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J. Phys. Chem. B 1998, 102, 7506-7509
COMMENTS Distribution of Temperature in Globular Molecules, Cells, or Droplets in Temperature-Jump, Sound Velocity, and Pulsed LASER Experiments Bengt No¨lting Prussian PriVate Institute of Technology at Berlin, Go¨ rschstrasse 40, D-13187 Berlin, Germany ReceiVed: October 29, 1997; In Final Form: April 22, 1998 Introduction A wide range of practically important heat diffusion phenomena may approximately be modeled by the scenario of the heat exchange of homogeneous, spherical bodies that are heated in a constant energy flow and are in heat exchange with a large reservoir of solvent: This situation may occur in LASER spectroscopic experiments1-3 (Figure 1) where the knowledge about local heating effects is important and in rapid kinetic measurements using temperature-jumping (T-jumping)4-6 where it is important to know how fast the molecules reach the desired temperature. In particular, there is a strong current interest in the extension of the time scale of protein folding studies using T-jumping.7-11 The high structural and kinetic resolution of the early, fast folding events is expected to contribute significantly to the unraveling of the folding paradox,12-14 i.e., the question how in the course of the folding reaction the gigantic number of unfolded conformations can so rapidly be funneled into the unique native conformation. The maximum time resolution of the new method involving T-jumping of protein, starting from the cold-denatured state,7-11 is essentially given by the speed of temperature equilibration in the sample. An analytical solution of this heat diffusion problem has been given and applied on sound velocity measurements.15 Since the sound wave represents a pressure wave, a small temperature change of the protein solution will be involved.15,16 The theory predicts that under the usual experimental conditions there is a significant heat exchange between the interior of the molecule, its solvent shell, and the bulk solvent.15 Qualitatively, this has been pointed out earlier by Kharakoz and Sarvazyan, who called the estimated compressibility of the interior of the protein molecule pseudoadiabatic.16 Since usually there is only a very small difference between adiabatic and isothermal compressibility of proteins,16 this prediction could not be confirmed experimentally yet. Here it is shown how the same theory can be applied on a wide range of important experiments, for example, in which intense LASER pulses or rapid electrical discharges are used. The limits given by the analytical solution can now be compared with direct experimental evidence,7,8,17 showing a good agreement. It is concluded that kinetic folding studies which use T-jumping may safely be extended to the time scale below 100 ps.
Figure 1. Pulsed LASER experiments on spherical bodies, e.g., molecules, cell organelles, cells, micelles, droplets, etc. The sample is excited by an intense LASER beam. Chemical or physical changes may be monitored with a second beam (not shown). Question: what is the temperature change in the sample?
Temperature Distribution in the Sample The temperature change relative to the bulk solvent, ∆T, as function of time, t, and radius, r, of an approximately homogeneous, spherical body that is heated in a constant energy flow is given by eq 1 (see eq 18 in ref 15):
∆T(rr0,t) ) θ(I+ - I- + J+ - J-)
(1b)
I( ) (1 + w(2)Φ(w() + (2π)-0.5w( exp(-0.5w(2) - w(2 (1c) J( ) 2ξ (-w(2Φ(w() + (1 - w(2)(2π)-0.5w(-1 exp(-0.5w(2) + w(2)w(r0/(3r) (1d) w( ) |1 ( r/r0|/ξ
(1e)
ξ ) (2Λt/FCr02)1/2
(1f)
where Λ, F, C, and r0 are the heat conductivity, density, specific heat capacity, and radius of the body, respectively, and Φ is the Gauss function. θ is the increase in temperature that would occur in the absence of thermal diffusion. Equation 1 is valid for an approximately constant product ΛF-1C-1 within the sample and no significant heat radiation, heat convection, or shock waves which may occur for very high ∆T. It may be applied on a variety of objects and sources of heating, such as pressure-induced heating, but also heating caused by light absorption or an electrical discharge. Temperature profiles according to eq 1 for different values of the parameter ξ are shown in Figure 2. As one can see, for ξ , 1 there is little heat exchange with the bulk solvent, and for ξ . 1 there is significant exchange with the solvent. Results Sound Velocity and Compressibility Measurements. Sound velocity measurements are an important tool for the determination of compressibility,18-21 solvation,16,22-24 and conformational relaxation25-30 of protein molecules. Adiabatic com-
S1089-5647(98)00020-0 CCC: $15.00 © 1998 American Chemical Society Published on Web 08/19/1998
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J. Phys. Chem. B, Vol. 102, No. 38, 1998 7507
Figure 2. Temperature profiles according to eq 1 for different values of the parameter ξ ) (2Λt/(FCr02))0.5, where Λ, F, and C are the heat conductivity, density, and specific heat capacity of the molecule, respectively, and t is the time, are shown. The temperature difference relative to the bulk solvent is ∆T; θ is the increase in temperature that would occur in the absence of thermal diffusion; r and r0 are the radius under consideration and the radius of the spherical body, respectively.
pressibility, β, density, F, and sound velocity, u, of a solution are related to each other by the Laplace equation:31
β-1 ) Fu2
(2)
The picture of the physical properties of the protein molecule that emerges from recent progress is as follows:16 1. The interior of protein molecules has an average compressibility of about (14 ( 3) × 10-11 Pa-1, which is 3 times less than that of water and comparable to that of soft organic solids. 2. The molecule is surrounded by a hard solvent shell which comprises several hundred water molecules that are significantly changed in their physical and chemical properties, in particular hydrogen bonding and structure, relative to bulk water. The average compressibility of the solvent shell is 35 × 10-11 Pa-1, 20% less than that of bulk water. The parameter ξ in eq 1 increases with time and decreases with the size of the molecule. At the beginning of the compression or expansion half-cycle, corresponding to a small ξ, the pressure-induced temperature change of the protein molecule is proportional to the time and as large as in the absence of thermal diffusion, i.e., ∆T ) θ (Figure 2). Since the thermodynamic parameters of protein differ from that of water, a temperature difference between the interior of the protein molecule and the water will build up. For large ξ, corresponding to a small radius of the molecule and a large time, the further growth of the temperature difference, ∆T, of the interior of the protein molecule relative to the solvent is significantly slowed by heat diffusion, i.e., ∆T , θ. For the usual conditions of sound velocity measurement of proteins, r0 < 10 nm, C ≈ 1300 J kg-1 K-1, F ≈ 1400 kg m-3, Λ ≈ 0.17 J m-1 s-1 K-1, and sound frequencies of less than 100 MHz, one finds ξ . 1 and thus significant heat exchange of the molecule with bulk solvent. Because a system is adiabatic when no heat exchange occurs between the system and its surrounding,32 usually the protein molecule is not under adiabatic conditions,1 even though the whole protein solution is still under adiabatic conditions, unless there is significant heat exchange of the solution with the walls of the cell. Kharakoz and Sarvazyan pointed out that their
Figure 3. Snapshot of a sound wave passing through a sample cell that contains a protein molecule. The temperature profile of the sample is shown with a solid line. The dashed line represents the profile that would be obtained if heat transfer is neglected. Three types of heat exchange that may cause a deviation from adiabatic conditions of the interior of the protein molecule are illustrated: (1) At the usually applied low and medium frequencies significant heat transfer occurs between the interior of the protein molecule, its solvent shell, and the solvent. (2) At unusually low frequencies there may be a significant heat transfer between protein solution and the walls of the sample cell. (3) Significant heat transfer between the half-wave of compression and the half-wave of expansion is only possible at exceedingly high frequencies.
estimate of the intrinsic compressibility is a pseudoadiabatic value.16 For a two-component system of solvent and solute, the effect of thermal diffusion on the compressibility may be accounted for in a thermal scattering theory which has been validated for bodies as small as 130 nm.33 For small proteins it may be necessary to consider as a third component the solvent shell which has a relatively larger contribution to the overall sound velocity and compressibility than typically for much larger bodies. Pinfield and Povey mentioned a further possible source for a deviation from adiabatic conditions: At high sound frequencies there may be a heat transfer from the half-cycle of expansion to the half-cycle of compression of the sound wave.33,34 This is because at very short sound wavelengths, heat diffusion can efficiently cross the small distance between half-wave of expansion and half-wave of compression. This effect occurs because the wavelength of sound changes linearly with the reciprocal of frequency, 1/f, but the thermal decay length, l, changes with the square root of 1/f:33
l ) (Λ/(πFCf))0.5
(3)
where Λ, F, and C are the heat conductivity, density, and specific heat capacity, respectively. However, under the usual conditions of compressibility measurement this mechanism does not apply.33 In water at 20 °C this mechanism would cause a deviation from adiabatic conditions at a frequency of above roughly 1013 Hz.33 The different causes for a departure from adiabatic conditions in compressibility measurements via sound velocity are illustrated in Figure 3 which represents a snapshot of a cross section through the sample cell. Only the heat exchange between protein and solvent appears to be significant under the usual conditions of measurement. It should be pointed out that for most proteins isothermal and adiabatic compressibility differ only slightly, and there appears to be no significant effect on former conclusions. Estimates of the compressibility of the interior of the protein molecule from sound velocity data16 are in good agreement with those from mechanic measurements of protein crystals35 or from the packing density in crystals.36 LASER Spectroscopic Experiments. Equation 1 applies to experiments in which an intense LASER beam is used to provide an optical trigger of chemical or physical changes3,37-42 in a suspension of approximately homogeneous, spherical bodies that contain a light-absorbing dye in case of no strong extinction
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Figure 4. In temperature-jump experiments, rapid protein folding under essentially isothermal conditions may be studied in the time scale from about 20 ps to seconds. The folding pathway of barstar, the inhibitor of the ribonuclease barnase, is shown. The ribbon represents the part of the structure that is significantly consolidated. The early events of folding of barstar are characterized by a nucleation in and around helix1 in about 500 µs. Then further structure grows step by step around the nucleus.
over the length of the single absorbing bodies. In each case one needs to calculate the parameter θ, i.e., the temperature which would occur in the absence of any heat diffusion, now given by eq 4:
θ ) 2.3Ioct/(FC)
(4)
where Io, , and c are the light intensity (in W m-2), decimal molar extinction coefficient of the dye, and concentration of the dye, respectively. For short light pulses, i.e., ξ , 1, there is no significant heat exchange with the environment and ∆T(r
