Parametric sensitivity analysis of avian pancreatic polypeptide (APP)

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PROTEINS: Structure, Function, and Genetics 23:218-232 (1995)

Parametric Sensitivity Analysis of Avian Pancreatic Polypeptide (APP) Hong Zhang,' Chung F. Wong? Tom Thacher? and Herschel Rabitz' 'Department of Chemistry, Princeton University, Princeton, New Jersey 08544; 'Department of Physiology and Bionhvsics., Mount Sinai School of Medicine, New York, New York 10029-6574; and "Biosym Technologies, Znc., Sun Diego, California 92121 1

"

ABSTRACT Computer simulations utilizing a classical force field have been widely used to study biomolecular properties. It is important to identify the key force field parameters or structural groups controlling the molecular properties. In the present paper the sensitivity analysis method is applied to study how various partial charges and solvation parameters affect the equilibrium structure and free energy of avian pancreatic polypeptide (APP). The general shape of A P P is characterized by its three principal moments of inertia. A molecular dynamics simulation of A P P was carried o u t with the OPLS/Amber force field and a continuum model of solvation energy. The analysis pinpoints the parameters which have the largest (or smallest) impact o n the protein equilibrium structure (i.e., the moments of inertia) or free energy. A display of the protein with its atoms colored according to their sensitivities illustrates the patterns of the interactions responsible for the protein stability. The results suggest that the electrostatic interactions play a more dominant role in protein stability t h a n the part of the solvation effect modeled b y the atomic solvation parameters. 8 1995 Wiley-Liss, Inc.

Key words: protein conformation, protein stability, sensitivity analysis, avian pancreatic polypeptide (APP), molecular dynamics simulation, OPLS/Amber force field, continuum solvation model INTRODUCTION Much work has been done to study the free energies of ligand binding to biomolecules and to predict the conformation of biologically important molecules, using computer simulations such as Monte Carlo (MC) and molecular dynamics (MD).1-5 Computer simulations employ a simple approximate classical force field to describe various interactions among atoms. By studying how the results of computer simulations are affected by various force field parameters, one is able to find those parameters which (1)may be further improved to deliver more 0 1995 WILEY-LISS, INC.

reliable simulation results, and (2) may be most influential upon certain properties of a molecule, thus pointing out the atoms or groups of atoms which play dominant roles in determining the properties. The knowledge gained may be useful for understanding functions of biomolecules, and designing new drug molecules. Molecular simulations often require very timeconsuming computations. Due t o the large number of parameters involved, it is difficult to employ brute force simulations to study how changes of various force field parameters will affect the conformation or other properties of a biomolecule. In the present paper, we employ the more efficient sensitivity analysis technique recently extended to the biomolecular modeling The changes in conformation or any observable are characterized by their first or higher order derivatives, called sensitivities, with respect to the force field parameters. The sensitivities can be efficiently calculated by a single MD (or MC) simulation. The first-order sensitivities are the linear response of the properties of a molecule to changes in the force field parameters, and the second or higher order sensitivities represent the nonlinear responses of the system. Therefore, the sensitivities provide important information on how the molecular properties depend on various parameters, as illustrated in the present paper and other The conformation of a biomolecule is generally believed to be directly related to its biological functions. An incorrectly folded protein, for example, can cause malfunctions and diseases. Efforts are being made to determine what interactions (or force field parameters) are responsible for the unique conformation in which a biomolecule exists. In this paper, the conformation of the pancreatic hormone, avian pancreatic polypeptide (APP), is studied using an MD simulation. APP is a small globular protein with 36 amino acid residues and a compact and stable conformation. It can exist as a monomer though

Received November 16, 1994; revision accepted April 11, 1995. Address reprint requests to Herschel Rabitz, Department of Chemistry, Princeton University, Princeton, N J 08544.

219

PARAMETRIC SENSITIVITY OF APP

it often forms a dimer. Its crystal structure has been refined to 0.98 hl by X-ray experiments," and its conformation in solution has also been The crystal structure consists of a polyproline-like helix running from residues 1 to 8, packed against the hydrophobic face of an a-helix that extends from residues 13 t o 31. Residues 9 to 12 form a p-turn. The more reflexible C-terminus residues (32-36) do not form part of the a-helix, and it (as well as the N-terminus) has been suggested to be responsible for APP bi0a~tivities.l~ The principal moments of inertia of APP are studied, since they characterize the compactness of a molecule around the three principal axes, providing a general picture of APP conformation. The free energy of the APP and the average rms deviation of the simulated structure from the crystal structure are also of interest. The sensitivities of these quantities with respect to the partial charges, the atomic solvation parameters, and the dielectric constant are calculated and analyzed. As a result the key parametersiinteractions most influential to these observables are identified. THE MODEL AND THE COMPUTATIONAL PROCEDURES Molecular Dynamics Simulation The crystal structure of APP refined to 0.98 A by Glover et a1.l' was obtained from the Brookhaven Protein Data Bank. The OPLSAmber force field233 with a continuum solvation term was adopted. The total potential energy is expressed as follows:

where the terms stand, respectively, for energies with respect to bond lengths, bond angles, dihedral torsion angles and the improper torsion angles, the van der Waals interaction, the coulomb interaction, and the solvation term. The symbols have their usual meanings. With the continuum model of solvation applied here, u iis the atomic solvation parameter for atom i, and Siis the solvent accessible surface area of atom i. Hasel et al.'s approximate analytic method1* (which speeds up the calculations of the solvent accessible surface area) is used t o evaluate the solvent accessible surface areas using a probe radius of 1.4 hl. The atomic solvation parameters are given by Schiffer et al.15 The effective dielectric constant is taken as the distance between an atom pair. There is no finite cutoff distance for calculating nonbonded interactions. There are N = 368 atomslextended atoms (reduced from 581 atoms in the all atom model) in the model of APP. The mo-

lecular simulation program DISCOVER by Biosym Technologies, 1nc.l' is used with the necessary alterations made for the current studies. At the initial stage, the crystal structure was energy minimized using the steepest descent method. The minimized structure has an rms deviation of 0.12 hl from the crystal structure. Starting from the minimized structure, an MD simulation is run for 1ps each at 100 K and 200 K, and then 3 ps at 300 K. During this period, the temperature is scaled to the given value a t each step. The MD simulation continues up to 420 ps with the heat bath method,17 with a relaxation time of 0.1 ps. The step size in the MD simulation is 1 fs and structures are saved every 10 fs. The Leapfrog method is used for time integral in MD simulation. Calculating the Moments of Inertia Given the Cartesian coordinates for a structure of APP, we first calculate the center of mass r, = CrilN, where r, is the Cartesian coordiantes of the ith atom. Here and in the following, in calculating the moments of inertia, we have set the masses of all atoms t o one. This is because we are only interested in characteristic measures of the overall geometric distributions of the atoms. We then define the modified coordinates r:

=

ri - r,

(2)

and the elements of the moment of inertia matrix A is given by

A12 = A21 =

-C(x:y'J

(6)

~

1 =3 ~

3 = 1

-C(x\z;)

(7)

~

2 =3 ~

3 = 2

-C(y:z:)

(8)

The moments of inertia around the three principal axes are given by the solutions of the secular equation IA - XJ = 0 (9) The three principal moments of inertia (Ii,i = 1, 2, 3) are then always arranged from large to small, i.e., I , 2 I , 2 Z3.The summation of the three moments gives twice of the gyration radius R , ( 3 Cri2)of the molecule: I, + I , I3 = 2RG. All three principal axes pass through the center of mass, and since the APPs structure is ellipsoid like, the axes for I , and I , would roughly be perpendicular t o the axis of the a-helix, while I , would be parallel to that axis. These moments of inertia characterize the compactness of APP around these axes.

+

220

H. ZHANG ET AL.

Ensemble Averages and Sensitivity Calculations The ensemble average of a physical quantity H only depending on coordinates is given by

--2

r

-500

I

I

I

I

,

I

I

I

I

I

I

I

I

I

I

50

100

I

I

I

I

I

I

I

I

I

400

450

I

.

I

-550

-600 x

-650 fi

JHe- pv({rz))d{r,} =

(10)

Je-Pv({rL8d{rL}

The parametric sensitivity of with respect to a potential parameter a is given by

;

4

5

!2

and a second order sensitivity by

aH aV

5 -

4.5 3.5

%

3

2

1.5

l i

H

O

-

tl

-

4 -

=: 4

aH aV

-850 I 5.5,

-.

0

150 200 250 300 350 Time of MD Simulation (ps)

Fig. 1. (a) The potential energy of APP vs the MD simulation time. The energy plotted is the averaged value over 500 fs. (b) The rms deviation of all heavy atoms in APP from the crystal structure vs the MD simulation time.

The observed value of H is , and its sensitivities to a and y are to be studied. Here, p = l/k,T, where Fz, is the Boltzmann constant and T is the absolute temperature. If the moments of inertia or the rms deviation are the observables and the partial charges and solvation parameters are the parameters, the first term in Eq. (11)and the first five terms in Eq. (12) vanish. The free energy of the molecule is given by

A

=

-

l/p ln[B

x Je-Pv({rL”d{rL}l (13)

where B is a constant. The free energy sensitivities to a potential parameter a can be calculated by

aA

(14)

aa

and its second order sensitivities to a and y by,

a2A aaay -=

(”>

aaay

(yd.av av)

(15)

All the sensitivities calculated in the present paper are normalized as (d/da) x a and (a2i d a d y ) x ay. Thus, sensitivities with respect to different parameters can be compared.

Using the ergodic hypotheses, the ensemble average are calculated by

(H)= C ~ ~ I H ~ / N T

(16)

where Hkis calculated using the kth structure in the MD simulation and NT is the total number of MD structures used.

Force Field Parameters to Be Studied The force field parameters to be studied are the OPLS partial charges q , and the solvation parameters cr, of each atom. The solvation parameters for different atom types C, N/O, N’, and 0- ( = 0.0325, -0.0175, -0.2 17, -0.280 kcal/mol, respectively) were obtained by Schiffer et al.15 The solvation parameters were optimized to yield the smallest rms deviation between the crystal and the simulated structures of rat trypsin after a 40 ps MD simulation. The first order sensitivities with respect to each individual charge and cr, are calculated, as well as the second order sensitivities d2/aq,dcrk. Another parameter studied is a in the dielectric constant (E,,, = a x where R , , is the distance between an atom pair i-j.). a = 1 is chosen in the current study, but a = 4 has also been used by others.

RESULTS AND DISCUSSIONS The potential energy of APP as a function of the MD simulation time is plotted in Figure la. It ap-

PARAMETRIC SENSITIVITY OF APP First

1

31.0 I

I ".

.0

Second

,

I

I

l

, l

1

60 80 100 120 Time of MD Simulation (ps)

40

-I

I

I

1

Third

20

-

140

-

160

-

Fig. 2. The ensemble average of the three principal moments of inertia vs the MD simulation time (indicating the size of the ensemble) in the analysis period, after 250 ps MD simulation.

TABLE I. The Sensitivities of the Moments of Inertia With Respect to a* I, 3908

1-2

13

-845.8

5660 -1355 -493.5 157.8 *For each moment Z,,the first column lists the sensitivity coefficients averaged over 140 ps, and the second column lists the value minus the one averaged over the first 70 ps.

pears that the APP molecule has settled down into a potential well a t around 250 ps. Figure l b shows the rms deviation of all heavy atoms of the APP molecule from the crystal structure during the MD simulation. The rms deviation for the backbone atoms (not shown) varies similarly around a smaller value of -3.5 A. It should be mentioned that in the crystal structure APPs form dimer and each of APP molecules binds a zinc ion, which in part stabilizes the

221

APP conformation. Since we have a single APP molecule without the zinc ion in our MD simulation, the large rms difference may be expected. Approximately, 150 ps of the MD trajectory after the initial 250 ps simulation is used in our analysis. Most of the rms deviation has occurred in the early stage of the MD simulation as indicated in Figure lb. The average structure of the APP over the period for analysis is calculated by averaging APP structures of heavy atoms in the MD trajectory superimposed with the crystal structure. In the average structure, the a-helical structure is maintained except some distortions in Pro-13 and Val-14. Both the p-turn width and shape are changed. This result is similar to the MD simulation by Kruger et a1.l' In the crystal structure there is a zinc ion bound to Gly-1, to the side chain of Asn-23, and to the side chain of His-34 of three different APP molecules. Without the zinc ion in our MD simulation, the N-terminus residue Gly-1 is bent over to interact with the sidechains of Asn-23 and Tyr-36, and the side chain of Tyr-27 interacts with Ser-3 and Gln-4. These interactions seem to be responsible for stabilizing the structure of the N-terminal residues (as supported by the sensitivity analysis results later). The strand containing the three prolines has a large rms deviation from the crystal structure, which is due to the shrinking and distorting of the polyproline strand. The deviation is much more pronounced than that in Kruger et a1.k work. The C-terminus residue Tyr-36 bends over to interact with the N-terminus residue Gly-1, in contrast to the crystal structure and the simulated one by Kruger et al. in which Tyr-36 is exposed to the solvents. As a result, the C-terminus residues have the largest rms deviation (see Fig. 6b) from the crystal structure. The side chain of Phe-20 rotates during the MD simulation, similar to that of Kruger et a1.l' As a result the six-membered ring in the average structure becomes a small ring as shown in Figure 3 (translational motions of the ring would keep the average ring size unchanged).

The Principal Moments of Inertia Figure 2 plots the ensemble averages of the three moments of inertia vs the MD simulation time. The values of the moments of inertia (II- Z, and ZIJ2 > Z3) clearly indicate the ellipsoid like shape of the APP structure. The center of mass is located near residue Asn-23. The axis for I3 is almost parallel to the a-helix. The axis for I , points (in a small angle with the plane formed by a-helix and the polyproline strand) toward the polyproline strand of the APP. Table I lists the sensitivities of these moments with respect to a. There are two values listed for each sensitivity. The first is calculated based on the total 140 ps of the trajectory, and the second is the difference between this value and the one calculated using the first half (70 ps) of that trajectory. The second number gives a measure of the statistical

222

H.ZHANG ET AL.

Fig. 3. A display of APP with atoms colored according to the sensitivities of the gyration radius with respect to the atomic solvation parameter IT, of each atoms. A darker blue represents a larger positive sensitivity, and a darker red indicates a larger negative sensitivity. The white color indicates the zero sensitivity. The

green line is a ribbon diagram of the APP structural motif, and the blue axes indicate the three principal axes of inertia. The x, y, and z axes correspond to /3,/2,and I,. The y-axis is noted in the figure, and the x-axis points to the right lower corner and the other axis is the z-axis.

fluctuations in these sensitivities. The data show that 13,which characterizes the width of APP, is not very sensitive to a. In contrast, a small increase in a (the scaling factor of the dielectric constant) would increase both I , and 12,i.e., an increase in electrostatic interactions would shrink the protein in the longitudinal direction while keeping the width unchanged. It seems that the hydrogen bond interactions within the a-helix may play an important role for the effect. The gyration radius is also found to decrease. The sensitivities of the gyration radius with respect to uiof each individual atom are displayed in Figure 3. The protein conformation in the figure is the average structure over the trajectory for analy-

sis, and the atoms are colored according t o the values of their sensitivities. The largest sensitivities of the moments of inertia are listed in Table 11. Only a few of the atoms have relatively large sensitivities. They are on the molecular surface and exposed to solvents. The effect of increasing the magnitude of ui is equivalent to pulling (or pushing) those colored atoms in Figure 3 outward (or inward) to increase1 decrease the solvent accessible surface areas.

Fig. 4. (a) A display of APP with the atoms colored according to the sensitivities of gyration radius of APP with respect to partial charges of each atom. The others follow the same rule as in Figure 3. (b) A close look at C-terminus residues.

Fig. 4.

224

H. ZHANG ET AL.

TABLE 11. The Largest Sensitivities of the Moments of Inertia with Respect to u j of an Individual Atom* Atom name CG(Pro2) CG2(Thr6) CG(Asp-10) OD2(Asp-l0) OD1(Asp-11) OD2(Asp-ll) OD3(Asp-16) OD2(Asp-16) CD l(Leul7) OD2(Asp-22) CD l(Leu-24) CGl(Va1-30) NHl(Arg + 33) O(Tyrc-36)

mi

12

11

0.0325 0.0325 0.0325 -0.28 -0.28 -0.28 -0.28 -0.28 0.0325 -0.28 0.0325 0.0325 -0.217 -0.28

-7.80 -0.48 -3.21 1.56 5.46 -0.91 17.32 18.15 13.95 1.50 0.41 - 10.94 11.70 3.71

-0.49 8.03 2.49 -7.16 3.01 19.66 -1.73 -3.50 2.80 10.15 7.42 0.62 8.03 -2.16

I3

-2.34 9.12 2.93 -16.2 9.08 34.54 -0.49 -3.06 3.12 3.72 12.45 -0.68 6.48 -6.83

5.45 -26.5 -10.57 31.03 -13.01 -49.07 10.87 11.11 0.64 2.20 - 1.97 -1.23 7.43 10.80

-10.63 16.64 -0.46 -7.160 4.5 15.43 5.34 6.79 -3.14 -0.41 -2.27 -8.06 -1.23 2.04

-0.88 4.08 -1.81 3.59 -3.69 -9.50 -1.20 -0.22 -3.01 -0.36 -4.58 0.10 -0.59 3.95

*Follow the convention in Table I.

TABLE 111. The Sensitivities of the Moments of Inertia With Respect to uiof Different Atom Tvr>es* ~

Atom types C NIO N+ 0-

1,

-195.6 36.7 37.5 71.5

I3

12

-165.7 21.5 9.1 40.3

38.5 29.6 22.3 10.7

50.5 -4.3 8.0 14.9

-

157.5 -6.7 5.8 2.0

-39.9 4.7 -3.0 -11.5

*Same convention as in Table I.

TABLE IV. The Largest Sensitivities of the Moments of Inertia With Respect to Partial Charges* Atom name O(Ser-3) HN(G1n-4) C(G1n-4) O(G1n-4) HE21(Gln-4) O(Tyr-7) O(Pro-13) OE2(Gl~-15) O(Arg+ 19) HH21(Arg+ 19) OH(Tyr-27)

-357.0 312.6 -153.0 293.4 - 174.0 - 114.5 -97.1 -243.3 -664.9 -86.9 125.4

13

12

11

Qi

-0.50 0.37 0.50 -0.50 0.425 -0.50 -0.50 -0.80 -0.50 0.46 -0.70

-295.8 256.3 -137.9 325.6 -409.6 144.0 138.3 -131.9 -3.6 86.3 316.8

793.2 -974.7 594.1 -1241 950.1 -525.6 -522.4 558.0 -311.2 -580.5 -824.1

-373.8 369.6 -201.5 471.9 -466.2 259.5 352.1 -436.5 -92.9 407.2 436.2

-653.9 732.8 -484.3 946.7 -701.7 10.4 115.2 0.3 -31.4 130.8 583.7

-129.8 83.1 -88.5 121.7 -190.2 -170.7 -148.1 283.5 165.6 - 195.4 158.3

*Same convention as in Table I.

As anticipated, a n increase in the magnitude of u i of carbon atoms will decrease the gyration radius while an increase in the magnitude of ui of charged oxygen atoms will expand the molecule as shown in Table 111. The data also indicate that with increasing hydrophobicity for C, the APP shrinks simultaneously in all directions, while with increasing magnitude of uifor 0-, the APP enlarges mostly in the longitudinal direction. The sensitivities of the gyration radius with respect to partial charges are plotted in Figure 4. Again atoms are colored according to the values of their sensitivities. Generally, there are two possible patterns in which atoms are colored. First, if a n

atom pair strongly interacts with one another that is influential to the gyration radius, then by increasing the magnitude of either charges, the interaction will be strengthened in the same way (the interaction energy is proportional to the product of the charges). Therefore, the moments of inertia will be equally sensitive to either charges, and both atoms will be colored the same. The second possible pattern is that one atom interacts with a group of atoms whose Coulomb interactions among themselves have an opposite effect (if they have the same or very weak effect, then the color pattern could be the same as for the pair interactions mentioned above) to the observables than their interactions with the

PARAMETRIC SENSITIVITY OF APP

225

400 200

aa.Ra’q;QC < aq,acc

0

. .

-200

-400

100

.-.

!-

I

-I I

I

I

I

I

I

I

I

I

I

I

I

I

I

150

200

250

300

350

400

50

-50 -100 300

200 100 P

aqiauz, QPN+

0

-100

-200 -300

0

50

100

Atom Number Fig. 5. The second-order sensitivities of the gyration radius (solid lines). The dashed lines give the differences between these sensitivities and those sensitivities calculated using the first 70 ps MD trajectory.

226

H. ZHANG ET AL.

individual atom. As a result, a charge increase in the individual atom may have different effect to the gyration radius (or other observables), and thus the atom may be colored darker or lighter than the atoms within the group. A picture of atoms colored according to their sensitivities as shown in Figure 4 displays vividly the distribution of the charge interactions within the APP which are important for stabilizing the conformation. The figure shows that the pair interactions seem often to be more important than the three or more body interactions. The figure also shows the hydrogen bondings within the a-helix stabilize the conformation. The hydrogen bonds between C-terminus residues Arg + 33 and Thr-32 with Leu-28 and Asn-29, and between Gly-1 and Tyr-36 are important for stabilizing the C-terminus residues. The N-terminus residue Gly-1 (and likely the polyproline strand) seems to be stabilized by interacting with the side chains of Tyr-36 and Asn-23. The h-bond between H(N) and O(C) of Tyr-7 seems to be important for stabilizing the N-terminus backbone. The details of the other important interactions can also be recognized in the figure. The sensitivity data also show that the moments of inertia are most sensitive to the dielectric constant a,and then the partial charges. They are much less sensitive to the atomic solvation parameters. This suggests that the electrostatic interactions play a more dominant role in stabilizing the APP conformation than the solvation effects modeled by the atomic solvation parameters. The moments of inertia converge much faster than their sensitivities in our study. This is understandable based on Eqs. (10) and (11)(aH/aa = 0, aHlar = 0), which show that the sensitivities are given by the differences between two similar quantities (each of which is much larger than the sensitivities in the current case, especially in the case of the moment of inertia sensitivities). The second-order sensitivities are even more slowly convergent than the first-order ones. Indeed, while the general shape of the protein is relatively stable in thermal equilibrium, some parts of the protein may have large fluctuations. Consequently, the sensitivities with respect to the parameters in the flexible parts of the protein may be converging more slowly. Table IV lists the largest sensitivities with respect to charges. Some of them display large fluctuation. The APP with atoms colored according to their statistical fluctuations can be displayed (not shown). We found that the interactions in the N-terminus between the side chains of Asn-23 and Thr-6 and O(C) of Ser-3, and between O(C) and H(N) of Gln-4, have the largest fluctuation. Because the sensitivities t o parameters in the N-terminus converge most slowly, the N-terminus may be a more mobile part of the molecule. The second-order sensitivities d2/aa,dq, (K = 1, 2, 3, 4 for C,N/O,N+,O-) show the correlations

v.----J 4.22

w"

4.2 0

80

lo0

20

40

I

I

I

I

I

I

I

I

I

15

20

60

120

140

160

I

I

I

I

I

I

I

25

30

35

40

Time of MD Simulation (ps)

z rr!

f

1.1

1 0.9

.s

g

0.8

.O

0.7

a m

5

's

0.6 0.5 0.4

I 10

0.3

0

5

Residue Number

Fig. 6. (a) The ensemble average of the rms deviation of all heavy atoms in APP from the crystal structure as a function of the MD simulation time. (b) The rms deviations of the average structure as a function of residue. (c) The rms fluctuations from the average structure as a function of residue.

(balance) between the solvation and coulomb interactions in affecting the observables. Though the second-order sensitivities have quite large fluctuations as shown in Figure 5, one can still get some qualitative information. The sensitivities involving N/O and N+ are generally smaller than those involving C and 0-. The largest second-order sensitivities are around 400. The first order sensitivities with respect to the partial charges are around 500 (see Table IV), and those with respect to the atomic solvation parameters are also of order of 200 (see Table 111).This suggests that relatively speaking, the response of the moments of inertia to the changes in the atomic solvation parameters may be modulated by the changes in the partial charges. However, the response of the moments of inertia to the partial

227

PARAMETRIC SENSITIVITY OF APP

TABLE V. The Largest Sensitivities of the rms Deviation With ResDect to u: of Individual Atom* Atom name CGZ(Thr-6) OD2(Asp-10) ODl(Asp-11) ODZ(Asp-11) OEl(G1u-15) OE2(Glu-15) CDl(Leu-17) CEl(Phe-20) ODl(Asp-22)

uI

,0325 -.28 -28 -.28 -.28 -.28

,0325 ,0325 -.28

Final value -0.0054 0.0066 -0.0059 -0.0157 -0.0048 -0.0037 -0.0045 -0.0033 -0.0033

Fluctuation 0.0005 -0.0013 0.0016 0.0049 0.0002 0.0002 0.0001 -0.0006 -0.0003

*The first column lists the sensitivity coefficient averaged over 150 ps, and the second column lists the value minus the one averaged over the first 75 ps.

charges are not as much affected by the changes in the atomic solvation parameters.

The RMS Deviation From the Crystal Structure The ensemble average of the rms deviation of the heavy atoms from the crystal structure is plotted in Figure 6a. The rms deviations of the average struc-

TABLE VI. The Sensitivities of the rms Deviation With Respect to ui of Different Atom Types* Atom types C N/O N+ 0-

Final value -0.0025 -0.0022 -0.0038 -0.0273

*Same convention as in Table V.

ture from the crystal structure and the rms fluctuations as a function of residue are plotted in Figure 6b and c, respectively. The rms fluctuation of a residue is calculated by superimposing the positions of all heavy atoms of the structures in the MD trajectory with those in the average structure and taking a n average over the trajectory for analysis. Clearly, the polyproline strand and the C-terminus have the largest rms deviations. The rms fluctuations indicate the mobility of the different parts of the APP structure during the MD simulation. Despite the large rms deviation for the C-terminus residues from the crystal structure, the C-terminus residues are quite stabilized as shown by their small rms

Fig. 7. A display of APP with atoms colored according to the sensitivities of the rms deviation heavy atoms in APP with respect to the partial charges of each atom. The other rules follow those in Figure 3. of all

Fluctuation 0.0001 -0.0012 -0.0005 0.0033

228

H. ZHANG ET AL.

TABLE VII. The Largest Sensitivities of the rms Deviation With Respect to Partial Charges* Atom name O(Ser-3) N(G1n-4) HN(G1n-4) C(G1n-4) O(G1n-4) NE2(Gln-4) HE21(Gln-4) O(Pro-13) HN(G1u-15) OE2(Gl~-15) HH21(Arg+ 19) OH(Tyr-27)

qi

-0.50 -0.57 0.37 0.50 -0.50 -0.85 0.425 -0.50 0.37 -0.80 0.46 -0.70

Final value 0.25 0.13 -0.27 0.16 -0.32 -0.16 0.24 -0.11 0.17 0.25 -0.16 -0.33

Fluctuation -0.02 -0.01 0.02 -0.01 0.03 -0.00 0.01 0.04 -0.03 -0.06 0.06 0.02

*Same convention as in Table V.

fluctuations. On the other hand, the N-terminus residues, the residues at the turn location, and the residues Phe-20, Tyr-21 are most mobile. By plotting the rms fluctuations as function of atom (data not shown), we find that the backbone atoms of Phe-20 and Tyr-21 have small fluctuation, and the flexible side chains are responsible for the large rms fluctuations. 100

I

I

I

TABLE VIII. The Sensitivities of the Free Energy With Respect to uiof Different Atom Types* Atom type C N/O N+

Final value 55.0 -7.5 -6.4

C-

Fluctuation -0.04 -0.01 0.02

-37.5

0.26

*Same convention as in Table V.

The largest sensitivities of the rms deviation with respect to cri of an individual atom are given in Table V and the sensitivities with respect to uiof different atom types are listed in Table VI. One can see that the rms deviation is most sensitive to only a small set of solvation parameters. The rms deviation is most sensitive to the atomic solvation parameter of charged oxygens. The sensitivity of the rms deviation with respect to the scaling factor of the dielectric constant a is 0.07 with a fluctuation of -0.09. The large negative fluctuation indicates the sensitivity is continuously decreasing with the MD simulation time (as seen by plotting the sensitivity with MD time, not shown). The final value for the sensiI

I

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Final Value Fluctuation

50 v)

a,

F (d

5 0 1 .

c

F

...........

0

a,

c

W a,

2

LL

c

0 v)

a, ..-c5 .c .-

-50

v)

K

a,

v)

-100

-150 0

50

100

150

200 Atom Number

250

300

Fig. 8. The sensitivity of the free energy of the APP with respect to the partial charges.

350

400

PARAMETRIC SENSITIVITY OF APP

229

Fig. 9. A display of APP with atoms colored according to the sensitivities of the free energy of the APP with respect to the partial charges of each atom. The other rules follow those in Figure 3.

tivity is much larger than those to the solvation parameters. Figure 7 shows the sensitivities of the rms deviation with respect to the partial charges of the atoms, where the atoms are colored according to their sensitivities. The plot explicitly shows two regions in the APP that present the largest sensitivities. The first region is the turn location around residue Val14; and the second is around residues Ser-3, Gln-4, and Tyr-27. The charges in these two regions are not particularly special compared with other regions of the molecule. It seems that the rms deviation is sensitive in the second region because the N-terminus is flexible and has a large deviation from the crystal structure in the absence of the stabilizing zinc ion in the MD simulation. These results suggest that the interactions among the atoms colored in this region

are important for stabilizing the structure of the N-terminus and the polyproline strand. At the first region, Val-14 is at the beginning of the a-helix, where there are large structural distortions as suggested by the large rms fluctuation a t Pro-13 and Val-14. The sensitivity results suggest that Coulomb interactions there may be responsible for stabilizing the helix and the proline strand. Table VII lists the largest sensitivities of the rms deviation with respect to the partial charges. Their values are much larger than those with respect to the atomic solvation parameters. Similar to the sensitivities of the moments of inertia, this suggests that electrostatic interactions play a more dominant role in determining the fine structure of the protein than the solvation interactions modeled by the atomic solvation parameters.

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a= aqiauc

qiac

l

Oe5

7

t

2 -

I

I

I

I

I

I

I

I

I

I

I

I

I

I

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I

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I

I

I

I

50

100

300

350

400

1 -

-4

'

0

150 200 250 Atom Number

Fig. 10. The second-order sensitivities of the free energy (solid lines). The dashed lines give the differences between these sensitivities and those sensitivities calculated using the first 75 ps MD trajectory.

23 1

PARAMETRIC SENSITIVITY OF APP

The Free Energy of the APP The sensitivities of the free energy with respect to the ui are listed in Table VIII, and the sensitivities to the partial charges are plotted in Figure 8. Figure 9 shows clearly those electrostatic interactions which are most influential to the free energy. The free energy is very sensitive to the dielectric constant a with a sensitivity equal to 1455 and a fluctuation of -2.6. To a much less extent than a, the free energy is more or less equally sensitive to the solvation parameters and partial charges. These features are different from that for the rms deviation and the moments of inertia, where the structural observables are much less sensitive to the solvation parameters. In addition, the fluctuations in the free energy sensitivities than those found in are much smaller (less than 1%) the sensitivities of the rms deviation and moments of inertia. There may be two reasons for such small fluctuations. First, from Eq. (10) one can see that if aHlaa 0, the sensitivities are calculated based on the disturbance in the Boltzmann distribution caused by a change in a force field parameter (the sensitivities are given by the difference between two similar quantities). However, if aHlaa f 0 as in Eq. (131, the sensitivities are calculated based on the ensemble average of aviaa, which may be a larger number and is not equal to zero even if the distribution is not altered, thus they are less dependent on the statistical fluctuations. Second, it seems that after 250 ps MD simulation the free energy of the molecule has been well equilibrated. However, there exist parts of the APP structure which may move around without causing large energy fluctuation. Compared with the sensitivities with respect to the moments of inertia and the rms deviation, the results also show that a force field parameter which is influential to the free energy of a molecule is not necessarily influential to a structural observable. In other words, a force field may be good for a structural study but not for energy analysis, or vice versa. The second order sensitivities d2Alau,dqi ( k = 1, 2, 3, 4 for C, N/O, N f , and 0-l are also calculated (see Fig. 10). They have slightly larger fluctuations especially for those related with uc.In general, the free energy is most sensitive to uo-, less to u, and uN+,and least sensitive to u ~related , ~second-order sensitivities. The first three groups of sensitivities have a magnitude of order of 1, which are much smaller than the first order sensitivities which are in the order of 50. This indicates that the first order sensitivities can more reliably predict the trend on how the free energy would be altered when there are finite changes in both the partial charges and the atomic solvation parameters.

+

SUMMARY In conclusion, the present paper demonstrates that sensitivity analysis can be used to study protein

conformation and stability. Similar to the early work of Kruger et al.,” the flexibility in the main chain C-terminus as derived from the B-values of the X-ray structure is not pronounced. Instead, our results show that the N-terminus, the turn structure and the side chain of the residues Phe-20 and Tyr-21 are the most flexible parts of the protein. The result may reflect the fact that no zinc ion was present to stabilize the protein structure, especially the N-terminus, in our MD simulations. The various parameters and atoms which are influential t o the APP conformation and free energy are identified. Our results also show that the electrostatic interactions play a more dominant role in the stability of protein conformation than the part of the solvation effect modeled by the atomic solvation parameters. Comparisons between the sensitivities of the structural observables and of the free energy show that a force field parameter which is influential to the free energy is not necessarily influential upon a structural observable. Therefore, a force field can be accurate enough for a structural study yet still not good for an energy analysis, or vice versa. The current work shows how atoms interact with each other to affect the properties of a protein. In rational molecular design, the sensitivity analysis of the rms deviation of a molecular structure from a certain target structure may be useful for pointing out directions for designing molecules. The sensitivity analysis of the free energy would be particularly useful in a ligand binding study where the free energy of binding may need to be improved and the sensitivity analysis results can suggest how this may be achieved. In summary, sensitivity analysis is useful for further improving force field parameters, better understanding protein functions and bioactivities, and for rational drug design.

ACKNOWLEDGMENTS We thank Professor C. Still for making his surface-area code available. We also thank the NIH, the AFOSR, and the Bristol-Myers Squibb Corp. for partial support of this research.

REFERENCES 1. Beveridge, D. L., Jorgensen, W. L. (eds).Annals of the New York Academy of Sciences, 482, 1986. 2. Weiner, S. J., Kollman, P. A., Case, D. A., Singh, U. C., Ghio, C., Alagona, G., Profeta, S., Jr., Weiner, P. A new force field for molecular mechanical simulation of nucleic acid and proteins. J. Am. Chem. SOC.106:765-784, 1984; Weiner, S. J., Kollman, P. A., Nguyen, D. T., Case, D. A. An all atom force field for simulations of proteins and nucleic acids. J. Comput. Chem. 7930-252, 1986. 3. Jorgensen, W. L., Tirado-Rives, J. The OPLS potential functions for proteins. Energy minimizations for crystals of cyclic peptides and crambin, J. Am. Chem. SOC.110: 1657-1666, 1988. 4. McCammon, J. A., Havey, S. C. “Molecular Dynamics of Proteins and Nucleic Acids.” New York: Cambridge Univ. Press, 1987. 5. Beveridge, D. L., DiCapua, F. M. Free energy via molecular simulation: Applications to chemical and biomolecular systems. Annu. Rev. Biophys. Chem. 18:431-492, 1989.

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6. Susnow, R., Schutt, C., Rabitz, H., Nachbar, R. B. Jr. Sensitivity of molecular structure to intramolecular potentials. J. Phys. Chem. 958585-8597, 1991; Study of amide structure through sensitivity analysis. J. Phys. Chem. 95: 10662-10676,1991, 7. Wong, C. F. Systematic sensitivity analyses in free energy perturbation calculations. J. Am. Chem. SOC.113:32083209, 1991; Wong, C. F., Rabitz, H. Sensitivity analysis and principal component analysis in free energy calculations. J . Phys. Chem. 959628-9630, 1991. 8. Thacher, T. S., Hagler, A. T., Rabitz, H. Application of sensitivity analysis to the establishment of intermolecular potential functions. J. Am. Chem. SOC.113:2020-2033, 1991. 9. Pearlman, D. A. Free energy derivatives: a new method for probing the convergence problem in free energy calculations, J . Comput. Chem. 15:105-123, 1994. 10. Glover, I., Haneef, I., Pitts, J., Wood, S., Moss, D., Tickle, I., Blundell, T. Conformational flexibility in a small globular hormone-X-ray analysis of avian pancreatic-polypeptide at 0.98 A resolution. Biopolymer 22:293-304, 1983. 11. Kriiger, P., Strassburger, W., Wollmer, A., van Gunsteren, W. F. A comparison of the structure and dynamics of avian pancreatic polypeptide hormone in solution and in the crystal. Eur. Biophys. J. 13:77-88, 1985.

12. Strassburger, W., Glatten, U., Wollmer, A., Fleischhauer, J., Mercola, D. A,, Blundell, T. L., Glover, I., Pitts, J. E., Tickle, I. J., and Wood, S. P. Calculated tyrosyl circular dichroism of proteins. FEBS Lett. 139:295-299, 1982. 13. McLean, L., Buck, S. H., Krstenansky, J . L. Examination of the role of the amphipathic alpha-helix in the interaction of neuropeptide-Y and active cyclic analogs with cellmembrane receptors and dimyristoylphosphatidylcholine. Biochemistry 29:2016-2022, 1990. Becksickinger, A., Gaida. W.. Schnorrenbere. G.. Lane. R.. June. F. Neuropeptide-Y-identification2the'bindiudg site. 12;J. Peptide 36522-530, 1990. 14. Hasel, W., Hendrickson, T. F., Still, W. C. A rapid approximation to the solvent accessible surface areas of atoms. Tetrahedron Comput. Methodol. 1:103-116, 1988. 15. Schiffer, C. A., Caldwell, J. W., Kollman, P. A,, and Stroud, R. M. Protein structure prediction with a combined solvation free energy-molecular mechanics force field. Mol. Simul. 10:121-149, 1993. 16. Biosym Technologies, 9685 Scranton Road, San Diego, CA 92121-2777. Phone: (619)458-9990. 17. Berendsen, H. J. C., Postma, J. P. M., van Gunsteren, W. F., DiNola, A., Haak, J . R. Molecular dynamics with coupling to a n external bath. J. Chem. Phys. 81:3684-3690, 1984.

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