detection and prevention :

                [LOOPreduc int (sqrt(N))] [FRAMe int (10)]
                [TOLOscill real (.15)] [PROJincr real (2.)] [MAXOscil int (4)]

miscrefine  :== miscellaneous :

                [DELTa real (-e-7)]  [DISPlay int (0)]
                [NTANgent int (3)]  [REMOvemod int (0)]

linextrem   :== one-dimensional line-extremization :

                [BRAKetstep real (.1)] [BRKScale real (2.)] [LXEVal int (8)]
                [BRKMagnif real (5.)] [FIRStep real (.05)] [EXITmode int (3)]
                [TOLMax real (e-4)]  [TOLGrad real (.05)] [ATOMax real (0.75)]
                [TOLStep real (e-10)] [TOLEne real (e-7)]

ori         :== coordinate orientation :

                [ ORIEnt | NOORient ] , default is NOORient.

CHARMM command variables set by the CPR command

?SADI       :== index of the highest fully refined saddle-point
?SADO       :== order along the path of that saddle-point
?SADE       :== energy of that saddle-point

Note about file-names

Examples of allowed names are

   ~/subdir/file.crd
  ../subdir/file.crd
   ./subdir/file.crd
     subdir/file.crd

In case the file is already open, it must be closed before entering TReK, otherwise the “TRAJ READ” command will fail.

TReK Main Command

Before invoking TReK for the first time, the following must have been done in CHARMM :

  • The molecular system was GENErated.
  • One structure was read into the main CHARMM coordinate-set (for ex. the thoroughly minimized structure of the reactant conformer).
  • Optional: the desired Images were set-up, Image Centering turned off.
  • Optional: the desired atoms were fixed with CONS FIX SELECT ... END.
  • Optional: the desired QM regions and link-atoms were setup.
  • The FASTER mode was set to a value compatible with the non-bond settings.
  • AND at least one ENERgy call was made, with the above settings being the same as used for the minimization of the path end-points.

Once TReK has been launched, a typical session would involve :

  • Reading the structures of the initial path, and if applicable, declare already refined saddle-points (see typic1).
  • Refining the path with CPR or SCM commands (see respective sections).
  • Saving the resulting path (see typic2).
  • Optional: Printing the energy profile along the current path (see section “TRAJECTORY ANALYZE”).
  • Optional: Copying a specified path-point to the CHARMM coordinate-set, for further analysis (see typic3).
  • Leaving TReK with QUIT or END.

END !!!

Exits the TReK program, back into the CHARMM command processor, without terminating the TReK session. All TReK data-structures are maintained in memory, allowing to re-enter TReK to continue refinement. In that case, none of the keywords of the TREK main command may be used (such as MAXPoints).

QUIT !!!

Terminates the TReK session, erasing all its data structures and freeing all its memory. It allows to re-start TReK as if it had never been called before.

Optional keywords on the TReK command-line are :

MAXPoints int !!!

When entering the TReK program from CHARMM, it is possible to specify the maximum number of structures that will make up a path during a given CPR or SDP refinement. If this value is reached during refinement and even more path-points are needed, then the refinement will stop gracefully, allowing to save the path with a TRAJECTORY WRITE command. Later, restart TReK with a larger value for MAXPOINTS to continue the refinement.

From experience of transitions in proteins, about 20-30 points are needed for each saddle-point. For ex. if less than 5 saddle-points are expected along the path, set MAXPoints to ~150. Large conformational transitions in proteins (ex. hemoglobin T to R) can have more than 100 saddle-points.

Using TReK with IMAGES

TReK supports IMAGES in CHARMM. Before starting TReK,

  • the image centering MUST be turned off : IMAGE FIXED SELE all END
  • the image updating should be set to automatic : NBOND IMGFRQ -1
[ { XAXI | YAXI | ZAXI } [ ROTAtion ] ] keywords :

When the only involved symmetry operation is a translation along a single axis and/or a rotation along a single axis, then this axis and/or that rotation must be declared explicitly upon starting TReK. Currently, only the three main axis are supported.

If there are translations in more than one dimension (e.g. a membrane), then these keywords are not necessary and should NOT be used.

Examples:

  1. A periodic DNA helix along the y-axis requires : TREK YAXI ROT
  2. An dimeric protein with C2 symmetry (one monomer per image) : TREK ZAXI ROT
  3. A periodic chain along the x-axis, without rotational symmetry : TREK XAXI
  4. A lipid-bilayer with periodic boundaries : TREK
  5. A crystal : TREK

SCALE keyword

It is possible to find the minimum-energy path on a “stretched” energy surface, where each atomic coordinate is multiplied by a scaling factor greater or equal to 1. This allows for example to express the path in terms of the mass-weighted coordinates.

Coordinate-scaling is activated by issuing the SCALe keyword on the TReK command. By default, the scaling-factors are taken for each atom-coordinate from the X,Y,Z values stored in the main coordinate-set. Adding the COMP keyword will result in these values being taken from the comparison coordinate set. Alternatively, if the WMAIn keyword is added, the scaling factors are taken from the WMAIN vector and the X,Y and Z dimensions of each atom are given the same scaling-factor. For ex. using WMAIN = sqrt(MASS) will yield the mass-weighted coordinates.

Use the SCALAR command, see SCALar : commands to manipulate scalar atom properties, to set the appropriate values of the X,Y,Z coordinate-vectors or of the WMAIN vector, before starting TReK. The scaling-factors should not be modified after starting TReK.

All RMS(distances) printed from within TReK will use the scaled coordinates, for ex. the reaction-coordinate LAMBDA printed by the “TRAJ ANALyze” command. However, with COPY or “TRAJ WRITE” the coordinates are re-scaled back to normal.

When coordinate-scaling is activated, re-orientation is automatically disabled during the CPR refinement (as if issuing the NOORient keyword), but if desired, orienting can still be used during the reading of the initial structures by “TRAJ READ”.

Usage

Many useful suggestions for setting-up a path refinement for a large transition in a protein are given in the article entitled “Automated computation of low-energy pathways for complex rearrangements in proteins: application to the conformational switch of RasP21”, by F. Noe, F. Ille, J.C. Smith & S. Fischer, Proteins 59, p.534-544 (2005).

Using TReK with QM/MM

TReK can be used to find paths and saddle-points on combined quantum/empirical energy potentials. For this to work well, some default settings must be modified with the CPR command (see section on “Numerical energy potentials”).

Preparing the path end-points !!!

Make sure that the internal coordinates of atoms that are not significantly involved in the reaction are the same for the reactant and the product.

For example, a phenyl side-chain has two rotationally equivalent orientations, in which the delta1 and epsilon1 atoms are exchanged with the delta2 and epsilon2 atoms.

When the reactant and the product coordinate sets are from different origins (for example from different X-ray structures) or the result from different calculations (for example separated by a long dynamics run or an HBUILD), then for every group or side-chain that has equivalent (for ex. Phe, Tyr, Asp, Glu, -NH2, -NH3, -CH3 and -CH2- ) or nearly equivalent orientations (for ex. Val, Leu, Lys, Arg), it is worth checking that the corresponding atoms are named consistently in the reactant and product coordinates. Otherwise, CPR will proceed with refining all the transition paths associated with these changes in equivalent positions (rotating the Phe ring in the example above, or exchanging the H’s of a -CH2- group), which is very CPU time consuming and will make it difficult to analyze the reaction one is interested in.

Warning

In general, it is more efficient to use end-point structures that are very well energy-minimized, for ex. with RMS(gradient) < 0.0001 .

This is best achieved by minimizing them with the Newton-Raphson (ABNR) method in CHARMM.

For molecules with many local minima (ex. proteins), it is advisable to run some quenched molecular dynamics (MD): take structures at periodic intervals along a MD trajectory and minimize them. Use the ones with lowest energy as end-states of the reaction. An alternative is to do annealing MD (i.e. slowly decrease the temperature).

Using end-points which are poorly minimized or that are stuck in high-energy local minima will result in a path whose energy-profile “hangs” between the end-states, without clear activation barriers.

Fixing atoms !!!

When the reaction involves only a sub-region of the molecular system (e.g. catalysis in a protein), then fixing the atoms that do not really participate in the reaction speeds up the path-calculation significantly:

If some atoms have been fixed, then the fixed atoms of the reactant, of the product and of all initial intermediates MUST have exactly the same Cartesian coordinates. If they differ, then TReK will set the fixed atom coordinates of all structures to those of the reactant and give a warning !

To find out how to fix as many atoms as possible without biasing the transition, progressively add more distant atoms to the moving region and recompute the path. When the energy-barrier does not change significantly with more atoms, the final set of moving atoms defines the molecular region that really participates in the reaction.

Input

The TReK program has its own I/O commands.

TRAJECTORY READ !!!

The purpose of this command is to read the initial path to be refined. In its simplest form it consists of the two end-points of the reaction. If desired, one or several intermediate points can also be read. The points are read in sequence, as they would appear along the reaction:

  1. Reactant structure.
  2. Intermediate(s).
  3. Product structure.

This sequence of structures defines the initial-guess for the path, which is build by interpolating linearly in Cartesian coordinates from one structure to the next. If the two end-points are identical (for ex. when studying the 360 degree rotation of a side-chain), then at least 2 additional intermediates must be provided (obviously, since it is impossible to define a linear interpolation path between identical structures) ! More generally, any two structures adjacent along the initial path are not allowed to be identical.

Intermediate(s) should also be provided when direct linear interpolation between the two end-states would result in initial path structures which have extreme energies. For ex. the 180 degree flip of a phenyl side-chain. In that example, an intermediate could be constructed by rigidly rotating the Phe ring by 90 degrees. Alternatively, build the initial path by combined interpolation in Cartesian and Internal coordinates as described in the next section (see ringrotate).

Varying the initial path allows to search for different minimum energy paths: Using different intermediates will direct the refinement toward different and sometimes better paths. This is analogous to single structure optimization, where a structure of global minimum energy is searched by starting many minimizations from different initial conformers.

The command is also used to read a partially refined path, whose refinement is to be continued (i.e. the initial path is the partially refined path).

There are two ways to read coordinate-sets :

  1. From a series of formated CHARMM coordinate files.

AND/OR

  1. From an unformatted CHARMM dynamics trajectory file.

The “TRAJECTORY READ” command can be issued several times consecutively (in either of these two forms). The successive path-points will be appended to constitute the input-path used by CPR, SCM, CROS or SDP.

When reading from an unformatted dynamics trajectory file (for example the output of an earlier refinement), it is possible to restrict the input to subsection of the trajectory file with the BEGIN, SKIP, STOP keywords (see examples below in Example CPR input-files !!!).

Once a path refinement has been started (for ex. by issuing the CPR command), no more path-points can be read and the “TRAJECTORY READ” command is disabled.

Reference structure for orienting !!!

If there are no fixed atoms (see above), then it is advisable to remove the rotational and translational degrees of freedom. This is done by aligning the path-points onto a reference structure (so as to minimize the RMS-difference in coordinates between each path-point and this reference structure, as would the CHARMM command “COOR ORIEnt RMS”). This is achieved by issuing the “ORIEnt” keyword on the “TRAJECTORY READ” command line. This is generally recommended for the path-points of the initial guess-path. It is not recommended for the path-points of a partially refined path, whose refinement is being continued. The default setting is “NOORient”.

By default, the first structure that is read is used as the reactant structure (i.e. 1st path-point) AND is also used as the reference structure for orienting.

Alternatively (recommended procedure), the reference structure can be read explicitly, by specifying the “REFERence” keyword along with the “TRAJ READ” command. In that case, the first structure to be read becomes the reference point, but does NOT belong to the path. This is useful when the refinement of a very long path is split into several refinements of path subsections, each of which should align path-points onto a unique reference structure.

When saving the path to a file (TRAJ WRITE NAME ...), the reference point is NOT written as part of that path.

The reference point can only be specified once, before other path-points are read. If desired, the reference point can be centered and oriented according to its principal axis (in the same way as would “COOR ORIEnt”), by specifying the “ORIEnt” keyword with the “TRAJ READ REFER” command.

Path-points that are subsequently added or modified by a path-algorithm (CPR, SCM) remain essentially aligned with the other path-points. They can be re-aligned onto the reference point by specifying the “ORIEnt” keyword on the command-line of the respective algorithm, although this is not recommended in most cases.

If some atoms have been fixed, or if there are crystal images, then all coordinate re-orienting is automatically disabled (the reference point can still be read, but it will be ignored).

Declaring saddle-points !!!

After the initial-path has been read, if some path-points had already been refined to saddle-points in previous CPR runs, it is necessary to declare these path-points as saddle-points before using commands like “CPR”, “SCM” or “TRAJ DECR”.

The purpose of this is : - Before CPR, to avoid that CPR must re-refine those points all over again (very CPU consuming). - Before “SCM”, to leave the saddle-points unchanged. - Before “TRAJ DECR”, to prevent that saddle-points get removed.

There are two ways to declare path-points as saddle-points :

  1. The “SADDLE” command.

    For example if the saddle-points are in sequential position n1, n2, ... along the path (do not confuse the nX with the point-indices IDX of the previous CPR-run!), then the command to issue is

    SADDLE n1 n2 ...

    The command can be re-issued on successive lines, if necessary. Ignore the warning printed after each “SADDLE” command (see Bugs).

  2. An alternative to using the SADDLE command, is to read the restart-file corresponding to the current path (see CPRcmd). This is done on the CPR command-line.

    • If preparing to run a CPR calculation, then the restart-file should only be read at the moment of issuing the first CPR command.

    • If preparing to use another tool (e.g. SCM, etc.), then the CPR command can be called with “NCYCLE 0”, so that no CPR-cycles are performed, but the command is only used to read the restart-file :

      CPR NCYCLE 0  RESTART filename

      This flags the saddle-points according to the LinMin array of the restart-file : all points which are listed with LinMin < 0 are considered as saddle-points (no checking is done to verify that these points really qualify as saddle-points).

      Note

      If a restart-file is used, do not declare saddle-points twice by listing them again with the “SADDLE” command.

Create initial path by interpolating sidechains in IC

This section describes a method for generating an initial path for CPR when the two end-structures of the protein are so different that the default interpolation in Cartesian coordinates gives poor results. The method combines Cartesian and IC interpolation with sidechain shrinking. It has been described in detail in: Proteins 59, p.534-544 (2005).

The tool to generate this initial path consists of a PERL-script (called vector.pl) and a CHARMM-script that uses standard commands. These scripts are located in the CHARMM-tree subdirectory ./support/trek/initial_path/

Summary

Interpolation in Internal Coordinates (IC) describes well the torsional transitions between rotameric states of the sidechains. However, when IC-interpolation is applied to the backbone-torsion angles, it can lead to severe disruption of the backbone fold. In contrast, Cartesian interpolation approximately preserves the backbone fold in most cases (if the protein remains compact during the transition), but often leads to extreme deformation of the sidechains. Therefore, the two interpolation methods are combined here: First, the backbone atoms are interpolated in Cartesian coordinates so as to preserve the backbone fold, and then the sidechain atoms are built onto the interpolated backbone, using internal coordinates that are interpolated between the values of the end-states.

The initial path build by this “combined interpolation” can be further improved, so as to avoid unrealistic events such as bond-crossing or ring-penetration. These are effectively avoided by shrinking all sidechains in each intermediate point of the initial path. This is achieved by reducing all bond lengths of the sidechains (for ex. to half the original size) before building them onto the backbone. At first glance, this approach would seem to be unphysical and would result in energetically unfavorable paths. However, the minimization process applied to the path-points during the CPR computation rapidly restores the shrunken sidechains to their normal size, while undesirable events no longer occur in the resulting path.

vector.pl

For both the combined interpolation and the shrinking, the PERL script “vector.pl” is called from the CHARMM-script “comb_interpol.inp”.

vector.pl reads two IC tables from CHARMM and calculates the shortest change between each pair of torsion-angles in these IC-tables, for ex. -120 –> +120 : delta(angle) = 120 (and not 240). It writes out the resulting delta(angles) in an IC table, which can then be read into CHARMM. The script-parameters are displayed when executing it without any parameter or with the parameter -h.

comb_interpol.inp

Is the main CHARMM input-file. Calls

  • user_def.str
  • vector.pl
  • mini_interpol.str (optional)
  • make_traj.str (optional)

user_def.str

Here, the user must define all necessary settings. For. example some recommended values are :

SET steps =  20    ! Put 20 intermediate structures in the initial path.
SET shrink = 0.5   ! Halves the size of the sidechains.

mini_interpol.str

Optional, use only if CPR cannot deal with un-minimized initial intermediates. If requested, it is called at the end of comb_interpol.inp. Does a quick and constrained minimization of each frame.

make_traj.inp

Creates a CHARMM trajectory from the interpolated frames for viewing in VMD. An example of the resulting initial path (4 intermediates, shrinking to 50%) is available under ./support/trek/initial_path/frames/

Remarks

If there are no fixed atoms, then the product state is “COOR ORIENTed” upon the reactant state.

When doing CPR on the so-obtained initial path, to avoid that many of the initial intermediate path-points are deleted right away by CPR, use a larger value of INTERpol for a ~500 CPR-cycles, for ex.:

CPR NCYCLE 500 INTER 10 ...

Conjugate Peak Refinement (CPR)

Description of the algorithm

Provided with an initial guess for the transition path between a reactant and a product state, CPR refines the minimum-energy path and accurately determines all saddle-points of the high-dimensional energy surface along that path, in systems with up to several thousands of degrees of freedom. CPR uses the configuration energy and its first derivative.

Ref.: S.Fischer and M.Karplus, Chem. Phys. Letters 194, p.252-261, (1992).

The method is suited for the study of large-scale conformational change in macromolecules using empirical energy functions, or for simulating enzymatic reactions with the help of combined quantum/classical (QM/MM) force-fields. Because non-relevant saddle-points are abundant on the energy surface of even small molecules, CPR’s reliability in finding the transition-state that connects the desired reactant and product makes it useful also for the study of reactions with pure QM potentials.

An essential feature of the way CPR finds a path is that it does not drive (steer) the reaction along some pre-defined reaction coordinate, which would have been pre-defined as a function of the degrees of freedom. Rather, CPR self-consistently optimizes a chain of points describing the path, without applying any constraints that would affect the shape or the length of that path. The primary focus of the algorithm is on accurately determining the saddle-points along the path. Path-points which are not saddle-points are optimized only to the extent that the energy along the path segments connecting these points decreases monotonously from the saddle-point to its adjacent local minima. The purpose of points between saddle-points is to ensure the continuity of the path, not to find the bottom of the energy valley, which can be obtained more efficiently afterwards with the SCM method or by steepest descent on each side of the saddle-point(s). The number of path-points is not fixed during the refinement, but is allowed to increase and/or decrease. This enables CPR to accommodate any degree of complexity of the underlying energy surface, including the presence of multiple saddle-points along the path.

The method is a heuristic procedure, where in each CPR cycle:

  • the highest local energy maximum along the path (called the “peak”) is determined by stepping along the path and computing the energy at each step (this is referred to as “scanning” the path).
  • the path is modified by either improving, removing or inserting one path- point, so that the new path avoids the peak.

Improved as well as newly inserted points are optimized by a controlled conjugate-gradient minimization, which prevents the point from falling into local minima along the path and which converges to the saddle-point if the path is crossing a saddle-region of the energy surface. The path is fully refined when, after a number of such CPR cycles, the only remaining energy-peaks along the path are the exact saddle-points.

The settings of the algorithm are independent of molecular size and the nature of the reaction.

Before invoking CPR for the first time !!!

The following must have been done within TReK :

  • Reading the coordinate sets (at least two) of the initial path. The reactant and the product conformations must be very well minimized, with a RMS(gradient) < 0.001 . The initial path-points can be read either from a series of CHARMM coordinate files (ASCII format) or from a binary CHARMM dynamics trajectoryry file (see cpr1invok).
  • Flag those path-points that are already known saddle-points with the SADDLE command, or use a CPR restart-file, so that they do not get refined again (see :ref: cpr2invok <trek_input>).
  • Optional: Setting the desired amount of information printed out during refinement with the VERBOSE command. Also, CHARMM warnings about distorted angles, etc., which are abundant in the early phases of a path refinement, can be suppressed with “WRNLev 0”.

CPR command !!!

This command starts the path-refinement. The number of CPR-cycles is specified with NCYCLE. It can also be limited by specifying an approximate number of energy-calls NECAlls, which will not be exceeded. NECAlls allows better control than NCYCLE to determine the amount of CPU-time before exit, because the number of energy-calls made during one CPR-cycle can vary a lot from one cycle to another.

Tentative saddle-points are improved until their RMS-gradient reaches the value specified with the SADGrad keyword, and this for a successive number of line-minimizations equal to SADMini.

In the initial phase of a refinement, only a moderate amount of CPU-time is spent on refining individual saddle-points. This is achieved by setting the default value of SADMini to SQRT(n), (where n is the number of moving atoms) instead to 3n-1. This allows to refine the overall path and the approximate saddle-points. If desired, the exact saddle-points can be ultra-optimized in subsequent CPR runs, by issuing the CPR command with the SADDLE keyword (see below), the latter being more CPU-time consuming.

If the switch HIGHSAD is used, then the refinement will stop as soon as the highest tentative saddle-point has been found. This allows to get a quick estimate of the overall barrier height along the path, for example to decide whether it is worthwhile to proceed with refining of the rest of the path.

The default values of all CPR settings (including the values of SADGrad & SADMini) were optimized for the use with analytical energy and gradient functions, and are independent of the number of atoms or the type of reaction. So in general, they do not need to be modified. Note that all values are remembered from one CPR call to the next, as well as when temporarily exiting TReK with the END command.

Upon exit, CPR prints a summary of the energy and location of any saddle-points found so far. It also summarizes the location of some of the deepest local minima along the path, if any are present. If all energy-peaks along the path are saddle-points (according to the SADGrad and SADMini criteria, see above), then the printed output will contain the words “FULLY refined”. If the printout says “NOT fully refined”, continue with more CPR refinement-cycles.

The CPR command can be issued several times in the same TReK session, for example to allow for frequent saving of the path. Once a path reaches the status of “FULLY refined”, the remaining CPR commands are ignored, costing no extra CPU-time.

The energy-profile along the current path, and the location of the saddle-points can be obtained with the “TRAJ ANALysis” command, or by looking at the content of the TReK restart-file (see eneprof).

Saving the path !!!

After the path has undergone a number of refinement cycles, save the whole trajectory to a standard CHARMM dynamics file (binary format). This is done with the TRAJECTORY WRITE command (see cprsav). This saving should be done frequently, since a given CPR-cycle can be unpredictably long (unless the NECAlls keyword is used).

Ultra optimizing the saddle-points !!!

Once an overall path has been refined with the default settings for SADGrd and SADMini, it is possible to continue the refinement of the saddle-points, to make sure that they are true first-order saddle-points.

The definition of a first-order saddle-point:

  • it has a vanishing gradient, and
  • it has exactly one negative eigenvalue of its 2nd derivative matrix (this can be checked with the VIBRAN module of CHARMM).

CPR flags a path-point as a first-order saddle-point when:

  1. its energy is a local maximum along the path, and
  2. it is the result of 3n-1 successive line-minimizations (n = number of atoms) which all gave a gradient smaller than SADGrad.

Note

the final RMS-gradients at the saddle-point will be much smaller than SADGrad, because of the larger number (SADMini) of line-minimizations that will be performed.

The CPU-time required to do the 3n-1 line-minimizations is large. In practice, the criterion SADMini = 3n-1 is necessary only for small molecules (n < 300 atoms). For larger proteins, it usually suffices that SADMini ~ 1000, which brings the RMS(gradient) in the range < 10^-6 , where the limits of numerical accuracy of the computer are reached anyway.

In addition to SADGrad and SADMini, a number of other CPR settings have slightly different optimal values for ultra-optimizing the saddle-points. For convenience, they are all modified simultaneously by adding the “SADDLE” switch to the CPR command (see example below). The only effect of this switch is to modify the default settings for the following keywords :

[SADGrad e-3] [SADMini 1000] [TOL1proj .5] [TOL2proj .5]
[NTANgent 6]  [ATOMax 0]     [DELTa e-7]   [NOORient]  [LOOPreduc 0]

Specifying another value for any these keywords in addition to the SADDLE switch has precedence over these values. For ex. “SADDle SADMini 500” will use SADMini = 500, not 1000.

Note

Do not confuse this CPR switch with the “SADDLE” command used to declare saddle-points (see cprsad).

Reading the restart-file !!!

A CPR restart-file is written each time a path is saved (see cprrestart). It can then be read the first time the CPR command is given during a TReK session.

The restart-file serves two purposes :

  1. It allows to flag already refined saddle-points, so that they don’t get refined again at the beginning of each CPR-run. Refined saddle-points satisfy to the criterion LinMin(i) = SADMini , where LinMin(i) counts how many successive times the RMS-gradient at the path-point was less than SADGrd. To distinguish flagged saddle-points in the restart-file (or in the printout of “TRAJ ANAL”), they are given a negative sign, i.e. LinMin(i) = -SADMini (see restartinfo).

    Path-points that have been declared tentatively as saddle-points are also listed with a negative LinMin(i) value, but |LinMin| < SADMini. If such points are to be left untouched in the next CPR-run, set their LinMin to -999999 in the restart-file (not really worth the trouble, since usually it takes only 1 CPR-cycle to flag them as saddle-point again).

  2. It allows to save CPU-time upon initialization of CPR, by taking the information about the energy along the path from the restart-file, rather than having to rescan the whole path. However, when the relative energy change along a segment is less than 10^-6, this segment will be rescanned anyway to protect against numerical accuracy problems when writing/reading trajectory files.

    For each path-point, the energy and RMS(gradient) is listed, one line per point, on the first half of the line. Listed on the second half of the line is the scanning information for the segment preceding the path-point, such as the segment-length LenSeg(i).

It is imperative that the content of the restart-file correspond to the initial path that was read with “TRAJ READ NAME” in TReK, otherwise unpredictable behavior will occur.

If some segments in the restart-file do not have the correct scanning information, for example when two paths and their restart-files have been spliced, the user can force the re-scanning of the joining segments by setting their StepSiz(i) field to any value < 0 .

The re-scanning of the whole path can also be forced by using the SCAN keyword, in which case only the LinMin(i) information of the restart-file will be used to declare previously refined saddle-points. For example, this is required when a path that underwent a number of SCM-cycles is refined further with CPR.

Note

  • If any of the numeric fields contain “*”, TReK dies ungracefully.
  • Atom-coordinates in CHARMM-trajectory are writen and read with only 32-bit precision (i.e. REAL*4), whereas they are stored in memory in 64-bit precision (i.e. REAL*8). Just like an MD-run, the progress of a path refinement by CPR is very sensitive to small changes in the initial conditions. Therefore, stopping/restarting a refinement will result in somewhat different paths than an uninterrupted refinement.

Example CPR input-files !!!

* Beginning a CPR refinement.
*
! Generate the system :
STRE gene.str
! Fix atoms, if desired :
CONS FIX SELECT nomov END
! Read a coordinate-set and call the energy :
OPEN UNIT 13 READ FORM NAME  coord_file1.crd
READ COOR CARDS UNIT 13
CLOSE UNIT 13
! This energy-call is mandatory (use desired non-bond settings) !!! :
ENER IHBFR 0 ...

! Ready to start TReK :
TREK
   ! Read the reference-coordinates onto which the path will be oriented
   ! (it will not be part of the path) :
   TRAJ READ REFErence   !  NOORient (optional)
     coord_file0.crd
   DONE

   ! Reading the initial-guess path (orient only structures that are
   ! put into the initial path):
   TRAJ READ ORIEnt
     coord_file1.crd     ! this is the reactant structure.
     coord_file2.crd
     coord_file3.crd
   DONE
   ! Adding some points from a dynamics trajectory :
   TRAJ READ NAME  md.dcd  BEGIN 20 SKIP 50 ORIENT
   ! Adding more points :
   TRAJ READ ORIENT
     coord_file4.crd
     coord_file5.crd     ! this is the product structure.
   DONE

   ! Optional: check the energy-profile of the initial path:
   TRAJ ANAL

   ! Starting refinement :
   CPR NCYCLE 100
   ! Writing the path to a trajectory file :
   TRAJ WRITE NAME  path.dcd  RESTART path.cpr

   ! Continue refining, while frequently saving the improved path :
   CPR NCYCLE 100
   TRAJ WRITE NAME  path.dcd  RESTART path.cpr
         .
         .
         .
   CPR NCYCLE 100
   TRAJ WRITE NAME  path.dcd  RESTART path.cpr

   ! Optional: check the energy-profile of the final path:
   TRAJ ANAL
   ! Copy the 3rd point along the path to the main-coordinate set (for ex.):
   COPY ORDER 3
   ! Returning to CHARMM :
QUIT

! Continuing the CPR refinement.
! Re-initialize TReK with more space for path-points :
TREK  MAXPoints 200
   TRAJ READ REFErence  NOORient
     coord_file0.crd
   DONE
   ! Reading the previously saved CPR trajectory (do not orient !!!) :
   TRAJ READ NAME  path.dcd

   ! Re-starting CPR, repeat :
   CPR NECAlls 10000   RESTART path.cpr
   TRAJ WRITE NAME  path.dcd  RESTART path.cpr
         .
         .
         .

   ! Refining 1st-order saddle-points, repeat :
   CPR NECAlls 10000 SADDLE
   TRAJ WRITE NAME  path2.dcd  RESTART path2.cpr
         .
         .
         .

   ! Copying the saddle (if already found) to the main-coordinate set :
   COPY SADDLE
   ! Temporarily returning to CHARMM :
END

! Analyzing the saddle-point:
IC FILL
PRINT IC

RETURN

Numerical and quantum energy potentials !!!

If the energy and/or its derivative is computed numerically (for. ex. quantum potential or finite difference gradient), some CPR settings (TOLMax, TOLStep, TOLEne) must be relaxed, otherwise the lack of precision in the energy/gradient would perturb the convergence of CPR.

DELTA should be increased, so that it is larger than the one-dimensional finite-difference step used to compute the derivatives (see below for a description of DELTA).

Example of practical values when using SCF quantum-potentials :

CPR DELTA -1.0e-4 TOLMax 0.1 TOLStep 1.0e-5 TOLEne 1.0e-5

If the SADDle keyword is added (i.e. 1st-order saddle-points are to be optimized), relax the saddle-point convergence criterion. A recommended value is SADGrad 0.01 .

Because high-energy regions of conformational space are explored in the early phase of a CPR refinement, it often happens that quantum-methods fail to reach SCF convergence. This can be aleviated by introducing intermediates into the initial path, so that the linear interpolations between path-points do not generate structures that are too distorted. These intermediates do not need to very good, since they will be removed automatically by CPR later during the refinment. Also, it sometimes helps to do the few first CPR cycles (until enough “good” intermediates are added by CPR itself) with orthogonal (rather than congugate) line-minimizations. This is achieved by using the “DELTA 0” setting. For. example :

TREK
   CPR NECALLS 100  -
       DELTA 0.0 TOL1proj 3.0  FIRStep 0.001 BRKMagn 2.0   -
       EXIT -3  TOLMax 0.15 TOLGrad 0.1 TOLStep 0.001 TOLEne 0.01

   TRAJ WRITE NAME  ugly.dcd
QUIT

Do not forget to “QUIT” TReK after using these settings (not “END” !), so that all CPR-settings are returned to their efficient values.

Scanning of the path

CPR searches for energy-maxima along the path (peaks) by “stepping” along the path and evaluating the energy at every step. In TReK, the step-size varies along the path. This is a major change relative to the implementation of CPR in TRAVEL, where the step-size was constant along the path. In TReK, the step-size is a fraction 1/m of the length of each path-segment (which varies along the path).

The keyword INTERpol specifies the value of m (m>1). The default value is m = 3. A larger value is more robust and results in a path with more points, requiring more computations. m = 2 saves CPU-time (useful for QM potentials) and is sufficient in most cases.

Boundaries for the step-size can be enforced, with the lowest value specified with the STPLow keyword (default is 0), and the largest value with the keyword STEPsize (see below).

Summary of main CPR settings

NCYCLE = maximum number of CPR-cycles to be performed.

NECAlls = maximum number of energy-calls to be done.

SADGrd  = RMS-Gradient in successive line-minimizations leading to a saddle.
          Don't set lower than 10^-3, since the final RMS-gradient will be
       much lower anyway due to the many (SADMini) conjugate
          line-minimizations.

SADMini = Number of successive line-minimizations with
          RMS(gradient) < SADGrd, that qualify a path-point to be
          considered as a saddle-point.  Synonymous to SADCyc.

DISPlay = Nb. of path-points for which the energy & gradient are printed
          each CPR-cycle (0 by default).  Only useful for small molecules.

Internal CPR-settings (usually do not need to be tampered with)

DELTA    = Finite difference step [in Angstroem] along a path-segment
           for obtaining the 1-dimensional 2nd-derivative used in the
           construction of directions conjugate to that path segment.

           - DELTA = 0 : the set of directions is constructed orthogonally
             to the path segment, not conjugate.

           - DELTA < 0 : the finite difference step is equal to |DELTA|.
             This is the default. When the gradient vector is computed by
             finite-difference or with QM-energy, make sure that |DELTA| is
             larger than the finite-difference step used for 1st derivatives
             or the numerical accuracy of the QM method.

           - DELTA > 0 : the finite difference step is equal to the distance
             of the scanning point on the segment to the maximized point
             on the segment, so that the value of DELTA is only used as step
             when the line-maximization has failed to move the scanning
             point.  This is the default when ultra-optimizing saddle-points.

STEPsiz  = Upper limit on the size (in Angstrom) of the steps taken
           when searching for energy peaks along the path.
           By default equals the RMS-distance between reactant and product,
           divided by NGRId+1  (NGRId = 6 by default).
           Of course, if the reactant and the product are the same structure
           (for example in the case of a full group rotation), then NGRId
           makes no sense and the STEPsiz value can be specified explicitly.

NTANGENT = Number of energy probes on path tangent in search for local max.

TOL1PROJ = Gradient projection tolerance when refining a path-point.
TOL2PROJ = Gradient projection tolerance when adding a path-point.

LINMax   = Maximum number of line minimizations per CPR-cycle.

FRAME    = Frame-length (in number of cycles) for oscillation-detection
           (maximum allowed value is 20).

TOLOSCIL = Oscillation tolerance ratio for energy or gradient.
           - TOLOSCIL = 0 : oscillation will not be detected.

PROJINCR = Increase factor in TOL1prj and TOL2prj when oscillation occurs.

MAXOSCIL = Maximum tolerated number of oscillation-occurrences.

REMOVEMOD= m.  If m> 0 , do m line-minimizations for adjacent minimum
           upon point removal. If m <= 0 , add minimum only to avoid looping.

Line-extremization settings. (usually do not need to be tampered with)

These settings are common to the CPR and SCM commands. Their default values have been optimized for the use with analytical energy and derivative functions, assuming 64-bit precision real numbers (i.e. REAL*8).

If non-analytical energy and/or derivative functions are used, the following values should be made less stringent (see “Numerical energy potentials”).

TOLMAX   = Tolerated gradient, 1-dim. maximizations.
TOLGRAD  = Tolerated gradient, 1-dim. minimizations.
TOLSTEP  = Smallest 1-dim. extremization step, in Angstrom.
TOLENE   = Smallest fractional energy change.

Internal line-extremization settings (do not tamper) :

BRAKETST = Maximal 1-dim. bracketing step, in Angstrom.
FIRSTEP  = First bracketing step.
BRKSCALE = Dynamic bracketing Scaling factor.
BRKMAGNI = Bracket magnification-limit factor.
EXITMOD  = Exit-Mode: exit with respect to TOLGRA, or also TOLStep & TOLEne.
LXEVAL   = Max. number of energy-evaluations during each line-extremization.
ATOMax   = Max. tolerated motion of any atom during a line-maximization.
           When the initial guess for a path is very poor or some of its
           points are very strained, some atoms can experience excessive
           forces, so that they get displaced too much during the
           line-minimizations (some even get pushed out of the structure !).
           To avoid this, ATOMax is set to 0.75 Angstrom during the
           early phases of CPR or SCM refinements.
           If ATOMax = 0, then no limit is imposed on atomic motions.

Obsolete CPR features

In the previous implementation of CPR called TRAVel, the step-size was not variable along the path, but was assigned a constant value specified with the STEPsiz keyword.

For the purpose of backward compatibility, this still can be done by setting INTERpol < 1, although this is NOT ADVISABLE.

Reduction of STEPsiz (division by SQRT(2) ) then occurs every LOOPred occurrence of algorithm-looping (unless LOOPred is set to 0). For ex. if LOOP=4, then the reduced STEPsiz will be 1/2 its original value after 8 occurrences of looping. When INTErpol > 1, the default value of LOOPREd is 0 .

Output

TRAJECTORY WRITE !!!

This command writes the path to a file in the format of a standard CHARMM dynamics trajectory (binary format). This file can then be read with the “TRAJECTORY READ NAME” command (see output) in order to continue the refinement, or be treated like any normal CHARMM trajectory, for ex. for visualization with programs like VMD.

Content of the restart-file

When writing a path, a CPR restart-file is also written (in ASCII format). This file contains information about the path, if writing takes place right after a CPR command. If writing after other commands (for ex. “TRAJ INCR” or SCM), information such as the energies written to the restart-file are not pertinent, only the LinMin column (used to flag saddle-points) is meaningful.

The restart-file can be read on the CPR command-line in later TReK sessions (not with the “TRAJ READ ..” command), for ex. when continuing the path refinement (see readrestart).

Writing the restart-file does not cost CPU-time, since the data is taken from the internal data-structures of TReK. Because the lines of this file are 160 characters long, it is easier to view it with the “less” utility, using the -S key: “less -S restart.cpr” .

The header of the restart-file looks as follows:

    * CPR RESTART-FILE (FORMAT 1.1),  CORRESPONDING TO THE PATH IN FILE :
    *                name_of_file_saved_with_traj_write
    *                        title lines ...
    *                            ...
    *
                      THE CPR PARAMETERS WERE :
SADGRD    SADMini   STEPSZ  REDLUP   DELTAS  LORIEN  PRTOL1   PRTOL2    INTERP
5.0000E-2      4  9.1980E-2    0   1.000E-07    T    1.000     3.000       3

The title-lines start with an “*”, can have any content and appear in unlimited number. By default, the second title-line contains the name of the corresponding trajectory-file, as specified when it was saved.

In addition to the title, the header shows the value that some CPR settings had in last CPR run. These header-lines do not start with a “*” and are mandatory.

After the header, the path is described one path-segment per line. The first part of the line lists information meaningful to the user. It is the same as printed by the “TRAJ ANAL” command (described below) :

N   IDX  Lambda  X(N)-Xref  Energy  Grad  LinMin  Curvat

The second part of the line lists internal CPR data. This information is used by the CPR algorithm when continuing a refinement (see scaninfo):

StepSiz   LenSeg   StpMax   MaxEnergy   NxMn  PvMn  Up  Dwn

This information describes the path-segment from point (N-1) to N, i.e. the segment PRECEDING the point listed on that line (the values for the first path-point are dummy-values) :

StepSiz = The step-size last used to scan the segment.
          Setting it < 0 will force the rescan of that one segment.
LenSeg  = Length of the segment.
StpMax  = Number of steps to the peak on the segment, counting from point N-1.
MaxEner = Energy of the peak on the segment (if any).
NxMn    = Number of steps to the first local minimum, counting from point N-1.
PvMn    = Number of steps to the last  local minimum, counting from point N-1.
Up      = .TRUE. if the energy ends with an increase on that segment.
Dwn     = .TRUE. if the energy starts with a decrease on that segment.

TRAJECTORY ANALYZE !!!

This command re-calculates and print for every path-point the following values in column-format. To get an energy profile along the path, plot “Energy” versus “Lambda”. Saddle-points are identified by a negative value in the column “LinMin”.

Column:
.......
N          = The order of the point along the path.  It is the number that
             is referred to by the SADDLE command, or with BEGIN and STOP keywords.

Idx        = The identifier number given to each point in the last CPR
             calculation.  It has no meaning for the next runs.  It should
             not be used to identify saddle-points with the SADDLE command.
             Idx is given a negative sign when the structure has not been
             modified (for ex. by CPR) since the last time it was read
             with the "TRAJ READ" command.

              N
Lambda     = SUM{ |X(i) - X(i-1)| } / SQRT(3n)  ,
             i=2

             where X(i) is the coordinate-set of path-point i (n atoms).
             This is the total path-length up to path-point N.
             It is the intrinsic reaction-coordinate.  Quantities such as
             the energy, etc., should be plotted as a function of Lambda,
             not N.  Units are Angstrom (see *note lambda: Bugs).

X(N)-Xref  = The RMS-difference between the coordinates of the point N and
             the reference-structure. Units are Angstrom
             (see *note rmsdiff: Bugs).

Energy     = The energy (in kcal/mol)

Grad       = The RMS-gradient of the energy (in kcal/(mol*angstrom)).

LinMin     = A negative number indicates a tentative saddle-point :

             -SADMini  the path-point satisfies the SADGrd & SADMini criteria
                      for being a saddle-point (see *note analsad: CPRcmd).
                      The saddle-point is sure to be first-order
                      if SADMini was set to 3*n-1 (n = number of moving atoms),
                      and SADGrd was set < .05.

             -M       the path-point has satisfied the condition
                      RMS(gradient)<SADGrd in M successive line-minimizations,
                      M < SADMini, yet it is already flagged as a saddle-point
                      because the numerical machine precision has been reached,
                      so that all requested SADMini line-minimizations could
                      not be performed. Generally is a true saddle-point
                      nevertheless.

             -999999  the point has been declared initially as a saddle-point
                      with the SADDLE command.

              0       a normal path-point.

             +M       the path-point has satisfied the condition
                      RMS(gradient)<SADGrd in M successive line-minimizations,
                      M < SADMini.  Not yet a saddle-point, but might become
                      one at a later stage.

             +999999  a path-point declared as a saddle-point with the SADDLE
                      command, but another (lower) local energy max. is still
                      present within the path-segment starting at this point.

Curvat     = The path curvature, as measured by the angle (in degrees) of the
             two path segments joining at this point and divided by half the
             sum of the two adjacent segment-lengths (in Angstrom).
          Ex.: zero means the path is linear.

Grad/Proj  = The ratio of the gradient to its projection onto the plane
             orthogonal to the path at this point. Always > 1 . The larger,
             the better the path follows the gradient (can be improved by
             running SCM for small molecules).

“Curvat” and “Grad/Proj” are not very meaningful for large molecules.

With the SCAN switch, the analysis is extended to points lying along the path-segments, built by linear interpolation between the explicit path-points. The size of the interpolation-steps is specified with the STEP keyword (default is the STEPsize value last used with CPR).

COPY [COMP]

One structure of the path can be copied to the CHARMM main (or comparison) coordinate set, where it will be available for further manipulation after exiting the TReK program.

The point can be selected according to its order number N along the path (ORDER N) or according to its current index (INDEX idx).

It is also possible to select the currently highest saddle-point along the path by using the SADDLE switch, if such a saddle-point(s) has already been refined or declared.

Miscellaneous tools

TRAJECTORY INCREASE or TRAJECTORY DECREASE

This command allows to increase or to decrease the density of points along the path.

If increasing, new points will be inserted by interpolation between the existing points, so that the distance between adjacent points corresponds to the specified STEPsize.

Conversely, if decreasing, points will be removed if they are closer than STEPsize to the point preceding them along the path. This allows to generate a path with nearly equidistant points, which is required for SCM.

“TRAJ DECREAse” will not delete previously refined or declared saddle-points (unless the “SADD” keyword is given), it will preserve path-points which are local minima along the path (unless the “MINI” keyword is given) and it will not delete path-points if this introduces energy peaks within the new path-segments (unless the “PEAK” keyword is given). This allows to use “TRAJ DECR” to remove all unneeded points from a path, for example before smoothing the path with SCM.

To avoid removing the saddle-points of an old path, declare them either with the command “SADDLE” or by reading the CPR restart-file corresponding to that path (see decr).

A sample input-script :

TREK MAXPP 200
  ! Reading the refined CPR-path :
  TRAJ READ NAME   fully_refined.dcd

  ! Flagging the saddle-points previously found by CPR :
  CPR NCYCLE 0  RESTART fully_refined.cpr

  ! Remove as many points as possible (i.e. large STEP values), but
  ! preserve minima & saddle-points and don't introduce energy peaks:
  TRAJ DECREAse STEP 999.9

  ! Get a path with points separated approx. by 1/10 of the
  ! distance between reactant and product, do not preserve
  ! minima or saddle-points and allow new energy peaks:
  TRAJ DECREAse NGRID 10  MINI SADDLE PEAK
  TRAJ WRITE NAME ngrid10.traj  RESTART ngrid10.cpr


  TRAJ WRITE NAME shorter.traj  RESTART shorter.cpr
QUIT

Miscellaneous

From inside TReK, it is possible to use all the CHARMM commands handled by the SUBROUTINE MISCOM(), such as STREAM, GOTO, LABEL, SET, INCR, DECR, etc. See Miscellaneous Commands for more details.

VERBOSE int

Sets the amount of details being printed out. Values 0-2 are useful for production runs (default is 2), values 3-6 should only be used for debugging.

The print-level variable PRNLEV is set to 3 when entering TReK, but this can be overridden by issuing the PRNLevel command. This is useful to prevent abundant warnings when the structure is very distorted, which is frequent in the early phase of a path refinement. Setting “WRNL 0” is also useful in this respect.

Synchronous Chain Minimization

SCM is an algorithm that smooths the shape of an existing path and brings it closer to the bottom of the adiabatic valley. Unlike CPR, SCM does not change the number of points along the path.

SCM

Before issuing this command, an initial path must be available. The density of path-points along the initial path must be high enough to give a relatively continuous description of the shape of the path. Also, the path-points should be approx. equidistant along the path.

The best use of SCM is to start from a path resulting from a CPR refinement. Alternatively, the initial path could be generated by a straight line-interpolation (see the “TRAJ INCRease” command) between the reactant and product, but this is very inefficient computationally.

SCM works by synchronously minimizing all its path-points, under the constraint that the points move within hyper-planes orthogonal to the path. These planes are updated no more often than every MINUpdate cycles of conjugate minimization (per point), but no later than after MAXUpdate of such cycles.

The number of conjugate line-minimizations done between plane-updates on a given point is controlled by the value taken by the angle between the two path segments joining at that point. This angle varies from 0.0 degrees (when the path is linear) to 180.0 degrees (when the path is reversing its direction). The difference of SCM with other locally updated plane (LUP) implementations, is that successive conjugate line-minimizations are continued only as long as this angle is decreasing. When this angle is increasing, minimization of that point is stopped once the angle exceeds a specified value ANGLe, although at least MINUpdate line-minimizations are done no matter what the behavior of this angle. When the setting of ANGLE=180 , SCM becomes equivalent to the LUP method by Ron Elber. In that case, and upon extensive minimization, the path may converge towards a discontinuous state and some path-points may fall into the reactant and/or the product states.

SCM has converged when the projection of the gradient onto the path is less than PROJTOL at every point. This can be quite time consuming, so it is recommended to force an exit from SCM by specifying the maximum number of global cycles NCYCLE. In every SCM-cycle, there will be between MINUPDATE and MAXUPDATE line-minimizations done at each point, so that NCYCLE should be kept small to allow for periodic saving of the path (see scmsav).

Path-points that are already saddle-points should be declared (see scmdecl, command “SADDLE”), so that they are kept unchanged during the SCM refinement.

See extrm for a description of the line-extremization keywords.

Example input-file

* Smoothing a refined path with SCM.
*
! Set the desired segment-length in the "equidistant" path,
! for ex. 1/30 of the total path-length :
set finalstep 0.02
! Generate the system :
STRE gene.str
! Fix atoms, if desired
CONS FIX SELECT nomov END
! Read any coordinate set and call the energy :
OPEN UNIT 13 READ FORM NAME  system.crd
READ COOR CARDS UNIT 13
CLOSE UNIT 13
! This energy-call is mandatory (use desired non-bond settings) !!! :
ENER IHBFR 0
! Start TReK :
TREK
   ! Read the coordinate-set upon which all path-points will be oriented :
   TRAJ READ REFErence  NOORient
     coord_file0.crd
   DONE
   ! Reading the path previously refined with CPR :
   TRAJ READ NAME  path.dcd

   ! Dummy CPR-command (since NCYCle=0), only to get the saddle-point(s)
   ! declared from the restart-file :
   CPR NCYCLE 0  RESTART saddle.cpr

   ! Trick to make points along the path nearly equidistant (increasing then
   ! decreasing the density of points), so that SCM works better :
   CALC incrstep = @finalstep / 10
   TRAJ INCREASE STEP @incrstep
   ! Keep the saddle-points :
   TRAJ DECREASE STEP @finalstep MINI PEAK

   ! Do a few SCM-cycles:
   SCM NCYCLE 100  ORIEnt
   ! Writing out the path to a trajectory file :
   TRAJ WRITE NAME  path.dcd  RESTART path.cpr

   ! Repeat and save :
   SCM NCYCLE 100  ORIEnt
   TRAJ WRITE NAME  path.dcd  RESTART path.cpr
         .
         .
         .
   ! Get the final energy-profile (after SCM runs, the restart-file holds
   ! invalid energies, only its saddle-point flag LinMin(i) is valid) :
   TRAJ ANAL
QUIT
RETURN

Combining SCM and CPR

The SCM and CPR commands can be given in alternation within a same TReK session. This allows to get a path that is both smooth in shape and has no energy peaks along the path-segments.

It is recommended to run CPR on the path resulting from SCM-runs, because the SCM-runs may have introduced new energy peaks along the path, which will be removed (or refined into saddle-points) by CPR.

The restart-file written after the SCM runs still indicates which points are saddle-points, but it no longer has the proper energy scanning information. Therefore, the “SCAN” keyword must be added to the first CPR command, to insure that the path gets searched for energy-peaks.

CPR and SCM runs can be alternated in this way until the last CPR-run does not modify the path anymore. This then means that the previous SCM-run did not introduce any new peaks, so that the resulting path is completed.

Steepest Descent Path

SDP

The SDP command provides a carefully controlled descent along the energy valley, down from a 1st-order saddle-point (obtained from CPR, for example). This is a rather slow procedure, practical only for small systems. For large systems, in it recommended to smooth the existing CPR-path with the SCM method (see smooth).

Four modes of descent are available :

Mode 1 :   The size of the step down the gradient is reduced until the angle
           between the path and the gradient is less than the value specified
           by ANGLE. This mode is the truest to the definition of the adiabatic
           path, but it also is very slow. It is recommended only for very
           small molecules.

Mode 2 :   The step is taken along the gradient until the new gradient is
           orthogonal to it (= strict steepest descent).

Mode 3 :   The step along the gradient is taken as large as possible, as long
           as the energy keeps decreasing (= loose steepest descent).

Mode 4 :   The steps are those of a strict conjugate-gradient descent.
           While not following the exact bottom of the adiabatic valley
           from step to step, the average path obtained by saving path-points
           after several steps is not significantly worse than the path
           obtained by saving after several steps in Mode 1.
           This mode is the fastest and the one recommended for large
           molecules. If a very accurate adiabatic path is demanded, the
           resulting path can be further improved with the SCM method.
           This mode is the default.

The input to SDP is a chain of points which straddles the saddle. This chain can be created with the “CROSS” command, for example (see next). Up to NREACTANT and NPRODUCT points are then added to the chain, on the reactant and product side respectively. The total number of points can not exceed MAXP, though, so when using SDP, it is advisable to start TReK with a large enough value for MAXP. A new point is added to the path every time the sum of the steps taken since the last addition reaches SAVDISTANCE. When the path enters a region where the energy-gradient is less than specified with MINGRAD, SDP stops extending the path on that side of the saddle-point, provided that it already did at least MINCYCLE steps (to allow moving away from the saddle region, where the gradient is also vanishingly small). See linextr for a description of the line-extremization keywords.

CROSsing

This command is used to prepare a minimal path-chain for input to SDP from a path refined with CPR. A fully refined saddle-point and two points lying on each side of it are the required input to CROS. The two surrounding points serve as initial guess for the saddle-point crossing direction. The output of CROSS is also a chain with three points, the second of which is the unmodified saddle-point, while the two surrounding points are located close to the saddle and near the bottom of the adiabatic valley. This is simply achieved by stepping down along the gradient in Mode 1 (see descmod for a description of the descent modes), after having taken the first step on each side of the saddle-point in direction of the two initial surrounding path-points.

The keywords of CROSS are :

MINStep int (50)   = The minimum number of steps that are taken, before the new
                     point adjacent to the saddle (on either side) is saved.
MINDist real (e-4) = The minimum distance moved from the saddle, before the new
                     point adjacent to the saddle (on either side) is saved.
FIRStep real       = The size of the first step away from the saddle in
                     direction of the initial adjacent path-point.
                     By default, this size is 1/20th of the distance from
                     the saddle to the respective initial point.
ANGL real (20)     = The angle between the stepping direction and the gradient,
                     which is a setting of the steepest-descent in Mode 1.

By reiterating the CROS command, these points are gradually improved and yield the reactive mode at the top of the barrier. CROS is NOT designed to refine a saddle point, which instead must be provided to it.

If the path provided to CROS has more than three points, it is assumed that this path is the result of a CPR refinement during the same TReK session and all points are deleted, except for the highest saddle-point and its two direct neighbors (provided that CPR had already fully refined the saddle-point, otherwise the CROS command is ignored).

Sample input-file

* Getting the steepest-descent path.
*
! Generate the system :
STRE gene.str
! Fix atoms, if desired
CONS FIX SELECT nomov END
! Read any coordinate set and call the energy :
OPEN UNIT 13 READ FORM NAME  system.crd
READ COOR CARDS UNIT 13
CLOSE UNIT 13
! This energy-call is mandatory (use desired non-bond settings) !!! :
ENER IHBFR 0
! Start TReK with more space to put new path-points :
TREK  MAXPoints 200
   ! Reading the saddle-point and two neighbors from a fully refined CPR path :
   TRAJ READ NAME  cpr_path.dcd   BEGIN 11 STOP 13
   ! Analyzing the path :
   TRAJ ANAL
   ! Improving the barrier crossing mode :
   CROSsmode
   ! Repeat to improve further :
   CROSsmode
   ! Verify that the 2 surrounding points are close to the saddle-point :
   TRAJ ANAL
   ! Specify the distance between saved points (in Ang.) in steepest descent:
   SDP  SAVDist 0.05
   ! Writing out the path to a trajectory file :
   TRAJ WRITE NAME  path.dcd
   ! Smooth and further refine the path :
   SCM  NCYCle 5
   TRAJ WRITE NAME  path.dcd
   ! Analyzing the path :
   TRAJ ANAL
   ! Returning to CHARMM :
QUIT
RETURN

TReK known bugs

Definition of the RMS-difference in coordinates

Normally (and in CHARMM), the RMS-difference between the coordinates of two structures X1 and X2 is defined as follows (n = number of atoms) :

RMS = | X1 - X2 | / SQRT(n) ,

                          3n
where | X1 - X2 | = SQRT( SUM{ x1(i) - x2(i))^2 } )
                          i=1

But in TReK, all input and output of any value involving a distance in conformational space (for ex. LAMBDA or the curvature in “TRAJ ANAL”), the coordinate RMS-difference is computed as :

RMS'= | X1 - X2 | / SQRT(3*n)       , in other words

RMS = SQRT(3)*RMS'

Prematurely exceeding MAXP

In some molecular systems, it has happened (very rarely) that nearly identical points are added repeatedly one next to the other at the same location along the path, so that the maximum allowed number of path-points NMAXP is reached and the path-refinement stops. If this happens, simply continue the refinement after excising these extraneous points as well as a few additional path-points on each side along the path.

SADDLE command

A warning is printed after each “SADDLE” command: Ignore it ! The command is accepted anyway. After the CPR command has been issued, path-points that have been accepted as saddle-points are listed explicitly.