Ively. Note that right here we label these spline functions as the italic f to adhere to the literature convention, and they must not be confused together with the Cartesian atomic force vectors, which are labeled as bold non-italic f inside the text. Consequently, the spline function described in Eq.i (A4) consists of altogether mi = 2ngrid adjustable parameters that must be solved for the ithCV’s FM condition, i.e.,i i i i P i i i i F il(g1, g2, …, gmi) f ri, rk , f1, f1 , f2, f2 , …, fni i grid, fnii grid(A7a)whereJ Chem Theory Comput. Author manuscript; available in PMC 2022 August 10.Kim et al.Pagei i (g1, g2, …, gmii ) i = 2ngridi i i i f1, f1 , f2, f2 , …, fniigrid, fnii gridAuthor Manuscript Author Manuscript Author Manuscript Author Manuscript(A7b)In FM, the numerical remedy of Eq. (A3) is obtained at a stationary condition thati minimizes the objective function 2 with respect to the parameters gj(j = 1, two, …, mi): P i i i L N F il(g1, g2, …, gmi) d two 1 Ref P i i i 2 F il – F il(g1, g2, …, gmi) – = i i LN l i dgj gj =1 =(A8)=Using a short-hand notation:P i F il gj = P i i i F il(g1, g2, …, gmi) i gj(j = 1 , 2,…, mi)(A9)P at the same time because the certain functional form of F il defined in Eq. (A4), we’ve got: P i P F il fj = F il i g P F il f i2j -=A(A10a)j+P = F il gi2j +=B(A10b)P i P F il fj = F il gi = C 2j P F il f i(A10c)j+P = F il gi2j +=D(A10d)Right after rearrangement, Eq. (A8) may be written as:l = 1i =LNP i P F il gjF il =l = 1i =LNP i Ref F il jF il (j = 1 , 2,…, mi) g(A11)Now take into account a column vector g because the union of all parameters for the N collective variables,i g = gj (j = 1, two, …, mi; i = 1, two, …, N T (A12)where the superscript “T” denotes a transpose; the dimension M in the unionized parameter vector g is:J Chem Theory Comput. Author manuscript; accessible in PMC 2022 August ten.Kim et al.PageM=miiNAuthor Manuscript Author Manuscript Author Manuscript Author Manuscript(A13)Then Eq. (A8) can be written within a far more compact matrix kind: FP Together with the following identity: FP = FP g g Eq. (A14a) might be written as: FPT g (A14b) T P g F= FPT Ref g F(A14a)FP g = FP gT Ref g F(A14c)This is equivalent to solving the parameters set g to get a linear equation system: FPT gFP g = FP gT Ref g F(A15)Note that due to the overdetermined nature of Eq. (A15) in FM, its numerical answer may be obtained by QR decomposition45 or singular worth decomposition (SVD),45 with which Eq. (A3) would be satisfied in a least-square manner; for any perfect FM, the parametrized force correction in Eq. (A15) would restore the reference force correction exactly: FP g = FRef g For that reason we determine Eq. (A15) because the essential functioning equation for conducting FM in multidimensional CVs.Lumican/LUM, Mouse (HEK293, His) Note that in Eqs.MEM Non-essential Amino Acid Solution (100×) manufacturer (A146), FP and FRef are both NL-dimensional column vectors,(A16)FP is NL M matrix together with the major dimension (i.PMID:25046520 e., quantity of rows) getting NL, and g gis a M-dimensional column vector to be solved. For implementation objective, consistency of Eq. (A15) in its matrix form may be verified by analyzing the dimensionalities with the matrix operations involved: (M NL) FP T (NL M) FP (M 1)g = (M NL) FP T g g g (NL 1)FRef = M (A17)J Chem Theory Comput. Author manuscript; available in PMC 2022 August ten.Kim et al.PageFor readers who are thinking about far more facts concerning the implementation, a concrete example could be identified in SI.two, where we illustrate the matrix type of Eq. (A15) to get a two-bond CV case according to a certain set of sample and grid distributions.Author Manuscript.
