Why Proteins are Big: Length Scale Effects on Equilibria and Kinetics

Abstract

Proteins are polymers, and yet the language used in describing their thermodynamics and kinetics is most often that of small molecules. Using the terminology and mathematical descriptions of small molecules impedes understanding why proteins have evolved to be big in comparison. Many properties of the proteins should be interpreted as polymer behavior, and these arise because of the longer length scale of polymer dimensions. For example, entropic rubber elasticity arises only because of polymer properties, and understanding the separation of entropic and enthalpic contributions shows that the entropic contributions mostly reside within the polymer and enthalpy originates mostly at the site of small-molecule binding. Recognizing the physical chemistry of polymers in descriptions of proteins’ structure and function can add clarity to what might otherwise appear to be confusing or even paradoxical behavior. Two of these paradoxes include, first, highly selective binding that is, nevertheless, weak, and, second, small perturbations of an enzyme that cause large changes in reaction rates. Further, for larger structures such as proteins every thermodynamic measurement depends on the length scale of the structure. One reason is that the larger molecule can control up to thousands of waters resulting in collective movements with kcal sums of single-calorie-per-molecule solvent energy changes. In addition, the nature of covalent polypeptides commonly leads to multiple binding—i.e., multivalency—and the benefits of multivalent binding can be assessed semiquantitatively drawing from understanding the chelate effect in coordination chemistry. Such approaches clarify the origins, inter alia, of many low energies of protein denaturation, which lie in the range of only a few kcal mol⁻¹, and the difficulties in finding the structures of proteins in the multiple substates postulated within complex kinetic schemes. These models involving longer length scales can be used to elucidate why such observed behavior occurs, and can provide insight and clarity where the phenomena modeled employing experimentally inseparable translational, vibrational, and rotational entropy along with charge, dipole moment, hydration, hydrogen bonding, and van der Waals energies together obscure such origins. The short-distance, long distance separation does not include explaining any enzymatic lowering of activation energies due to stabilization of the intermediate(s) along the reaction path. However, well known small-molecule methods that treat electrostatics and bonding can be used to explain the local chemistry that contributes most of the changes in enthalpy and activation enthalpy for the process.

Appendices

Appendix 1

1.1 Relations of the Energies, Force Constants, Gaussian Widths, and Entropies

For the stretched, connected springs of Fig. 3,

$$ - k_{w} \,\Delta x_{w} = F_{w} = - F_{s} = k_{s} \,\Delta x_{s} $$

(15)

where the subscript w represents properties of the weak spring on the left, and subscript s represents the properties of the strong spring on the right. F is the force, k the force constant, and Δx the distance the point of connection resides from its position at the rest point of the unconnected spring. The signs of Δx_s and Δx_w are opposite.

The internal energies involved in stretching each spring from its rest point so the bond can be formed are, respectively,

$$ U_{w} = \tfrac{1}{2}k_{w\,} \Delta x_{w}^{2} ;\quad U_{s} = \tfrac{1}{2}k_{s\,} \Delta x_{s}^{2} $$

(16)

From Eq. 15, we know that Δx_s = − (k_w/k_s)Δx_w and, substituting for Δx_s in Eq. 16, then U_s = ½ (k ²_w /k_s) Δx ²_w . Putting Δx ²_w  = 2U_s (k_s/k ²_w ) into Eq. 16 and by rearranging get

$$ U_{w} = \frac{{k_{s\,} }}{{k_{w\,} }}\,U_{s} $$

(17)

which is Eq. 2 in the main text.

The quadratic potentials of Fig. 4 show the energies versus stretch position for a reactive group attached to a polypeptide and elastically restrained by it. This group has a greater probability of being at the center, unstretched position than at the ends. The quantitative probability distribution of the possible locations of the binding groups are derived next.

With the energy U of the binding group at a location Δx from the minimum of the polymer potential U is described by

$$ U = U_{0P} + \frac{1}{2}k\left( {\Delta x} \right)^{2} $$

(18)

where k is the Hookean force constant, Δx is the distance moved from the minimum of the potential, and U_0P is the energy minimum of the polymer potential. For the remainder of this derivation, we assign the value of U_0P = 0. To simplify the appearance of the equations, from now on, x is used in place of Δx.

The normalized probability distribution of the population over the parabola is

$$ \rho (x) = \left( {\sqrt {2\pi k_{B} T/k} } \right)^{ - 1} \exp \left( { - \frac{{kx^{2} }}{{2k_{B} T}}} \right) $$

(19)

where k_B is the Boltzmann constant, and k the force constant. This is a Gaussian centered at the energy minimum. This Gaussian distribution can be characterized by its standard deviation. By comparing Eq. 19 to the general equation for a normalized Gaussian distribution

$$ f(x) = \frac{1}{{\sigma \sqrt {2\pi } }}\,\,\exp \left( { - \frac{{x^{2} }}{{2\sigma^{2} }}} \right) $$

(20)

shows that the Gaussian’s characterizing width

$$ \sigma = \sqrt {\frac{{k_{B} T}}{k}} $$

(21)

Applying Eq. 21 for both springs of Fig. 4 at the same k_BT, we can find the three equalities of Eq. 22

$$ \sigma_{w}^{2} k_{w} = k_{B} T = \sigma_{s}^{2} k_{s} ;\quad \sqrt {\frac{{k_{s} }}{{k_{w} }}} = \frac{{\sigma_{w} }}{{\sigma_{s} }};\quad \frac{{k_{s} }}{{k_{w} }} = \frac{{\sigma_{w}^{2} }}{{\sigma_{s}^{2} }} $$

(22)

The σ values are a measure of 1-D displacement, so ΔS = − R ln (σ_w/σ_s) expresses the entropy change between the site before the bond is formed and when bonded. In addition, Eqs. 21 and 22 show that if the force constant changes, so does the width. So the protein’s structural entropy can be changed by stiffening the tether. Possible molecular mechanisms to do so include structuring the chain by intrachain hydrogen bonding or forming intramolecular bonds or bonding with adjacent molecules to shorten or strengthen the polymer spring.

In three dimensions, this change in σ transforms to a difference in accessible volume. For a reaction of bond formation, let the smaller, bonded volume accessible be V₁, and the initial, larger unbonded accessible volume be V₂. Then,

$$ \Delta S = R\,\;\ln \frac{{V_{2} }}{{V_{1} }} $$

which appears as Eqs. 3 and 10a and 10b. This result only holds for the quadratic approximation for a continuum system for each binding group [83] and the approximation of homogeneous distributions within the volumes. As the attachment becomes looser, such as on a “long” unstructured polypeptide, the gaussian distribution may need to be explicitly included.

Appendix 2

2.1 Another Experimental Example of Fast Bond Formation

Another experimental example for bond formation times (compared to relatively slow mass migration) is for bonds between gold atoms, where the preorganization is fulfilled by the proximity of weak clusters of Au(CN) ⁻₂ in water. Kim et al. [201] used femtosecond x-ray scattering to follow photoactivated bond formation between the gold atoms that reside at van der Wall distances within these Au(CN) ⁻₂ clusters. The bond formation occurred within about a picosecond over which time three adjacent golds having distances of 3.9 Å and 3.3 Å from the central gold and forming a 101° angle transformed to a linear structure with both bond distances 2.8 Å. (This bond-forming process is surprisingly fast when we realize the characteristic diffusion distance over that time at ambient temperature is around 0.6 Å.) This measurement of the bond-forming supports the fraction-of-an-Angstrom motion of bond formation/breaking in solution.

Appendix 3

3.1 More About Separating Entropy and Enthalpy and the Importance of k _B T

At the end of Sect. 2.3, it was noted that molecular structures can be altered by energies less than k_BT—an entropic mechanism—or altered by energies greater than k_BT that contribute enthalpy. For the quantitative support of that description, I will be switching back and forth between molecular lengthscale descriptions and descriptions using terms that ordinarily only apply to macroscopic systems. When descriptions involve molecular characteristics such as vibrational modes, the description should be interpreted as being that of a single molecule, which resides in a form that is an average of the ensemble from which the molecule is taken. Similarly, since a single molecule is assumed to represent the ensemble average, entropy will be used for single-molecule differences that are due to entropy on the macro scale. In addition, when thermodynamic terms are used on the molecular scale, enthalpy has no clear meaning since pressure-volume (PV) work is part of the enthalpy. Nevertheless since free energy involves both the enthalpy and entropy, let us use enthalpy as the term and ignore the difference between internal energy and enthalpy.

To clarify the nature of entropy, consider a protein residing in its equilibrium form that exhibits many low energy vibrations [202] (i.e., < 200 cm⁻¹ at 25 °C). Now add some energy within a few ps [42, 101, 203]. The added energy first partitions among these same vibrational modes and eventually also migrates and adds to the solvent’s vibrations, rotations, and translations that define the temperature. The added energy is eventually distributed in a volume so large relative to the protein that the temperature can be considered unchanged. This added energy cannot be recovered because no temperature difference remains which would be able to drive some energy back to, e.g., change the molecule’s structure. The energy distributed to these low-energy modes of the protein and of the water is, then, the entropic part; the entropy is the energy “dispersed” or “dissipated,” [204] which are useful terms to describe the progression of events described in this paragraph.

On the other hand, if the work going into the molecule is distributed into modes that are not occupied at k_BT (say, arbitrarily, 10 k_BT) or changes the Boltzmann occupancy of some modes at lower energies but still greater than k_BT, then the energy put in can be recovered by removal from the same modes, which drops the molecular energy to lower states. This energy can be recovered since the temperatures of those modes—as characterized by the Boltzmann distribution—are greater than the surroundings, which allows the energy to flow back. This recoverable energy is the enthalpic part.

While energy added to the protein can flow into modes that can be characterized as either entropic or enthalpic, the delineation between the two is not sharp [189]. The reason the boundary remains indistinct is that while the ambient temperature is the dividing line, the Boltzmann distribution indicates that energy states greater than k_BT are also occupied at the system temperature. (As mentioned earlier, a state with energy k_BT above ground is 37% as populated as the ground state, and one at 2k_BT is 14% as populated.) As a result, some normally unrecoverable energy (when ΔT = 0) can be recoverable by removing energy from the partially occupied higher-energy states. This blurring in energy may be reasonably assumed to be about 3 k_BT wide. If no states exist in that range, the enthalpy-entropy separation is sharp. For a protein in solution, this sharp separation is not possible.

One reason for this impossibility is that the longer the length scale of a vibration, the lower the vibrational energy levels that are available in the (molecule + solvent). This relationship can be seen in column 3 of Table 1. The lower energy vibrations are associated with larger groups of bonded atoms: examples are librations of groups of four atoms and vibrations that occur over so many atoms they are classified as acoustic modes [205,206,207]. This lengthscale-frequency relationship brings us full circle to the differences between distorting alkane chains being enthalpic (propane) and entropic (polyethylene) as discussed in Sect. 2.2. The long chain has low energy vibrational modes that can be excited by the random thermal motions of the surroundings. The short chain can have only the methyl rotations excited this way. The indistinct boundary between enthalpy and entropy is confirmed since chains with lengths greater than propane decrease the enthalpic contribution, and shortening the polyethylene eventually increases the enthalpic contribution until reaching some length of chain that has both mechanisms contributing equally to the restoring force. The boundary of enthalpic and entropic contributions is indistinct.