hamiltonian mapping demonstrates how the standard wham formalism can be modified and used to extrapolate information 
 about a biased hamiltonian from the unmodified reference hamiltonian , h0(x ) , where x represents the molecular coordinates .
wham is typically used along 
 with umbrella sampling , in which a set 
 of l additional restraining potentials , { v } = v1(x ) , 
 ... , vl(x ) , are added to h0(x ) in order to enhance conformational sampling in rarely visited regions 
 of phase space . in this case
, the modified hamiltonian takes the form1where {  } is the set 
 of coupling parameters 
 used for scaling individual biasing potentials , 0 = 1 , and h0(x ) = v0(x ) .
then , for r independent simulations , each performed at a temperature t = 1/kb ( where kb is the boltzmann constant ) and with differing 
 {  } , the bias is removed by solving the following pair of wham 
 equations self - consistently2and3 t may be constant or vary 
 across the r simulations .
)  is a particular 
 value along a progress variable of interest , nm is the total number of configurations in 
 the mth simulation , and nk({v}, ) is the histogram count 
 of configurations with { v } and . finally , fm is related to the helmholtz 
 free energy for the mth simulation .
in contrast to standard wham , in the hamiltonian mapping framework , 
 we proceed in the opposite direction ; we perform one or more simulations 
 of the system under the original hamiltonian h0(x ) and then add a bias to the statistical 
 averages computed from h0(x ) to examine thermodynamic behavior under a modified hamiltonian . 
 for this scenario ,
eq 2 is recast as4essentially , 
 the histogram counts nk now come from the sampling 
 of h0(x ) , and the sum 
 of the scaled restraining potentials in the denominator reduces to h0(x ) .
briefly , in 
 this model , each residue is represented by a single bead centered 
 at the c position and with the mass of the corresponding amino 
 acid . an additive potential describes the bonded and non - bonded interactions 
 between the beads .
virtual bonds and angles are defined by harmonic 
 potentials with reference values determined by the c coordinates 
 in the experimental structure . a potential for virtual dihedral angles 

is based upon backbone dihedral angle probability distributions from 
 the protein data bank ( pdb ) for the 400 possible amino acid pairs 
 and thus is independent of the specific topology of the system .
residue 
 pairs separated by three or more bonds interact through the following 
 potential5where rij is the distance between residues i and j , ij corresponds to 
 the distance between the two residues at which vij is minimum , and ij is the interaction strength at ij .
residues located in close proximity in the experimental structure 
 ( i.e. , forming a  native contact  ) interact favorably 
 with ij values based on the statistical 
 contact energies of miyazawa and jernigan , while residue pairs not in close proximity experience a slight repulsive 
 interaction that takes the form of a typical 126 lennard - jones 
 potential
. further description of the go-like model and details 
 of the simulation setup can be found in refs ( 25 ) and ( 8) , respectively .
we further 
 consider the non - bonded interaction potential ( eq 5 ) for native contacts between the idp and the receptor in 
 tuning the model to match the experimental kd .
for this set of intermolecular contacts , we scale the interaction 
 strength , ij , by a factor  
 so as to control the proportion of bound and unbound states throughout 
 simulation . from the fraction of unbound states , pu ,
the kd is computed as6where [ protein ] is the concentration of protein 
 in moles per liter ( 10 mol / l in the current study ) .
a configuration is considered unbound if zero intermolecular native 
 contacts are formed and if the distance between the centers of mass 
 of the idp and of the target protein is greater than the cutoff for 
 calculating the non - bonded interactions ( 25  ) .
we also note 
 that the strength of intramolecular contacts within the idp are tuned 
 in a similar manner to reproduce the experimental estimate of residual 
 helical structure in the unbound state .
 molecular recognition by intrinsically 
 disordered proteins ( idps ) 
 plays a central role in many critical cellular processes . toward achieving 
 detailed mechanistic understanding of idp  target interactions , 
 here 
 we employ the  hamiltonian mapping  
 methodology , 
 which is rooted in the weighted histogram analysis method ( wham ) , 
 for the fast and efficient calibration of structure - based models in 
 studies of idps . by performing reference simulations on a given hamiltonian 
 , 
 we illustrate for two model idps how this method can extrapolate thermodynamic 
 behavior under a range of modified hamiltonians , in this case representing 
 changes in the binding affinity ( kd ) of 
 the system . given sufficient conformational sampling in a single trajectory , 
 hamiltonian 
 mapping accurately reproduces kd values from direct simulation . 
 this method may be generally applied 
 to systems beyond idps in force field optimization and in describing 
 changes in thermodynamic behavior as a function of external conditions 
 for connection with experiment . 
Computational Methods