We designated the simulation system of pKID in the absence of KIX as the “pKID system” and that in the presence of KIX as the “pKID–KIX system.” In the NMR experiment on the pKID–KIX complex [27], the pKID sequence was longer than that deposited in PDB (residues 119–146) because unstructured regions are not deposited. We used the deposited pKID region for the simulation, which is the minimum sequence of pKID binding with KIX (residues 586–666).

We prepared the pKID system as follows. The coordinates of pKID were taken from the first model out of the 17 NMR ones. The pKID was immersed in a solvent sphere (called sphere 1; radius = 30 Å), which consists of water molecules with the density of 1 g/mL equilibrated at 300 K in advance. The mass center of pKID was set to the center of sphere 1, and water molecules overlapping with pKID were removed. To neutralize the net charge of the system and make consistency with the ionic strength of the NMR experiment, nine water molecules were selected randomly and replaced with five Cl⁻ and four Na⁺ ions. Finally, the pKID system consisted of 3473 water molecules, pKID, and nine ions. Although pKID was taken from the NMR model at this stage, the structure was randomized completely using a high-temperature simulation, as described later.

Next, we prepared the pKID–KIX system as follows. The coordinates of the two proteins were taken from the first NMR model again. They were immersed in a solvent sphere (called sphere 2; radius = 37 Å). The sphere 2 radius was sufficiently large to contain the entire KIX structure, as described later. The center of sphere 2 was set to the mass-center position of pKID of the NMR model. It is noteworthy that the center of sphere 2 is fixed in space (i.e., the center of sphere 2 is not fixed to pKID) when pKID is moving in the simulation: The translation and rotation of pKID are not restrained. Water molecules overlapping the proteins were removed. Eighteen randomly selected water molecules were replaced with nine Cl⁻ and nine Na⁺ ions. Then, the pKID–KIX system came to consist of 6166 water molecules, pKID, KIX, and 18 ions. This complex was dissociated by a high-temperature simulation, as shown later.

We used the AMBER-based hybrid force field for the proteins. This force field is the combination of AMBER force field parm94 (E₉₄) [31] and parm96 (E₉₆) [32] with a mixing rate ω as E = (1-ω)×E₉₄ +ω×E₉₆ . We set ω= 0.75 because our previous works indicated that ω= 0.75 reproduces the optimal secondary structure preference for some peptides [33,34]. The TIP3P model [35] was used for the water molecules. After energy minimization, we performed a high-temperature (700 K) canonical MD simulation for each of the pKID and pKID–KIX systems to generate the initial simulation conformations for the following TTP-McMD. Figure 1 shows that this temperature was sufficiently high to demolish the native conformation of pKID and to dissociate the native complex. Although pKID was able to move freely in the solvent spheres, the structure of KIX was weakly restrained in the simulation of the pKID–KIX system, as described later. We used the simulation program PRESTO ver. 3 [36]. The canonical MD and TTP-McMD simulations were done in the following condition: 1.0 fs time step, SHAKE algorithm [37] to constrain covalent bonds between heavy atoms and hydrogen atoms, and the cell-multipole expansion method [38] to compute the long-range electrostatic interactions. Throughout the simulation, we maintained the volume of sphere 1 and 2 by supplementing a harmonic potential on atoms flying out of the sphere to pull them into the sphere. Additionally, for the pKID–KIX system, the positions of the main-chain atoms (N, Cα, C, and O) of KIX residues (591–594, 597, 608–611, 617–621, 623–640, 646–648, 660–661) were restrained on those of the NMR model by a weak harmonic potential to maintain the KIX structure around the NMR model. No restraint was applied on pKID to diffuse it freely in sphere 2. Figure 1(a) and Figure 1(b) are the last snapshots of the high-temperature canonical MD simulations of the pKID and pKID-KIX systems, respectively, that are used for the initial structures in TTP-McMD simulations.

Before introducing TTP-McMD, we describe the conventional McMD, i.e., the single-run McMD. The single-run McMD is a canonical simulation at a temperature T (a constant-temperature method [40] was used as thermostat in this study) using the multicanonical energy E_mc instead of the original potential energy E of the system, as

Where n(E) is the density of states of the system, R signifies the gas constant, P_c(E,T) the probability distribution function of the canonical ensemble at T. If the simulation is sufficiently long, then the single-run McMD provides a flat distribution along the axis of E because the probability distribution is formally given by the following equation.

In that equation, Z_mc is the partition function for McMD, which can be regarded simply as a factor to normalize the distribution function P_mc To derive Equation (2), we used a formulation of statistical mechanics: P_c(E,T)=n(E)exp(-E/RT)/Z_c, where Z_c is a factor to normalize P_c(E,T) i.e., the partition function of the system). This flat energy distribution guarantees that the conformation can overcome the energy barriers and visit low energy conformational regions during the simulation. We refer to the entire conformational ensemble obtained from McMD as a “multicanonical ensemble.” Then we can reconstruct a canonical energy distribution P_c at any target temperature T_tag from P_mc as

To derive Equation (3), we used Equation (2) in the following form: n(E)=Z_mcP_mc(E)exp(E_mc/RT). The canonical conformational ensemble at T_tag is constructed by assigning the probability P_mc(E,T_tag) to all conformations in the multicanonical ensemble.

McMD uses the probability density function P_c(E,T) in Equation (1), which is generally unknown a priori (before simulation). Then, we must construct it self-consistently through iterative simulation runs, during which P_c(E,T) converges to a precise function. TTP-McMD takes an advantage over the single-run McMD [19,20]. TTP-McMD is technically equivalent with performing independent multiple runs of single-run McMD starting from various initial conformations. The multiple trajectories generated are integrated simply into an ensemble. It is noteworthy that low-energy conformations (low-energy basins) distribute widely in the conformational space, which are spaced by high-energy barriers. Then, a single-run McMD takes a long flight while overcoming the barriers to reach the low-energy basins. On the other hand, the multicanonical algorithm tries to ensure the flat distribution along the energy axis (Equation (2)) so that the conformation is expected to exist evenly in both the low-energy and high-energy regions. This evenness might cause a difficulty of single-run McMD, i.e., no convergence. The connected multiple runs (TTP-McMD), which are spread in the conformational space, are equivalent to a long trajectory, each part of which flights and searches the low-energy basins. Consequently, TTP-McMD provides the convergence of conformational ensemble faster than the single-run McMD.

The initial conformations for TTP-McMD were those shown in Figure 1(a) and (b), which were generated from the high-temperature canonical MD simulations, as described above. In this study, we performed 256 multiple runs for each system. To obtain the accurate P_c(E,T) in Equation (1), we performed the simulations iteratively. First, we performed a canonical simulation at 700 K and generated the probability P_c(E,700K) for each system. This probability distribution is restricted around the peak position (designated as E_high) of P_c(E,700K), and is accurate only in the narrow region. We designate this energy range as [E₀,E_high] (E₀<E_high), where E₀ is a lower limit of the accurately sampled energy region. Second, we performed the first TTP-McMD at 700 K with the multicanonical energy E_mc, where P_c(E,700K) obtained above was used for P_c(E,T) in Equation (1). In the simulation, we set artificial energy walls at E₀ and E_high so that the conformation did not escape out of the range [E₀,E_high]. This simulation produced a flat energy distribution P_mc(E) only in [E₀,E_high]. Then, using Equation (3) we reconstructed the probability P_c(E,200K). Here, E_mc defined by E + RT ln P_c(E,200K) is effective only for multicanonical runs at 200 K. The reason for this temperature reset (i.e., 700 K → 200 K) is explained later. We again extrapolated P_c(E,200K) to an energy range as [E₁,E_high] (E₁<E₀), and performed multicanonical runs at 200 K to produce a flat energy distribution P_mc(E) in this extrapolated energy range. Multicanonical runs were performed iteratively until the sampled energy range reaches an energy (E_low) that corresponds to a temperature lower than a room temperature. After reaching E_low, we performed the final TTP-McMD to generate a flat energy distribution in [E_low,E_high]. We used 256 computing cores (intel Xeon X5365 3.0 GHz), each core executed one run of TTP-McMD. In the equilibration stage (i.e., the stage to estimate E_mc before the final sampling stage), the simulations were done for 21 ns and 23 ns in each of 256 trajectories (total of 5.4 μs and 5.9 μs) of the pKID and pKID-KIX system, respectively. The final runs were done for 8.51 ns in each of 256 trajectories (total time of 2.18 μs) and stored the snapshots every 5 ps for subsequent conformational analyses. The computation times for the equilibration stage were 33 and 76 days for the pKID and pKID-KIX system, respectively. Those for the final sampling stage were 13 and 28 days for the pKID and pKID-KIX system, respectively.

As described above, we reset the simulation temperature from 700 K to 200 K. The multicanonical energies at the two temperatures are, respectively, E_mc(700K)=700×R ln n(E) and E_mc(200K)=200×R ln n(E). Mathematically, the multicanonical ensembles from E_mc(700K) and E_mc(200K) are expected to be equivalent:

.

Consequently, the temperature reset is theoretically non-sense. However, the two simulations produce practically different ensembles. We use a polynomial function to approximate P_c(E,T) in Equation (1). The function form P_c(E,200K) is smoother than P_c(E,700K). Then, P_c(E,200K) is more suitable than P_c(E,700K) to define E_mc.

The TTP-McMD simulation for the pKID system is explored evenly in an energy range [−35000.0 kcal/mol, −23480.0 kcal/mol] that corresponds to a temperature range [300 K, 700 K]. From the multicanonical ensemble, we constructed a conformational ensemble at 315 K (denoted as “315K-ensemble”) consisting of 9922 conformations. It is noteworthy that 315 K is the temperature of the NMR experiment [27]. The average of energy E at 315 K was −34272.0 kcal/mol. The secondary structure content at each residue position is shown in Figure 2. Apparently, the α_A region preferred helix more than the α_B region does. This result agrees with the NMR observation [30]. However, the helix content rate was small: The content for α_A was below 50% and that for α_B was about 20%, and the unbound pKID was fluctuating among the helix and non-helix conformations at room temperature. Furthermore, the relative positioning between the α_A and α_B regions fluctuated highly in the 315K-ensemble. Consequently, the simulation confirmed that pKID in the unbound state is intrinsically disordered.

We investigated whether the conformation in the unbound state showed similarity with that in the native bound state (i.e., NMR structure). Picking up a sampled conformation from the 315 K-ensemble, the root-mean-square deviation (RMSD) was calculated for each of the α_A and α_B regions as follows: After superimposing Cα atoms of the α_A region onto those in each of the 17 NMR models, 17 RMSD values were calculated and the smallest RMSD of the 17 values, denoted as RMSD_αA, was selected. Similarly, the smallest RMSD, denoted as RMSD_αB, was calculated for the α_B region. The Cα-atomic RMSD for the entire pKID (residues 121–144) is denoted as RMSD_all, where raw coordinates of pKID were used for the RMSD computation without superposition. Figure 3 presents the probability distributions of RMSD_αA and RMSD_ΑB for all conformations in the 315K-ensemble. A non-negligible peak was found in the α_A region at small RMSD_αA: RMSD_αA ≈ 1.0 Å. Contrarily, the α_B region showed a small peak at RMSD_αB≈ 1.0 Å. These results suggest that the α_A and α_B regions have different mechanisms of coupled folding and binding: α_A might bind with KIX with a population-selection mechanism, where the structured fraction of α_A is used for binding to KIX. In contrast, α_B might bind to KIX with an induced-folding mechanism, where α_B is bindable to KIX with various conformations and then the native complex is formed.

The TTP-McMD simulation of the pKID–KIX system produced a flat energy distribution in an energy range [−64440.0 kcal/mol, −42500.0 kcal/mol], which corresponds to a temperature range [300 K, 700 K]. From the multicanonical ensemble, we derived the 315K-ensemble (8840 conformations). The free energy landscape for the α_A and α_B regions on binding to KIX was constructed as a function of two variables R_αA and R_αB defined as follows: and , where and respectively represent the position vectors of the centroids of the α_A and α_B regions in a sampled conformation, and and those in the reference structure (i.e., the NMR model), respectively. Using these distances as the reaction coordinates, we calculated the potential of mean force (PMF) as

where the probability P(R_αA,R_αB) was represented as the number of conformations counted in a fraction [i ± 0.5 Å, j ± 0.5 Å] (i,j = 0,1,2,∙∙∙) of [R_αA,R_αB] over the total conformations (8840) in the 315-K ensemble, R is the gas constant, and T is the temperature (315 K). Figure 4(a) shows the free energy landscape PMF, which was complex and ragged. We found a low free-energy fraction circled by green circle around (R_αA,R_αB)=(4 Å, 3 Å). We designate this fraction as the “native-like low-free-energy fraction.” This native-like fraction comprised conformations with RMSD_all<7.1 Å. The nearest-native structure, shown in Figure 4(b), had RMSD_all of 5.65 Å, whose position in Figure 4(a) is: (R_αA,R_αB)= (4.43 Å, 2.94 Å). This structure forms a partially disordered α_A helix and a well-ordered α_B helix.

We also found a non-native broad fraction with low free energy in Figure 4(a) (red circle): [R_αA,R_αB]= [5–13 Å, 0–7 Å]. We designate this fraction “non-native low-free-energy fraction”, where pKID attached KIX with various binding poses as exemplified in Figure 4(c). Because the two fractions were distinguishable in Figure 4(a), a free energy barrier exists between them. The non-native low-free-energy fraction suggests that pKID can bind to KIX with a structural diversity, and the generated various encounters reach the final native-like fraction across the free energy barrier. This result suggests that the high flexibility of pKID might help the pKID–KIX association because pKID binds to KIX without adopting a well-ordered structure. The final native-complex formation is completed after forming the various non-native complexes. Later in this report, we describe our examination of why the non-native low-free-energy fraction was larger than the native-like low-free-energy fraction in the free-energy landscape and why the α_A helix is partially disordered, even in the native-like low-free-energy fraction (Figure 4(b)).

We found pKID bound to another site of KIX in 323 snapshots of the 315K-ensemble (Figure 4(d)). This site is a binding site for the activation domain of the mixed lineage leukemia (MLL) transcription factor [42]. The MLL–KIX complex structure (Figure 4(e)) shows that a segment of MLL adopts helix and binds to KIX, and the other parts are unstructured. In all of these snapshots, the α_B region adopted helix to bind to the MLL binding site with the α_A region unstructured. The orientation of the α_B helix cylinder was approximately the same as that of the MLL segment in the MLL–KIX complex structure [43]. Consequently, the α_B region corresponds to the MLL segment. In fact, both the MLL-binding site and the genuine α_B-binding site on the KIX surface consist of hydrophobic amino acids. Furthermore, the hydrophobic residues in the α_B region and the MLL segment have similarity; they contains ϕ-x-x-ϕ-ϕ motif (ϕ = hydrophobic residue and x = any residue), which is conserved in many KIX binding proteins (see Figure 9 of reference [43]). It is particularly interesting that in the presence of MLL, pKID binds to KIX with the two-fold higher affinity than pKID in the absence of MLL [44]. Our simulation results suggest that MLL might facilitate the pKID binding to the genuine binding site via blocking the MLL binding site.

The orientations of the α_A and α_B regions with respect to the KIX framework were investigated, respectively, using inner products, I_αA and I_αB. The inner product I_αA was defined as

where vectors and respectively represent the unit vectors of vectors and : see Figure 1(c). The vector is pointing from the Cα-atomic position of the 124th residue to that of the 128th residue in a sampled conformation. The vector is defined in the same way for the NMR structure. For the orientation of α_B residues, I_αB was defined as

The unit vectors and were calculated similarly as and by replacing the residue numbers 124–134 and 128–141. When the inner products I_αA and I_αB are 1, the orientations are aligned as in the native bound form.

The inner products were calculated for 2476 conformations whose distances [R_αA,R_αB] satisfy R_αA≤13 Å and R_αB≤7 Å, which involves the native-like (green circle in Figure 4(a)) and non-native (red circle) low-free-energy fractions. The histogram for I_αB (Figure 5(b)) has the largest peak at 1, which indicates that the α_B region attaches to KIX with the same orientation as that in the final bound form in both low-free-energy fractions. We discussed the factors stabilizing this orientation later in Section 3.4. Recall that the α_B region less adopts helix than the α_A region in the isolated state (Figure 2) and that the α_B region seldom adopts native-complex form in the isolated state (Figure 3(b)). Consequently, it is likely that the α_B region binds to KIX with the right orientation and then the helix is formed. In contrast, the α_A region has a large variety in the orientation (Figure 5(a)). These results agree well with the experimental observation: The α_B orientation ordered well and the α_A orientation does less in the native complex form [27]. Furthermore, the simulation results are consistent with the experimental report that the α_B region has a stronger binding affinity than the α_A region to bind to KIX [25].

The folding of α_A and α_B regions upon binding was investigated by measuring residue contacts between pKID and KIX. An intermolecular contact was determined as the distance between the centers of side chains of two residues: If this distance is below 6.5 Å, then we judge that the two residues are contacting. We calculated the contacts (“native contacts”) in the 17 NMR models, where a contact is assigned to a residue pair if the pair is contacting in at least 8 of the 17 models. We found 30 native contacts: 8 between α_A and KIX residues, 18 between α_B and KIX residues, and 4 between the other residues in pKID and KIX residues. We designate contacts other than the native contacts found in simulation snapshots as “non-native contacts”.

We counted the quantities of the native and non-native contacts, respectively denoted as Nnc and Nnnc, for each conformation. Figure 6 presents the helix content rate in pKID at 315 K projected on the Nnc-Nnnc plane. Coupled folding and binding for α_B is explained well from Figure 6(c): In complexes where the native contacts are less formed, various non-native contacts are formed. With increasing native contacts, the non-native contacts decrease. Finally the full native contacts are formed with a few (ca. 5) non-native contacts. This result suggests that coupled folding and binding of the α_B region accords to the induced-fit mechanism, where the α_B region varies the conformation in the encounter complex to reach the native complex. Because the helix content rate of α_B is small in the unbound state (Figure 3(b)), the induced-fit mechanism has an advantage over the population-selection mechanism. In fact, the formation of the non-native contacts before the native contacts might facilitate the pKID–KIX association, as discussed earlier in the Introduction: the α_B region can bind to KIX via non-native contacts without waiting until a helix is formed.

However, the α_A folding depends less on Nnc than the α_B folding did. A few native contacts (ca. 4, which is 50% of the entire native contacts) were able to stabilize the binding of α_A to KIX. This result derives from the high directional fluctuations of α_A, as shown in Figure 5(a). This result agrees qualitatively with those of the NMR experiment [27], where the flexibility of α_A is greater than α_B. One can recognize that α_A has directional diversity by viewing the NMR models (PDB ID: 1kdx). Furthermore, the fragile contacts between α_A and KIX qualitatively support and agree with the experiment [25] showing that the binding affinity assigned to α_A is weaker than that to α_B. We were unable to specify which of the induced fit or population selection characterizes the α_A binding better, although Figure 3(a) supports the population selection. The large flexibility of α_A prevents us from answering which is likely. However, we emphasize that the dynamics [27] and the weak binding affinity [25] of α_A are characterized by its large flexibility.

We demonstrated the change of accessible surface area (ASA) on binding. We denote ASAs for the α_A and α_B regions respectively as ASA_αA and ASA_αB. The distributions of ASA_αA and ASA_αB are presented in Figure 7. The average of ASA_αB was 1136 Ų for the pKID system and 889 Ų for the pKID–KIX system. Consequently, the reduction of ASA upon binding is clear for α_B (Figure 7(b)). This reduction was caused mainly by hydrophobic contacts formed in the complex. However, for α_A the reduction was small (Figure 7(a)): the average ASA_αA was 1294 Ų in the pKID system and 1150 Ų in the pKID–KIX system. This small reduction occurs because the α_A region has a large helix probability in the pKID system (Figure 2) and the α_A-KIX interaction is weak (Figure 5(a)). These results are consistent with experimental observations [25], showing that α_A and α_B regions are characterized, respectively, by weak and strong binding affinities.

The averages of ASA_αA and ASA_αB over the 17 NMR models (complex form) are, respectively, 1093±72 Ų and 723±52 Ų. Consequently, the computed ASA is larger than the experimental value because the computed 315K-ensemble consisted of various complex conformations, as shown in the non-native low-free-energy fraction (Figure 2(c)) and even the conformations in the native-like low-free-energy fraction partially disordered as in Figure 4(b).

A hydrophobic interaction is a non-directional interaction. However, the α_B residues were able to bind KIX with the same orientation as in the native complex form (Figure 5(b)). This directional conservation is inferred from a spatial distribution of the phosphorylated SER (pS133), which is located in the middle of the α_A and α_B (Figure 1(d)). The pS133 interacted with a hydrophilic residue (Y658) and a positively charged residue (K662) of KIX. Figure 8(a) depicts that pS133 restrained the hydrophobic interaction of the α_B helix with KIX as in the native complex form. Figure 8(b) demonstrates the spatial distribution of the Cα-atomic position of pS133, where the iso-density contours were computed for conformations with R_αA ≤ 13 Å and R_αB ≤ 7 Å. Although the high-density site was slightly deviated from that in the NMR structure, the anchor effect of pS133 is clearly shown. Consequently, phosphorylation plays an important role for specifying the α_B-helix orientation (Figure 5(b)). We presume that this phosphorylation-induced restraint on the α_B helix increases the binding affinity more than non-phosphorylated KID [25,26].