Urry D.W. (Ed.) What Sustains Life? : Consilient Mechanisms for Protein-Based Machines and Materials

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5.8 Integration of Cooperativities Due to Apolar-Polar and Charge-Charge Repulsion

195

too much hydrophobic hydration, at a given

temperature above Tj, causes the domains

immediately to reassociate. Should an ionizable

group, say, a carboxyl (-COOH), within the dis-

sociating hydrophobic domain ionize to form

the carboxylate (-COO"), it must obtain its

required hydration by destructuring nascent

hydrophobic hydration. The result is an in-

crease in free energy of the charged hydropho-

bic domain. Now there are two possibilities.

Either there is sufficient proton in solution

that the carboxylate recovers its proton, or a

second carboxylate forms and cooperates

with the first in destroying hydrophobic hydra-

tion such that the free energy of each of the

two carboxylates is less than that of the lone

carboxylate. Now formation of each new

carboxylate does so with a lower free energy.

Accordingly, as shown in Figure 5.30, the first

carboxylate forms with a higher pKa, and each

subsequent carboxyl ionizes with a lower pKa.

The last carboxyl to form carboxylate com-

pletes the positive cooperativity for the model

protein.

5.7.9.5 Basis for the pKa Shift Remaining

After Complete Hydrophobic Dissociation

When considering the residual pKa shifts for

the model protein series above, we need to be

clear that even Model Protein i with no pheny-

lalanine (Phe, F) residues exhibits a mean pKa

shift to 4.5 so that the residual shift due solely

to the three Phe (F) residues of Model Protein

iii is only 0.3 units or 0.43 kcal/mole-Glu. On the

other hand, the total AGap for Model Protein v

with a ApKa of 2.87 would be

4.1

kcal/mole-Glu

when at a temperature sufficiently above the Tf

value. Another relevant point is that in Figure

5.25B a residual hydrophobic hydration

remains for Model Proteins i and ii with less

than two carboxylates per 100 residues, which

residual hydrophobic hydration remains as

ionization continues to 3.4 carboxylates per

100 residues. The understanding is that the

hydrophobic residues tend to arrange on the

backside of the model protein from the car-

boxylate groups and are thereby shielded by

the polymer chain from the otherwise over-

powering carboxylates. Thus derives the image

of an apolar-polar repulsion. It should be

emphasized that the carboxylates, even though

immersed in water, do not achieve full hydra-

tion when their pKa is greater than 4.

Accordingly, there results a residual pKa shift,

after complete hydrophobic dissociation, as the

result of the sequence-imposed proximity of the

charged and hydrophobic side chains.

5.8 Integration of

Cooperativities Due to

Apolar-Polar and

Charge-Charge Repulsion into

Acid-Base Titration Theory

5.8.1 Hydrophobically Associated and

Hydrophobically Dissociated: Two

States for Diverse Allostery

Over the last century, one of the more enig-

matic, yet fundamental properties exhibited by

protein-based machines of biology has been

positive cooperativity. So impressed was Monod

by this phenomena that he is reported to have

considered positive cooperativity "the second

secret of life," with the first secret being the

structure of DNA.^^"^ The initial example goes

back to the first decade of the twentieth century,

the binding of oxygen by hemoglobin.^^'^^^

Many remarkable researchers have made

exceptional contributions over many years—

Wyman,^^ Adair,^^^ of course Perutz,^^"^ and then

in the 1960s Monod and coworkers,^^^'^^^ as well

as Koshland, Nemethy, and Filmer.^^^

By means of mathematical formalism

steeped in symmetry of multisubunit globular

proteins, Monod and coworkers described the

cooperative binding of oxygen by hemoglobin,

which effort warranted Nobel recognition.^^^ In

the general perspective, each of the identical or

near identical globular subunits could exist in

two different states of order, hence Monod's

use of the term allostery. The physical basis for

the process was not specifically addressed,

however, other than to credit a "conformational

change." In fact, the place wherein we have

found a mechanism had been discounted, with

196

Consilient Mechanisms for Diverse Protein-based Machines

the perspective that "One may set aside the

simple problem of fibrous proteins. Being used

as scaffolding, shrouds and halyards, they fulfill

these requirements by adopting relatively

simple types of translational symmetries."^^^

Thus,

what we describe in this volume, using

model proteins of inherent translational sym-

metry, had not been anticipated.

Another element, taught by the consilient

mechanism that stands out as not having been

anticipated prior to the current work but con-

stitutes the scientific core of this volume, is

contained in the words of Gregorio Weber in

one of the currently outstanding treatments of

protein interactions, "A complete description of

the energetics of hemoglobin, or any other

oligomeric protein, is well-nigh impossible.

It would involve not only the determination

of the energetic couplings of any number of

Ugands with each other and with the subunit

interactions but also the variations of these

quantities with pH, temperature and pres-

sure."^^^

It is here that the consiHent mecha-

nism, a name for chronicling the comprehensive

hydrophobic effect, has something to con-

tribute. Not only the variables of chemical

potential (e.g., ligand binding and pH), temper-

ature, and pressure, but also the variables of

applied potential, mechanical force, and elec-

tromagnetic frequencies all sum (+ and -) to

give the resultant change in Gibbs free energy

for hydrophobic association/dissociation,

2|AG'HA, as depicted in Figure 5.33. This

constitutes a statement for a

AGHA

additivity

principle.

Here, by introducing AGHA, which by com-

paring ionized and neutral states, can be equiv-

alent to AGap(=2.3 RT ApKa) in the absence of

charge-charge repulsion, we bring positive

cooperativity into the formahsm for acid-base

titrations.

As appUed to our model proteins, this

represents a problem in allostery quite equiva-

lent to that of hemoglobin. In each repeating

unit of hemoglobin there are two states—

oxygen bound and oxygen free. For our model

proteins with an ionizable functional group in

each repeat, there are ionized and nonibnized

states.

In the case of hemoglobin, the principle

variable is the fraction of ligand bound Y, and

it enters in the form Y/(l - Y). This is equiva-

lent in the acid-base titration theory to the

degree of ionization, a, and it enters as a/(l -

a).

For another example, for model proteins

containing redox couples, there is the degree of

reduction, a', and it enters into the formalism

as a7(l - a'). Thus, the following formalism is

relevant, independent of whether one treats

ligand binding to allosteric proteins, the ioniza-

tion process in an acid-base titration curve, or

the reduction process in a potentiometric titra-

tion. In our view of positive cooperativity,

whether in the model protein-based machines

of our design or in the protein machines of

biology, in all cases two states—the hydropho-

bically associated state and the hydrophobically

dissociated state—provide the common

allosteric denominator with the fundamental

fraction being the degree of dissociated and

associated states.

5.8.2 The Henderson-Hasselbalch

Equation for Dilute Weak Acids

For an isolated polypeptide containing one ion-

izable residue such as an aspartic or glutamic

acid residue without significant hydrophobicity

in the remaining residues, the Henderson-

Hasselbalch equation applies. The derivation

begins with the statement of an equilibrium con-

stant,

Ko,

that describes the relationship between

two states, in our case -COOH and -COO".

Ko =[COO-][H^]/[COOH]:

g-AGo/RT _ j^Q-AGo/2.3RT

(5.14)

where AGo is the change in Gibbs free energy

on going from reactants and products. By

taking the logarithm to the base 10 and intro-

ducing the definitions that pH = -log[H^] and

pKo = -logKo, Equation (5.14) becomes

pH = pKo + log{[COO-]/[COOH]} (5.15)

With the definitions a = [COO-]/{[COOH] -H

[COO"]} for the fraction of ionized species and

(1 - a) = [COOH]/{[COOH] + [COO"]} for the

fraction of nonionized species, substitution

into Equation (5.15) results in the well-known

Henderson-Hasselbalch equation, that is,

pH = pKo + log[a/(l - a)] (5.16)

5.8 Integration of Cooperativities Due to Apolar-Polar and Charge-Charge Repulsion

197

FIGURE

5.33.

Representation of low- Associated

ering the onset temperature, Tt(b),

of the inverse temperature (phase)

transition for hydrophobic associa-

tion from the dissociated state at

37°C to 25°C, that is, to a tempera-

ture just sufficient to achieve

essentially complete hydrophobic

association. Hydrophobic association

results from the summation of all of Dissociated -|

the variables (+ and -) that con-

tribute to AGHA, i.e., SJAG'HA. This

summation represents the AG/^

additivity

principle.

Transition Zone

25°C 37°C

1, >

SiAG^HA

5.8.3 Comparison of the Henderson-

Hasselbalch Equation to Titration

Curves of Polyelectrolytes and

Model Proteins

Experience shows marked deviation from the

Henderson-Hasselbalch equation. There are

examples of acid-base titration curves with

curves broader than predicted by the Hender-

son-Hasselbalch equation, an effect called neg-

ative cooperativity, and there are examples of

acid-base titration curves with curves sharper

than predicted by the Henderson-Hasselbalch

equation, an effect called positive cooperativity.

These deviations from Equation (5.16) can

be demonstrated by cooperative interactions

involving the charged functional group.

5.8J.1 Negative Cooperativity of

Poly (methacrylie acid) and

Charge-Charge Repulsion

When the Gibbs free energy increases as each

subsequent charged forms, negative coopera-

tivity occurs as shown for polymethacrylic acid

in Figure 5.35A (see page 209).^^^ With poly-

methacrylic acid there occurs a carboxyl

function on every other backbone atom. As

carboxylates form to the extent of greater than

50%

ionized, charge-charge repulsion becomes

very significant. Even in water with its high bulk

dielectric constant, there is insufficient space

between negative charges to pack in water mol-

ecules with their large dipole moment to shield

between charges. The broadened curve of

Figure 5.35A results, where the experimental

data are compared with the prediction in

Equation (5.16), given as the solid line for the

Henderson-Hasselbalch equation. As discussed

in the next section, the chemical energy

required to go from the COOH state to the

COO"

state is proportional to the width of the

curve in pH units and also to the shift in pK.

5,8.3.2 Positive Cooperativity of Model

Proteins and Apolar-Polar Repulsion

Positive cooperativity results when a repulsive

Gibbs free energy exists on formation of the

first

COO",

a repulsion that the presence of the

first

COO"

relieves to some extent for the for-

mation of subsequent carboxylates. Then the

second and subsequent carboxylates form with

a steadily decreasing repulsive free energy. The

result is a steepened and narrower acid-base

titration curve. The titration curves of two

model proteins in Figure 5.35B (page 209) are

much steeper and narrower than the curve of

Equation (5.16). Importantly, the curves

become progressively steeper as the hydropho-

bicity of the model protein becomes greater.

The result, in fact, is that less of a change in

chemical energy is required to go from the

COOH state to the COO" state for the model

proteins than for the idealized dilute weak acid

case of the Henderson-Hasselbalch deriva-

tion.^^"^ Furthermore, the amount of chemical

energy required decreases as more pheny-

lalanine (Phe, F) residues, with the more-

198

Consilient Mechanisms for Diverse Protein-based Machines

hydrophobic R-group -CH2-C5H6, replace less-

hydrophobic valine (Val, V) residues, v^ith the

R-group -CH2(CH3)2.

The series of Model Proteins i, ii, iii, iv, and

in Table 5.5 with 0,2,3,4, and 5 F residues per

30-mer exhibits a systematic nonlinear increase

in steepness, that is, in positive cooperativity,

and an associated nonlinear increased pKa

shift, as plotted in Figure 5.34. The energy

required to convert from the COOH state to

the COO" state systematically in a supralinear

way becomes less and less, as more Phe residues

replace Val residues. The energy required to

convert from the hydrophobically dissociated

state of COO" to the hydrophobically associ-

ated (contracted) state of COOH becomes less,

as the model protein becomes more hydro-

phobic. The elastic-contractile protein-based

machine becomes more efficient as it becomes

more hydrophobic. The cooperativity of Model

Protein iv with a Hill coefficient of 2.6 is similar

to that of hemoglobin with a Hill coefficient of

2.8.

Remarkably, the Hill coefficient of 8 exhib-

ited by Model Protein v exceeds that of most

known allosteric proteins.

The formalism for introducing negative and

positive cooperativity into the Henderson-

Hasselbalch equation. Equation (5.16), follows.

5.8.4 Introduction of Cooperativity

into the Henderson-Hasselbalch

Equation for Polymers of

Linear Repeats

The formalisms for describing the electrostatic

interactions between charged groups follow

those of Harris and Rice,^^^ Overbeek,^^^ and

Katchalsky,^^^ and the repulsive apolar-polar

interactions are introduced in an analogous

way, but with a distinguishing feature between

the two. Interestingly, the pKa shifts for

charge-charge repulsion and apolar-polar

Hydrophobic-induced pK Shifts and Positive Cooperativity

1.00

0.75

0.50

0.25

0.00

pKa

Polymer I: 5.9

Polymer i: 4.5

Polymer ii: 4.8

Polymer iii: 5.2

Polymer iv: 5.6

Polymer v: 6.4

—i—•"—I—'—I—'—T"

12 3 4 5 6 7

—I—

2.7

1.47

1.6

1.9

2.6

8.0

—1—•—I—'•

11 12 13

FIGURE

5.34. Acid-base titration curves of the series

of elastic Model Proteins I and i through v of Table

5.5 that exhibit systematic increases in hydrophobic-

induced pK shifts and positive cooperativity result-

ing from competition for hydration between apolar

and polar groups. (Inset) Slope of the Henderson-

Hasselbalch equation with n = 1, and the slopes for

the Hill coefficients, n, of the positively cooperative

Model proteins

ii,

iii,

iv,

and

respectively. It may

be noted for comparison that the negatively cooper-

ative poly(methacrylic acid) (PMA) has a

slope,

0.5.

The respective pKa and n values for the model

proteins are listed within the figure. See text for

further discussion.

5.8 Integration of Cooperativities Due to Apolar-Polar and Charge-Charge Repulsion

199

repulsion are additive. The cooperativities, on

the other hand, are of opposite sign and can

cancel.

Thus,

when the free energy of interaction

represented by the pK shift and the free energy

of interaction represented by the change in

cooperativity are the same, as is the case in

Figure 5.29, the apolar-polar repulsive free

energy for hydration dominates, that is, there is

no significant charge-charge repulsion.

5.8.4.1 Multiple Acid-base Equilibria in

Protein-based Polymers with Widely

Sequence Displaced Functional Groups

For the set Model Proteins i through v in Table

5.5,

one glutamic acid residue (Glu, E) repeats

every 30 residues. This means that there are 95

backbone and side chain atoms between the

ionizable functions, and for Model Protein i

there are 36 such repeats. There then exist 36

equiUbrium constants, K; through Kj^, one for

the ionization of each carboxyl function. An

overall equiUbrium constant can be written,

K^^, which is the product of all 36. This can be

written in general as

[(COO-)J[HlV[(COOH)J (5.17)

Taking the logarithm to the base 10,

nlogK-nlog[H^] +

log{[(COO-)J/[(COOH)J} (5.18)

and again defining pH = -log[H^], pK = -logK,

dividing by n, and rearranging gives

pH = pK + (l/n)log{[(COO-)J/[(COOH)J}

(5.19)

5.8.4.2 Introduction of Effect of

Substantial Hydrophobic Hydration

As shown in Figure 5.34, in the interactions

among carboxylates while under the influence

of increasing numbers of hydrophobic residues,

pK ^ pKo, but rather pK = pKo + ApK, that is,

the pKa shifts of section 5.7.8. Furthermore, it

is assumed that when one chain starts to ionize,

due to the positive cooperativity, it completely

ionizes before the next chain starts such that, to

an adequate approximation, the experimentally

measurable degree of ionization, a ~

[(COO-)J/[(COOH)„] and n is replaced by a

phenomenological n to correspond with this

assumption. Equation (5.19) can now be

restated as,

pH = pKo + ApK + (l/n) log[a/(l - a)] (5.20)

Equation (5.20) retains the form of the

Henderson-Hasselbalch equation, but it has

been generalized to introduce pK shifts and

cooperativity into the expression. This pheno-

menological introduction of cooperativity is

analogous to the treatments describing the oxy-

genation of hemoglobin. The quantity n is

equivalent to a Hill coefficient.^^ Furthermore,

in point 4 of section 5.7.8.2, we noted for pK

shifts of about 1 that the Wyman equation,^^

AAGo = RT(1 - l/n)/a(l - a), when evaluated

at the pK at a = 0.5 and the relevant Hill

coef-

ficients

(n),

gave the same value for the energy

of interaction as that calculated for the pK

shift. In particular, by Equation (5.13), AGap =

2.3RTApK, and when evaluating at a = 0.5,

AAGo = 4RT(1 - l/n). Thus, as occurred in the

data in Figure 5.29 where AGap ~

AAGo,

the rela-

tionship between ApK and Hill coefficient, n,

becomes ApK = 1.74 (1 - l/n).This suggests that

positive cooperativity and hydrophobic-

induced pK shifts can be different representa-

tions of the same interaction energy.

Larger pK shifts, however, have been

observed than are calculable by the Wyman

equation. For example, a pK shift of 6 has been

observed for aspartic acid in a very hydropho-

bic polymer (see Figure 5.32), whereas from the

Wyman equation, ApK = 1.74 (1 - l/n), the ApK

could not exceed 1.74. Also with a reference pK

value of 4.0 for the Glu carboxyl, a pK shift of

2.4 is seen in Figure 5.34 with a corresponding

Hill coefficient of 8. For larger Hill coefficients

and pK shifts, the limitation of the correlation

is likely due to limiting approximations of the

Wyman equation that have been overcome, for

example, by the Monod treatment.

Additionally, as shown in Figure 5.31A, pK

shifts separate into an apolar-polar repulsive

component that can be relaxed by a change in

hydrophobic association and an apolar-polar

repulsive component that is dictated by

sequence and thereby retained even in the most

unfolded and disassembled state. Expectedly,

the pKa shift, relaxed by a disruption of all

200

Consilient Mechanisms for Diverse Protein-based Machines

hydrophobic association, is the component that

results in positive cooperativity and in Hill

coefficients of greater than 1.

To continue with the derivation, because K =

^-AG/RT ^ ^Q-AG/2.3RT^

j^gj^^

^ -AGo/2.3RT = -pKo,

and Equation (5.20) can be rev^ritten in terms

of changes in Gibbs free energy, AG, to give

pH =

AGo

/2.3RT +

AAGo

/2.3RT

+ (l/n)log[a/(l-a)]

(5.21)

Derivation of Equation (5.21) considers

hydrophobic-induced pK shifts and hydropho-

bic-induced increases in positive cooperativity.

In Figure 5.30, however, there is experimental

deUneation between the hydrophobic domain

where the above considerations dominate and

the electrostatic domain where charge-charge

repulsion dominates with a negative coopera-

tivity.

Thus we require an expression that would

properly include both effects.

5.8.5 Toward a Generalized

Acid-Base Titration Expression

5,8,5.1 Introduction of the Harris and

Rice Treatment of Cooperativity

Following Harris and Rice,^^^ the last term of

Equation (5.8) can be effectively restated as

log[a/(l - a)] + (aAG/aa)T/2.3RT, where

(3AG/9a)T = 0 at a = 0.5, to give

pH = pKo + ApK + log[a/(l - a)]

+ (^AG/aa)T/2.3RT (5.22)

that

is,

the last term of Equation (5.22) replaces

the Hill coefficient with a term that provides

for the change in Gibbs free energy as the

degree of ionization changes, (3AG/3a)T, which

changes the steepness of the curve from that

given by the simple Henderson-Hasselbalch

equation.

5,8.5.2 Introduction of the Effects of Both

the Charge-Charge Repulsion, AG^o and

Apolar-Polar Repulsive Free Energy of

Hydration, AG^^

As discussed above, there are two primary

mechanisms in aqueous dominated media for

pK shifts and changes in the steepness of the

experimental titration

curve.

These are referred

to as electrostatic-induced, principally charge-

charge repulsion (cc) and as hydrophobic-

induced, that is, the apolar-polar repulsive free

energy of hydration (ap). The general expres-

sion can now be written specifically to include

these two primary interactions as

pH = pKo + ApKcc + ApKap +

log[a/(l-a)] + {[((9AG/^a)J^.

+ [OAG/(9a)J^J/2.3RT

(5.23)

Although the ApKcc and ApKap values are of the

same sign and add, the partial derivatives are of

the opposite sign and counter each other. In par-

ticular, [(3AG/3a)T]c-c is the result of a negative

cooperativity with a curve that is broader than

given by the Henderson-Hasselbalch equation,

and [(3AG/3a)T]a-p is the result of a positive

cooperativity with curve that is steeper than

given by the Henderson-Hasselbalch equation.

5.8.5.3 Experimental Delineation ofAGcc

and AGap in Series of Model Proteins

The experimental titration data for the series of

Model Proteins I, i, ii, iii, iv, and \ are listed

in Table 5.5 and shown in Figure 5.34, where

the differential effects of charge-charge and

apolar-polar repulsion become apparent. For

Model Proteins i, ii, iii, iv, and v, for each sys-

tematic shift in pKa there occurs a correspond-

ing increase in Hill coefficient. Model Protein

I, however, does not fall within the series,

because it exhibits a pKa shift without an

increase in magnitude of the Hill coefficient.

Model Protein I and iv exhibit essentially the

same Hill coefficient of 2.7, whereas the pKa

values differ by 0.3 pH units.

On inspection of the sequence of Model

Protein

it is more hydrophobic with five F plus

two I residues having replaced seven less

hydrophobic V residues in the repeating 30

residue sequence. If this were the only determi-

nant it would be expected to have a greater pKa

and a larger Hill coefficient than Model Protein

v. Model Protein I, however, has two proximal

E residues with carboxylates separated by only

two residues, that

is,

-EPGE-. From the data in

5.8 Integration of Cooperativities Due to Apolar-Polar and Charge-Charge Repulsion

201

Figure 5.30, in poly(GEGIP) the Glu residues

are separated by four residues, -EGIPGE-, and

the pKa shift due to charge-charge repulsion is

>0.35 pH units. The charge-charge repulsion

component of the pKa shift for -EPGE- is

expected to be greater than for -EGIPGE-.

Accordingly, in the titration curve of Model

Protein I, when compared with those of Model

Proteins i, ii, iii, iv, and v, the differential effects

of charge-charge and apolar-polar repulsion

become apparent, and an iterative fitting

process using Equation (5.23) should allow

determination of AGcc and AGap in this mixed

case.

In doing so the recognition of the residual

pKa shift after complete unfolding, shown in

Figure 5.31, should also be delineated.

5.8.5.4 Positive Cooperativity as a

Fundamental Property of the Competition

for Hydration Between Apolar

(Hydrophobic) and Polar (e.g..

Charged) Species

From the analysis of the acid-base titration

data in Figures 5.30 through 5.34, positive coop-

erativity results from the apolar-polar repul-

sive free energy of hydration, that is, from the

competition for hydration between apolar

(hydrophobic) and polar (e.g., charged) species.

The general statement can be that the appear-

ance on the scene of the first polar, for example,

charged, species must do the work of destruc-

turing hydrophobic hydration in order to

achieve adequate hydration for

itself.

To put this into perspective, consider again

the Gibbs free energy for solubility, AG(solu-

bility) = AH - TAS, and recall the discussion of

Butler's findings (see section 5.1.3.3), where

insolubiUty results from the formation of too

much hydrophobic hydration. Too much hydro-

phobic hydration causes the positive (-TAS)

term to become larger than the negative

(exothermic) AH term, that is, when AG(solu-

bility) becomes positive and solubility is lost.

The insolubiUty comes in the form of the asso-

ciation of a pair of hydrophobic domains.

However, the pair of domains undergoes disso-

ciation and association fluctuations. Now, a

polar species can emerge (e.g., COOH -^

COO") proximal to an opening fluctuation, if it

can achieve adequate hydration by destructur-

ing the hydrophobic hydration. In doing so it

lowers the magnitude of the positive (-TAS)

term sufficiently to allow the dissociation to

stand. Even though there may yet be substan-

tial hydrophobic hydration covering the sur-

faces of the previously paired hydrophobic

domains, there is additional water, and the

emergence of the second polar species achieves

its hydration without having to destructure

quite so much hydrophobic hydration, and its

equilibrium constant is more favorable. This is

the description of positive cooperativity at the

molecular level arising from an apolar-polar

repulsive free energy of hydration.

To the best of my knowledge, the hemoglobin

oxygenation curve is historically the first

example of a biologically essential positive

cooperativity. Because of this, it becomes an

important objective to explore the phenome-

nology of hemoglobin's positive cooperativity

and compare it with that of the consilient

mechanism due to an apolar-polar repulsive

free energy of hydration (as is done in

Chapter 7) and, in fact, to do so for a number of

protein-based machines that exhibit positive

cooperativity.

5.8.5.5 Evaluation of [(dAG/da)Tjap Within

the Single Polymer, Model Protein v

From the plot in Figure 5.31A of the titration

of Model Protein v, the pKa of the first car-

boxyls to ionize may be taken as 7.0, whereas

that of the last carboxyls to ionize is about 5.7.

Therefore, in the process of driving hydropho-

bic disassociation of a relaxation or, inversely,

in the process of driving hydrophobic associa-

tion of a contraction, the pKa shifts by some

1.3 pH units. By Equation (5.13), AGap

= 2.3 RT ApKa - 2.3 x 1.987 x 310 x 1.3

= 1.8kcal/mole-carboxyl. Now, because

AGap = J[(9AG/3a)T]apda, we have that

/[(3AG/3a)T]apda « 1.8 kcal/mole-carboxyl. This

is an example wherein only the apolar-polar

repulsive free energy of hydration is relevant,

as the presence of a Glu (E) residue every 30

residues does not give rise to charge-charge

repulsion in the unfolded, state. Also, when

totally unfolded the carboxyl pKa is still shifted

202

Consilient Mechanisms for Diverse Protein-based Machines

from the unperturbed pH value of near 4.0.

Even with the model protein unfolded, there

still exists an apolar-polar repulsive free energy

AGap of 2.3 RT ApKa = 2.3 x 1.987 x 310(5.7 -

4.0) = 2.4kcal/mole-carboxyl. Even w^hen disas-

sembled and unfolded, the sequence of Model

Protein v does not allow the carboxylate

complete access to water unencumbered

by hydrophobic groups. Furthermore, the

maximum pKa shift experienced by the car-

boxyls in this hydrophobic model protein is

7.0 - 4.0 = 3.0, giving a AGap of 4.2kcal/mole-

carboxyl (see Figure 5.31A).

For Model Protein I, there occur 2 glutamic

acid residues (Glu, E) per 30 residues, and

importantly these are separated by only two

residues. In this case there exist significant

values for both AGap and AGcc- Recall that the

free energies add for the pKa shift but are of

the opposite sign for cooperativity. This being

the case, it is possible to write two equations

with the two unknowns, AGap and AGcc, and to

solve the values. In this case, approximate

values for AGap of 2.0kcal/mole-carboxylate

and AGcc of 0.7 kcal/mole-carboxylate pair can

be estimated.

Thus,

when laid out in the form of

Equation (5.23), it becomes possible to begin

separation of different contributions to the

Gibbs free energy of interaction.

5.8.5.6 Relationship of the Consilient

Mechanism to Cold Denaturation and to

Hydrophobic Dissociation on Being Made

More Polar

Hydrophobic association on raising the tem-

perature is the most fundamental aspect of the

consiUent mechanism, arising as it does from

the inverse temperature transition. An equiva-

lent statement would be that hydrophobic

dissociation on lowering the temperature is

fundamental to the consilient mechanism. His-

torically, this has been called cold denaturation

ofenzymes.^^ In our view, those protein systems

that associate on heating to physiological tem-

peratures in order to achieve a functional state

should be considered in terms of the consilient

mechanism.

In fact, any protein function that involves

cycling between hydrophobically associated

and dissociated states by whatever energy

input, for example, whether chemical or elec-

trochemical, should be considered in terms of

the consilient mechanism of the apolar-polar

repulsive free energy of hydration.

5.8.6 Toward a Generalized Theory

for Achieving Function of Protein-

based Machines Founded on

Hydrophobic Association

Equation (5.23) provides a general expression

for fitting the family of curves schematically

shown in Figure 5.19B and experimentally

demonstrated in Figure 5.20A-C. The relevant

energy conversions can be considered as

electro-chemical (A), chemo-mechanical (B),

and electro-mechanical (C). In each case, posi-

tive cooperativity is the result of increasing

hydrophobicity, and it would appear to apply to

the many energy conversions described in

section 5.6. Thus, it would seem necessary only

to cloak the expression in the correct energy

terms,

recognizing as in Equation (5.21) that

pH,

for example, is a free energy divided by 2.3

RT.

It would seem, whatever set of energies

drive function, that a complete description

could be achieved by properly including the

required terms in such an equation.

Hopefully, the above proposed integration of

energy conversion, involving the recognition of

a commonality of protein function that includes

an apolar-polar repulsive free energy of hydra-

tion, might bring us one step closer to the

daunting challenge recognized by Weber^^^ of

achieving a more "complete description of the

energetics" of protein-based machines.

5.9 On the Efficiency

of Consilient Protein-

based Machines

5.9.1 Introductory Remarks

5.9.1.1 The Central Role of the Gibbs Free

Energy for Hydrophobic Association in

Diverse Energy Conversions

The change in Gibbs free energy for hydropho-

bic association,

AGHA,

affects and is affected

by the forms of energy utilized by living organ-

5.9 On the Efficiency of Consilient Protein-based Machines

203

isms.

It originates from inverse temperature

transitions; it establishes the comprehensive

hydrophobic effect, and, because of the perva-

siveness of its role in catalyzing energy conver-

sion, the comprehensive hydrophobic effect

provides the basis for the consilient mechanism.

From our developing perspective and as

demonstrated in part below, it appears that

AGHA provides the foundation for the most effi-

cient mechanism of diverse energy conversion

available in aqueous systems.

5.9,1.2 The Design of Poised Elastic-

Contractile Protein-based Machines

The design of efficient protein-based machines

requires judicious choice of model protein com-

position based on the intended energy source

and operating conditions of temperature,

medium, and so on. Proper design poises the

system immediately to initiate function on

addition of the driving force, the input energy.

Of course, in recognizing this, protein function

by modulating hydrophobic association simply

depends in a complementary way on the many

variables of the Weber concern,^^^ for example,

ligand binding, subunit association, pH, tem-

perature, and pressure, as well as phosphoryla-

tion-dephosphorylation, oxidation-reduction,

absorption of light, and so forth.

From a practical standpoint, for example, Tt

would be set just above the transition zone, as

defined in Figure 5.5, such that the energy input

for which the system is designed can most effi-

ciently lower Tt through the transition zone. For

biological conditions Tt would be poised at

37° C, and the added energy would lower Tt the

width of the transition zone to about

25° C.

The

reaction would be represented by an equilib-

rium constant for hydrophobic association,

KHA, expressible as logKnA = -AGHA/2.3RT,

where AGHA/2.3RT would be plotted versus Y'\

the fraction of hydrophobic association in

analogy to Equation (5.23), and modified to

include cooperativity as shown for a protona-

tion/deprotonation reaction also treated

in Equation (5.23). The chemical couple

COOH/COO", is a specific example, but the

reaction can involve any of the energy conver-

sions represented in the Hexagon in Figure

5.22. Of course. Axiom 5 (see section

5.5.3)

provides a guiding principle for increasing

efficiency of energy conversion, which has the

effect of increasing positive cooperativity.

5.9.1.3

AGHA

Provides an Efficiency

Limit and Becomes the Coupling Process

in Energy Conversion by the

Consilient Mechanism

Interestingly, for each model protein composi-

tion and energy source the consilient mecha-

nism sets an efficiency limit,

CCM-

It is simply the

change in Gibbs free energy for hydrophobic

association, the output energy, divided by the

input energy, that is,

CCM

= AGnA/input energy.

For the coupling of functions, as, for example,

occurs in electro ^-^ chemical transduction,

reduction becomes the energy input that results

in an output energy of

AGHA,

a decrease in the

Gibbs free energy of hydrophobic association

due to an increase in hydrophobicity; the

AGHA

thus produced then becomes the energy input

for the output of a change in chemical energy.

In general, these energy conversions can be

reversible, and

AGHA

emerges as the coupUng

process in energy conversion by the consilient

mechanism.

To elaborate on an electro

<*-•

chemical trans-

duction example, the chemical group to be

reduced can be a biosynthetic modification of

vitamin B3 (niacin, shown in Figure 3.8A) or of

vitamin B2 (riboflavin, shown in Figure 3.8B).

Electrical energy is expended to reduce the

oxidized state of a modified vitamin B3 or B2

attached to the protein, making the protein

more oil-like and lowering the Gibbs free

energy for hydrophobic association,

AGHA,

sufficient to drive hydrophobic association.

This decrease in AGHA changes the free energy

of a carboxylate and causes it to take up a proton

(see Figure 5.20A and associated discussion).

Analyses of these elements are considered

below after a brief general definition of efficiency.

5.9.2 General Statement for Efficiency

of Energy Conversion

The usual symbol for efficiency is r|. The defin-

ition of r\ is the output energy divided by the

204

Consilient Mechanisms for Diverse Protein-based Machines

input energy times 100 so that the efficiency is

given as a percentage:

= (output energy/input energy) 100 (5.24)

5.9.3 Consilient Mechanism's

Efficiency Limit,

CCM,

for Model

Protein Composition and

Energy Source

The Gibbs free energy for hydrophobic associ-

ation,

AGHA,

may be viewed as an output energy

that results from a particular input energy

acting on a specific composition of protein-

based machine. Thus, we define an efficiency

limit for energy conversion by the consilient

mechanism,

CCM,

for a particular energy input

and model protein composition.

ecM =

AGHA

/input energy (5.25)

The same input energy can result in very

different efficiency limits depending on the

protein composition. Furthermore, the same

energy input, acting on the same polymer com-

position, can vary depending on the range of the

intensive variable. For example, for the compo-

sition (GVGVP)n, raising the temperature from

5° to 20° C results in no

AGHA

(see Figure 5.5A).

On the other hand, raising the temperature from

25° to 40° C results in hydrophobic association

that is useful in many ways including the output

of mechanical work. Alternatively, an energy

input acting to lower the value of Tt would have

little effect until Tt begins to enter the transition

zone,

which for most mammals would begin at

about 37° C and be complete by about 25° C (see

Figure 5.5B). In relation to Figure 5.5, Tt on the

X-axis could be plotted versus Y", the fraction of

hydrophobically associated

state.

With Tt above

37° C, Y'' would be zero; as Tt is lowered through

the transition zone, there would be a sigmoidal

increase in Y'' until a value of

were reached at

about 25°

Graphically this is simply the

sigmoid of Figure 5.33.

A prevalent example in biology involves a

chemical energy and its intensive variable of

chemical potential, |J., the change in Gibbs free

energy per mole of chemical. For the polymer

composition in Figure 5.12A, at low salt con-

centration the slope ATt(%)/A|x, apparent as

ATt(x)/AN, is steep, whereas at higher concen-

trations the slope is much less. By Equation

(5.9),

AGHA is larger for the low salt concentra-

tion range than the high concentration range.

Thus,

CcM varies significantly depending on the

effectiveness of the energy input in achieving a

AGHA-

The latter quantity is calculable by

employing Equation (5.10b),

AGHA(X)

[AHt(ref) - AHt(x)], and using relevant experi-

mental data, much of which are found in Luan

and Urry's review^^ entitled "Elastic, Plastic,

and Hydrogel-Forming Protein-based Poly-

mers"

in the Polymer Data Handbook. Key

results are listed in Table 5.4. Accordingly, in a

plot of Tt versus concentration it is the slope at

any point along the curve, and the calculated

AGHA per mole, that is, |XHA, would be for that

concentration. A good example of the impor-

tance of model protein composition is shown in

Figure 5.15 and the associated discussions of

relative effectiveness in section 5.4.3.2.

5.9.3.1 CaCk and NaCl as Chemical

Energy Inputs to an Uncharged Model

Protein to Drive Hydrophobic Association

We begin with NaCl, for which calorimetry

data, AHt, are available from Table I of

Luan et al.^^^ for this salt interacting with

poly(GVGVP) and used in Equation (5.10b).

Chemical energy is defined as A|iAn, where the

chemical potential,

A|LI,

is 2.3RT log(a7a")- The

a' and a" are the activities of the ions at two

dif-

ferent activities, a' and a'', which equal concen-

tration at sufficiently low ion concentrations,

[C].This assumption will be made here for sim-

plicity, but the activity coefficients required for

accurate calculations with salts are available.^^^

This situation is further complicated by the

determination of An when there is no func-

tional group being titrated. Accordingly the

numbers calculated are taken for a An of 1.

These complications are not significant for

comparison of the effect of different chloride

salts on uncharged polymers, but they do

become relevant when making comparisons

with data from charged model proteins.

For the concentration step from

0.1

N to

l.ON, ATt is 12.5 and

AGHA

AHt(0.1

N - l.ON)

= 0.5kcal/mole-pentamer. Note for the concen-