2767
J. Phys. Chem. 1993, 97, 2761-2713
Brownian Dynamics Simulation of Electrooptical Transients for Complex Macrodipoles Jan Antosiewicz Department of Biophysics, Warsaw University, 02-089 Warsaw, Poland
Dietmar Porschke' Max Planck Institut fur biophysikalische Chemie, 3400 Gottingen, Germany Received: September 8, 1992
Brownian dynamics is used to simulate electrooptical transients for macromolecules with various contributions to their dipole moment, in order to get information for the characterization of complex macrodipoles. Contributions from dipole fluctuations and from time-dependent polarizabilities are analyzed for simple protein models; part of the models is constructed to simulate experimental data for a-chymotrypsin. The time constants of electrooptical rise curves are shown to beindependent of the rateofdipole fluctuations but are strongly dependent on contributions from polarizabilities. Such contributions from polarizabilities may not be detectable from stationary levels of optical anisotropies measured at different electric field strengths. The dipole moments derived from anisotropies as a function of the field strength represent mean dipoles in the case of fast dipole fluctuations and root mean square dipoles in the case, when the time constants of fluctuations are larger than the rotational time constant. For molecules of known structure, the nature of dipole moments may be assigned by comparison of experimental and calculated limiting values of the linear dichroism. For a special case of a time-dependent polarizability, the stationary values of the dichroism are shown to be dependent on the magnitude of polarization time constants. A change of polarization time constants may induce changes in the field dependence of the stationary dichroism, which are indicated by a change of the orientation mechanism upon interpretation of the data by standard orientation functions. The influence of an external electric field on the distribution of protons in the case of a-chymotrypsin is shown to be small, but changes of electric moments due to field induced proton redistribution are expected to increase with the size of the protein.
Introduction The assignment of electrical parameters to biological macromolecules is a notoriously difficult problem. This is mainly due to the fact that biopolymers usually bear large numbersof charged residues, which are strongly coupled with each other due to electrostatic interactions. Furthermore, the charge state of most residues is dependent on chemical equilibria like protonation or binding of metal ions. Because of the intricate coupling of these different effects, both the experimental characterization and the theoretical analysis of the electrical parameters is very difficult. In the case of proteins, electric moments may be partly due to an inherent anisotropy in the distribution of charged residues.'-5 Someof these residues are subject to binding reactions and, thus, the electric moment may fluctuate.6 Furthermore, the binding reactions may be affected by external electric fields, leading to a very special component of the polarizability with a time dependence determined by the rates of binding reactions. Finally, the ion atmosphere around protein molecules may also contribute due to a particularly high polari~ability.~-I I Under favorable conditions, one of these contributions may be dominant: in the case of a-chymotrypsin, for example, it has been shown that the permanent dipole moment due to an intrinsic asymmetry of the charge distribution is the main contrib~tion.~However, each protein may be different and, thus, each case has to be analyzed separately. One of the conditions for such analysis is information on the influence of the various contributions on the experimental quantities. Due to the complex coupling of these effects, analytical predictions are quite difficult or appear to be impossible. For this reason we have used the numerical procedure of Brownian dynamics simulations to characterize the coupling of some effects, which areexpected tobe important in the caseof protein molecules. Brownian Dynamics Algorithm The program used for our simulations is based on the work by Ermak and McCammon'2 and is a modified version of an
algorithm described previ~usly;'~ thus, the following discussion is limited to some essentialsof the simulation procedure. A related algorithm has been described recently by Allison.14 The program calculates temporal changes of the reduced linear dichroism of a %o1utionn containing N independent identical particles under the influence of a rectangular electric field pulse of strength E. Initially the particles are distributed uniformly among 87r2orientationalstates. Then, an electric field E is applied for a time interval t , and the trajectories of the rotational motion for each particle are calculated during some time tab. We define the particle coordinate system (PCS)corresponding to the system of principal axes of its rotatio_naldiffusion tensor 0,. The tensors of the extinction coefficients t and of the polarizability (r, and the set of possible permanent dipole moments p; due to different distributions of charges characterizing the particle are assumed to be known in this coordinate system (tensors are denoted by , elements of tensors are given without but with corresponding indices; tensors without superscript refer to the PCS). It is assumed that particles may have each possible permanent dipole moment with equal probability, instead of the probability determined by the free energy associated with a given distribution of charges (cf. below). This assumption does not limit the main conclusions derived from our simulations. We calculate the reduced linear dichroism At/t: of the solution according to the following equation A
A ,
where 2is the mean absorbance of the solution;All and Al denote the absorbances by light polarized parallel and perpendicular to the external electric field, respectively. In the presence of a homogeneous external electric field, the absorbance of the solution shows cylindrical symmetry around the field vector. When the field is directed towards the z-axis of the laboratory coordinate
QQ22-3654/93/2091-~167%04.QQ/Q 0 1993 American Chemical Society
2768 The Journal of Physical Chemistry, Vol. 97,No. I I, 1993 system (LCS)
and the reduced dichroism is given by
(3) where tr i is the sum of the diagonal elements of the tensor (which is invariant), c the molar concentration of particles, and I the optical path length. In theaboveequations ( ) denotes theaverage value for all particles in the solution. When 0 is the rotation matrix from LCS to PCS, then
=p;p where i = O;LCSOTand P,' is the unit vector of the z-axis of the LCS; P 05,' denotes the components of this unit vector and, thus, the direction of the electric field in the PCS. Because the plane of the polarized light and the field vector are parallel, it is sufficient to know the orientation of the electric field unit vector in the PCS for the calculation of the reduced dichroism contribution from a given particle. The reduced linear dichroism of the solution is calculated as the average of all individual contributions. This average is calculated as a function of time after application of the electric field; the resulting data correspond to experimental dichroism rise curves. As previously, we use the rotation vector formalisml5~'6and determine subsequent orientations of the electric field vector in the PCS after each rotation step according to
E = ELCScos x - ir x ELcssin x + (1 - cos x)(t.ELCS)ir (5) where t is the unit vector indicating the direction of the rotation axis and x the rotation angle for the transformation from the LCS to the PCS, which defines the orientation of the particle in the LCS after the given number of Brownian rotational steps; the symbols. and X denote the scalar and the vector product of vectors, respectively. The rotation angle x, for an individual step is determined by
where At is the time step, b, the rotational diffusion te_nsor, PO the torque exerted on the particle given in the PCS, and Krnd(Af) the random rotational step. The subscript 0 indicates the value at the start of the rotation step and the symbol I I the magnitude of the vector. The three independent components of random rotational steps, contributing to Krnd(At),are sampled from one dimensional Gaussian distributions with a mean value zero and astandard deviation (2D,,,,At)1/2wherei = x,y,z. Each individual rotational step with a rotation angle xs is added to the rotation angle x,which has been obtained after the previous step, according to the rules for addition of rotation vectors (cf. ref 13). By this procedure we get the values x and Ti after the last Brownian step. The torque in the presence of an external Electric field may be due to some permanent dipole moment p and/or to some polarizability a
Po = -> IiiDr,ii/kT,where I is the tensor of the moment of inertia, and yet shorf enough to ensure reasonably constant values of the torque T within the time intervals.
Technical Details of the Models Because we want to compare at least part of our results with experimental data obtained for a-chymotrypsin, essential parameters of the model used for our simulations were assigned according to this protein. The parameters were derived from the crystal structure of the monomer A of the a-chymotrypsin dimer published by Tsukada and Blow.17 The crystal structure was converted into a bead model by substitution of each amino acid by one bead; the bead radius was calculated as the average of the overall dimensions of all amino acids with an addition of 2.8 A to account for hydration. The rotational diffusion tensor and the position of the -center of diffusion" were calculated according to procedures described by Garcia de la Torre and Bloomfield,I8 with a volume correction for the calculation of the rotational resistance tensorI9 according to our procedure.*O The extinction
The Journal of Physical Chemistry, Vol. 97, No. 11, 1993 2769
Complex Macrodipoles tensor and the dipole moment resulting from the presence of ionizable groups and peptide bonds were calculated as described previously.5 The resulting parameters are as follows (values refer to the coordinate system used by Tsukada and Blowi7): The tensor of the extinction coefficients c is 42740 8936 -18803
-( c=
8936 -18803 53354 10442 10442 52706
1
in units of M-1 cm-I. The principal values of rotational diffusion tensor Drji are
D,,xx= 0.8844 X lo7; Dr,yy= 0.9177 X lo7; D,,, = 0.9899
X
lo7
in units of s-1. These values and all resulting parameters are given at 20 OC using the viscosity of water at this temperature. The rotation matrix R to the principal axes of the rotational diffusion tensor reads 0.07834
0.79041 -0.60479 -0.963 10 -0.09740
(
R = 0.25750
-0.60755 -0.75361 -0.25092
1
Thevolume correction for calculations of the rotational resistance tensor20 was 17 000 A3. The position of the 'center of diffusion" is
CD = (3.34 -1.67 in units of
47.6)
A. The dipole moment at pH 7 is = (0.87124 1.1820 0.10247)
in units of lO-2? Gem. This dipole corresponds to 441.3 D units. According to Monte Carlo calculation^,^ the mean dipole moment of a-chymotrypsin is only slightly different from its root mean square dipole moment; i.e., fluctuations of the dipole are small. Because of this small difference and because of the 'noise" associated with results obtained by Brownian dynamics simulations (at a reasonable expense of computer time), a definite decision about the effect of dipole fluctuations on experimental results might not be possible with the original a-chymotrypsin parameters. Thus, we simply amplified the magnitude of fluctuations and used an artificial distribution of dipole moments, which was generated by the following procedure: the mean dipole moment obtained from the a-chymotrypsin crystal structure was multiplied-by a factor f; the first four possible dipole moments were just f p ; the other six pssible dipole moments were obtained by rotation of the dipole fp by f45O around the x-, y - , and z-axis of the PCS. The magnitude off was selected by the condition that the resulting mean dipole moment of_the 10 possible states corresponds to the original dipole moment JL cf= 1.13). A second ensemble of dipoles was generated by the same procedure but with the factor f = 4.0, in order to test the effect of a larger mean dipole moment at constant diffusion parameters. The root mean square dipole moment for the set of 10 dipole moments generated with the factor f = 1.13 is 499.2 D. The factor f = 4.0 leads to a mean dipole of 1556.3 D and a root mean square dipole of 1762.9 D. The limiting reduced dichroism calculated for the given extinction tensor and the mean dipole direction is (Ac/c)= = 0.246; the ensemble of 10 dipole moments leads to a mean value of the reduced limiting dichroism of 0.168. Experimental values for the polarizability of proteins in general and of a-chymotrypsin in particular are not available. Thus, we have to estimate reasonable values by comparison with other available data. The main contribution to the polarizability is expected to come from some field-induced redistribution of ions around the protein. A corresponding process has been measured in the case of double helical DNA. It is likely that the
polarizability of a protein is not much larger than that of a DNA with corresponding maximal dimensions. Because the diameter of a-chymotrypsin corresponds to the length of a DNA with 14 base pairs, we take as a first approximation the polarizability p = 1.OX 10-34C m2V-I extrapolated for this DNA froma measured chain lengthdependence2I Thisvalue was used for thesimulations with the time dependent polarizability. A slightly larger value, 1.67 X C m2 V-I, was also used; the induced dipole resulting from this polarizability at 100 kV/cm is comparable with the permanent one. Thus, in our simulations over the range of electric field strengths up to 100 kV/cm, the permanent dipole moment is dominant. The main contributions to the polarizability of macromolecules are known to be time dependent, but there is rather little information on the dynamics. When the polarization is due to a redistribution of protons between theacceptor sites of a protein, the fluctuation time constant rP may be described by general expressions for proton transfer (cf. ref 22)
where k+ is the rate constant for proton association, k- that for proton dissociation, and cp the concentration of protons or of proton donors like buffers. This time constant may vary considerably depending on, e.g., the pKvalue of the binding site, its accessibility, and the buffer concentration. This expectation is confirmed by widely different values reported in the literature ranging from a few nanoseconds over -100 ns and 0.5 ps to We may assume that the main much longer contributions come from sites a t the surface with relatively high reaction rates and, thus, use time constants in the nanosecond time range. The relaxation time for the redistribution of counterions in an ion atmosphere around the protein has been estimated as23
where c is the molar concentration of salt. When the salt concentration is 10 mM, this relaxation time is about 10 ns. The movement of counterions has also been described in terms of a surface c o n d ~ c t i v i t ythen ; ~ ~ the fluctuation time constant is given by
= a2/2D (13) where a is protein radius and D the surface diffusion coefficient of the counterions. For a protein with the size of tu-chymotrypsin, the relaxation time for counterion redistribution2?is in the range of 500 ns. In summary, polarization time constants cannot be predicted exactly, but we may assume that the polarization time constant TO (valid at E = 0) is in the range from a few nanoseconds to a microsecond. The parameter K of eq 8 has been found for the case of DNAlO in the range of lO-I4 m2 V2. We use K values in the same order of magnitude from 2.3 X lO-I4 to 11.5 X m2 V-2, which lead to T E values at 100 kV/cm, which are 10-105 times smaller than T O . In most cases the field-induced decrease of the polarization time constant is larger than expected from the corresponding increase of the dipole moment. This implies a larger change of the electric moment during activation than the corresponding change associated with the reaction, which may not be realistic for standard molecular systems. Nevertheless, we have included these simulations in order to demonstrate the special effects obtained in this limit. T,,
Evaluation of the Simulated Transients As shown by Benoit,30 electrooptical rise curves for molecules with cylindrical symmetry and with a dominant permanent dipole moment are characterized by two relaxation processes with
Antosiewicz and Porschke
2770 The Journal of Physical Chemistry, Vol. 97, No. 11, 1993
TABLE I: Mean Dipole Moments ( F ) , Root Mean Square Dipole Moments ( w 2 ) 'I2,and Mean Absolute Values of Ratios of Orientational Energy to Internal Energy, r = (IAEJAEA), Sampled in the Monte Carlo Procedure as a Funchon of External'Electric Field Strength for the Models of a-Chymotrypsin, of 14 bp DNA and of 84 bp DNA' E (kV/cm) 20 40 60 80 100
0 " '
loo
'
'
'
zoo
'
'
'
360'
'
'
LOO
( r 2 ) ' / 2r 442 445 0.2 443 445 446 448
446 448 449 451
0.4 0.7 0.9 1.2
(P)
10 20 30 39 51
84bp DNA
(r2)"2 r 78 80 84 87 93
0.5 1.0 1.4 1.9 2.0
( r ) (r2)'/2r 1500 3040 4570 6080 7640
1790 3190 4670 6150 7710
2.9 3.0 3.0 2.7 2.3
See text for details; dipole moments are given in debye units.
t [nsl
field 2 and describes the maximal external interaction of the dipole with the field, corresponding to parallel alignment of dipole and field vectors. The ensemble of various protonated states are sampled by a standard Monte Carlo algorithm. Our calculations (cf. Table I) show that the mean dipole moment and also the root mean square dipole moment of a-chymotrypsin are hardly affected by external electric fields in the range up to 100 kV/cm. Thus, in the case of a-chymotrypsin, dipole moments arising from some redistribution of protons by external electric fields is negligible compared to the relatively large intrinsic permanent dipole moment. The procedure used for a-chymotrypsin is similar to the one used previously for the case of DNA double helices. In both proteins and nucleic acids there are fluctuating ligands associated with charges: for proteins the dominant species are protons, whereas cations like Na+ are more important for DNA. Thus, we have compared the magnitude of the field induced effects for these different macromolecules. As shown in Table I, the contribution due to fluctuations is larger in the case of a DNA fragment with 14 base pairs than that observed for a-chymotrypsin. The maximal dimensions of this fragment are close to those of a-chymotrypsin. Much larger contributions from fluctuations are found for the case of a longer DNA fragment with 84 basepairs. Thus, as should beexpected, thecontributions from dipole fluctuations increase strongly with increasing molecular dimensions. Some additional information on the relative contributions to the free energy is available from the fluctuations of the different energy terms. The magnitude of these fluctuations has been averaged for t_he"e$?rn,al" electrostatic free energy corresponding to the term 1~ + aEI.IEI and the "internal" free energy corresponding to the remaining two terms of eq 14. The ratio of the fluctuations for the external and the internal energy ( IhEo/AE,I) remains below 1 in the case of a-chymotrypsin for field strengths up to 80 kV/cm. Thus, contributions due to the internal energy are dominant for a-chymotrypsin, In the case of the DNA double helices the ratio (IhE~/hEil) is much larger and the overall energy in the presence of high electric fields is dominated by the external contribution. Fluctuating Permanent Dipole Moments. Electrooptical rise curves are known to be very sensitive indicators for the electric parameters of the molecules under investigation. Thus, some influence resulting from the rate of dipole fluctuations may be expected. The "rate" of the electrooptical rise, however, does not show any detectable correlation with the rate of dipole fluctuations. All the rise curves simulated for the cases without fluctuations (A), with fast (B), and with slow (C) fluctuations could be fitted at very satisfactory accuracy by a superposition of two time constants 71 and 72 with an initial slope zero (cf. Figure 1). The larger one of these time constants 72 decreases strongly with increasing electric field strength, whereas 71 is almost independent of the electric field (cf. Figure 2). In the limit of low electric field strengths, the time constants correspond to the values predicted according to Benok30 In all cases the time constants are
- 6 1 ' " ' ~200 ' ~ 300' ' ~ L"00 ' ' ' ' ' ~ 1
14bp DNA
a-chymotrypsin (P)
.
100
t [nsl
Figure 1. Rise curve simulated according to model C for 2.9 X lo6particles at 40 kV/cm and at a time step of 0.9 ns; the resulting time constants are 19.6 and 39.3 ns.
amplitudes of opposite sign and an initial slope of zero, provided that the electric field strength is sufficiently low. One of the time constants corresponds to that for overall rotational diffusion and the other one is 3 times larger. Similar rise curves with an initial slope of zero, but without restrictions on the magnitude of the time constants are expected for molecules with a "slow" polarizability.1° Thus, our simulated dichroism rise curves were fitted by a superposition of two exponentials with an initial slope (cr. Figure 1). The reduced dichroism at the stationary state of orientation was calculated from the averaged values of the last 100-150 points of the dichroism rise curves. These values were calculated for different electric field strengths and the resulting set of data was then used for the evaluation of the limiting reduced dichroism together with the permanent dipole moment or the polarizability by least-squares fitting to thecorresponding orientation function.32
Results and Discussion External Field Effect on a Fluctuating Dipole. Because of the complexity of biological macromolecules we have to expect a large number of different effects, when these molecules are subjected to external electric fields. One of the potential effects is a redistribution of protons between proton acceptor sites induced by external electric fields. The redistribution could be driven by an increase of the dipole moment resulting from fluctuations, because states with high dipole moments are favoured in the presenceof high electric fields. However, it is difficult to predict whether this effect is large enough to compensate for the increase of the internal electrostatic energy associated with states having high dipole moments. We have tested the magnitude of these effects for the case of a-chymotrypsin by Monte Carlo calculations (cf. ref 5 ) and used for this purpose the following energy terms: I
(14) The first one describes the change of free energy due to the protonation of sites using pKvalues pKi,, characteristicof isolated sites in aqueous solution; pH denotes the pH value, kTthe thermal energy, and x , an occupation variable, which is 0 for free and 1 for occupied sites. The second term Ace' denotes the internal electrostatic energy arising from Coulomb interactions between charged sites; details for the calculation of ACClby a procedure according to Tanford and K i r k ~ o o dare ~ ~given in ref 5 (cf. also ref 34 and 35). The last termjs the-maximal value of the scalar product of the dipole vector ( F + a E ) with the external electric
The Journal of Physical Chemistry, Vol. 97, No. 11, 1993 2771
Complex Macrodipoles T
A
TABLE II: Permanent Dipole Moment Reduced Limiting Dichroism (At/& from Fits of Simulated Data with an Orientation Function for a Permanent Dipole, and the Ratio R of the Sums of Squared Residuals for Fits with a Permanent and an Induced Dipole Orientation Function for the Different Models (See Text for Details)'
;i1
lns3 40 30
I
I
I
20
0
40
60
I
BO
I
100
E [kV/cm] Figure 2. Relaxation time constants 71 and 72 from dichroism rise curves at different electric field strengths: Model A 71 0, 72 0;model B 71 A, 7 2 0 ;model C 71 72 X. Estimated accuracies for the 7 values are f3 ns at 30 kV/cm and *I ns at 100 kV/cm.
+,
0.081
& E
0.06-
0.04-
0.02
1I
k
J
0.00 0
I
I
I
I
I
20
40
60
80
100
E [kV/cm]
Figure 3. Reduced linear dichroism Ac/c of the a-chymotrypsin model with electrical parameters according to model C. The continuous line represents a fit according to an orientation function with a permanent dipole moment of 491 D and AL/C = 0.170; the dashed line represents an attempt to fit the data by an induced dipole model.
equivalent for the three models, although these models are characterized by different rates of fluctuations. The rate of fluctuations is reflected, however, in the stationary levels of the linear dichroism At/€. The A€/€ values simulated a t different electric field strengths were fitted to standard orientation functions: in all cases the data could be fitted by the orientation function for permanent dipoles a t high accuracy and could not be fitted by the orientation function for induced dipoles (cf. Figure 3). The dipole moments resulting from these fits correspond to the input value for model (A), the mean dipole moment for the case of fast fluctuations (B), and the root mean square dipole moment for the case of slow fluctuations (C) (cf. Table 11). The limiting value of the reduced dichroism (At/t), corresponds to the input value for model A, to the anisotropy of theextinctioncoefficient tensor in thedirection of the mean dipole moment for the model B, and the mean value of the limiting dichroism for each of the states of the model C (cf. Table 11). For intermediate rates of dipole fluctuations, the resulting parameters should be in the range between those obtained for the limit cases. Our present results obtained for electrooptical transients are in close analogy to those obtained by ScheiderZ2for the case of dielectric relaxation. By theoretical considerations Scheider showed that the time constants of dielectric relaxation are determined by the rotational diffusion coefficient and are not influenced by the rate of dipole fluctuations. The dielectric increment, however, is clearly dependent on the rate of fluctuations
model (A) dipole averaging within one time step (B)dipole changed every time step (C) dipole constant during time of electric pulse (D) A + inst polar in directn of dipole ( D l ) p = 1 . 0 ; O < E 5 100 ( D 2 ) p = 1 . 7 ; O S E S 100 (D3)p = 1.7; 0 5 E 5 200 (E) A + relax polarn, 7,) = 200 ns; K = 6.9 X IO l 4 (F)C + relaxn polarn T" = 200 ns; K = 6.9 X 10
(D)
(Ac/c)-
R
431
0.251
0.002
434 491
0.248 0.170
0.001
404 383 449 513 (1.08)
0.373 0.010 0.435 0.062 0.345 0.459 0.231 (0.108) 13.0
0.003
559 (1.17) 0.167 (0.083) 12.8
For the models E and F we present the parameters obtained from fits by the induced dipole orientation function in parentheses, because in these cases the induce dipole fit is much better (the polarizability is in units of 10-33 C m2 V-I). and corresponds to the mean dipole moment for the case of fast fluctuations and to the root mean square dipole moment for slow fluctuations. Thus, in this respect the dipole moments obtained by dielectric and by electrooptical measurements are equivalent toeach other. However, theelectrooptical measurements provide additional information in the form of the reduced limiting dichroism (A€/€)-. The (At/€).. valuemay beuseful todistinguish between different cases, provided that it can be determined independently. It is possible to measure this parameter directly in the crystalline state; it can also be calculated, if the crystal structure is available. Obviously, assignments are more difficult, when the chromophores are mobile and/or when their average positions are not equivalent in solution and in crystals. As may beexpected, the change of the time constants, obtained from the simulated rise curves, with the electric field strength is strongly dependent on the magnitude of the dipole moment. The magnitude of this dipole moment may be estimated from slopes of this dependence or from the electric field strength, where the time constants 71 and 72 cannot be resolved anymore. Effects Resulting from Instant and "Slowly" Relaxing Polarizabilities. Although some macromolecules like a-chymotrypsin exhibit a permanent dipole moment, it is obvious that there must be a contribution from some polarizability. As discussed already above, we do not have enough information on the magnitudes and/or time constants of such polarizabilities and, thus, the simulations presented below are simple tests for potential effects resulting from various polarizability terms. These tests are partly motivated as an attempt to fit the rise time constants observed for a-chymotrypsin. A combination of the permanent moment of a-chymotrypsin with a polarizability tensor having zero component in the direction of the mean dipole and two equal contributions in the remaining perpendicular directions lead to rise curves with clear inversions of the amplitude. This phenomenon has been discussed already36 and has not been analyzed further in our present investigation, partly because it had not been observed in the case of a-chymotrypsin. Simulations with polarizability components a d in the direction of the dipole vector and zero components in perpendicular directions show rise curves which appear to be relatively close to those observed for a-chymotrypsin, although we were not able to get quantitative agreement. A comparison of time constants simulated for two different ad-values with the experimental ones (cf. Figure 4) shows the limits of this approach. It is remarkable that the stationary values of the dichroism simulated up to field strengths of 100 kV/cm are fitted with a much higher accuracy by the orientation function for permanent dipoles, in spite of the
Antosiewicz and Porschke
2772 The Journal of Physical Chemistry, Vol. 97, No. 11, 1993 1
,,
O.lO]
I
0
10-
d 0 0 0
I I
I
20
40
A
o + oo +I 60
A
+ +
A
+ + I
A + +
80
A
0 100
E [kV/cm] Figure 4. Relaxation timeconstants zI and T2 from dichroism risecurves at different electric field strengths for the models DIand D 2 ; experimental data obtained for a-chymotrypsin T I 0, T2 0;model D I T I A,T2 V;model Dz T I +, T2 X .
TABLE III: Relaxation Times 7 1 and 72 (in ns) and the Stationary Dichroism (Ad&, 5o kV/em Obtained from Fits of Dichroism Transients Simulated According to Model E (Permanent Dipole Combined with a Time-Dependent Polarizability) for Different Values of Parameters 7 0 and K (Cf. E ~ 8-10)' s I(
xi014
T E (50 kV/cm)
2.3 4.6 0.56~0 0.3270
6.9 11.5 0.18~0 0.06~0
21 35 0.041
19 40 0.042
21 35 0.042
19 40 0.042
19 40 0.042
18 44 0.045
19 45 0.045
17 46 0.045
20 43 0.045
20 43 0.045
6.3 66 0.048
7.0 72 0.049
8.1 71 0.050
9.9 68 0.050
13 59 0.049
11 50 0.044
11
57 0.045
7.4 65 0.047
7.1 69 0.048
11 65 0.049
0 TO
0.00-
I
I
1
I
1
20
40
60
80
100
E [kV/cm] Figure 5. Reduced linear dichroism Ac/t of the a-chymotrypsin model m2 V2, with electrical parameters according to model E ( K = 6.9 X 7 0 = 100 ns). The continuous line represents a fit according to an orientation function with a polarizability of 1.08 X 10-33C m2 V-I and Acm/
