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

Подождите немного. Документ загружается.

5.7 What Is the Physical Basis for the ConsiUent Mechanism of Energy Conversion?

185

These interpretations were reinforced by the

observation in the same time period of stretch-

induced pKa shifts (discussed immediately

below) by insights of the T^based hydropho-

bicity scale (discussed above) and then by

hydrophobic-induced pKa shifts (as discussed

further below).

5,7.8.2 Stretch-inducedpKa Shifts

In 1937, using series of simple organic mole-

cules,

Butler^^ first observed that exposure of

CH2 groups to water resulted in an exothermic

reaction with formation of a unique water

around such oil-like groups. In

1970,

Weis-Fogh

and Andersen^^ reported the exothermic reac-

tion on stretching elastin and interpreted it in

terms of the exposure of hydrophobic groups

with formation of hydrophobic hydration.

Hoeve and Flory^^ further affirmed this in their

1974 publication. Accordingly, stretching of

elastin-like protein forces exposure of

hydrophobic groups to the surrounding water

with the result of an exothermic reaction of

hydrophobic hydration.

Consideration of the effects of stretching

begins with hydrophobically associated and

cross-linked elastomeric matrix composed of

the same protein-based polymer as used in the

calorimetry studies with poly[0.8(GVGVP),

0.2(GEGVP)]. As shown in Figure 5.29,^^

the acid-base titration curve of unstretched

matrix of cross-linked poly[0.8(GVGVP),

0.2(GEGVP)] gives a pKa of 3.99, given ade-

quate time for equilibration of medium with

matrix. Repeating the acid-base titration after

stretching on applying a load of 1 gram shows

that the pKa shifted to 4.84. More extensive

data from another polymer composition

appears in Figure

5.23.^^

The chemical potential

of base required to remove the proton from

COOH increases on stretching. This means that

the free energy for formation of the carboxy-

late,

COO", has increased. Such a result is quite

striking, because there is an increase in water in

the matrix on stretching, which one might

expect to facilitate ion formation. It would

seem that the kind of water that enters the

matrix on stretching is not suitable for hydra-

tion of a charged

species.

The type of water that

Stretch-Induced

pKg

Shift

Acid

- Base

Titration

poly[4(VPGVG),(VPGEG)]

3.00 3.60 4.20 4.80

5.40 6.00

FIGURE 5.29. Stretch-induced pKa shift within

hydrophobically associated elastic model protein

implies competition for hydration between carboxy-

late and hydrophobic groups. Despite water uptake

into the model elastic protein on stretching, the

COO"

experiences less accessible hydration. The

implication is that hydrophobic hydration, formed

due to exposed hydrophobic groups on extension,

is unsuited for COO" hydration. (Reproduced with

permission from Urry et al.^^)

enters the matrix on stretching is primarily

hydrophobic hydration.

This brings us to the following analyses and

conclusions:

Even though there is an increase in the

amount of water in the elastomer on stretching,

much of which must be hydrophobic hydration,

the formation of carboxylates has become disfa-

vored.

Therefore, water of hydrophobic hydra-

tion must not be suitable for hydration of

carboxylates.

(dli/df)n=aM).5

< 0 for the ATt-mechanism:

The chemical potential,

|Li,

is defined as RT In a,

where a is activity of proton in this case. In the

pH range of 4 to 5, proton concentration, [H^],

can be used in place of a, such that \i = RT

ln[H^].

Because the choice for acid-base titra-

tions is logarithm to the base 10, log, it can be

stated that |i = 2.3 RT log[H^, and because pH

= -log[H^, then

|LI

= -2.3 RT pH. At 50% ion-

ization, that is, a = 0.5,

|LI

= -2.3 RT pKa, such

that A|i(proton) = -2.3 RT ApKa. From Figure

5.29 it is clear that an increase in load from zero

186

Consilient Mechanisms for Diverse Protein-based Machines

to 1.0 gram represents an increase in force;

therefore, Af is positive. On the other hand, the

pH at the pKa increased from 3.99 to 4.84;

therefore, A|x(proton) is a negative quantity.

Thus,

it can be written that (3|Li/3f)n=a=o.5 < 0, that

is,

stretching results in an uptake of protons,

where n = a = 0.5 indicates evaluation at the

degree of ionization of 0.5 which is the pKa, the

point of 50% ionization.

(d[i/df)n=a=o.5

> 0 for the charge-charge

repulsion mechanism: Studying polyelectrolytes

with the example of poly(methacrylic acid),

[-CH-CCHsCOOH-Jn, Katchalsky et al.'^ in

1960 presented the argument that stretching

resulted in the release of protons, that is,

(9|i/3f)n=a=o.5 > 0. The description of the process

for cross-linked poly(methacryUc acid), with a

carboxyl function on every other backbone

atom, follows: (a) At 60% ionization,

charge-charge repulsion gives complete exten-

sion into stiff rods, (b) At 0 to 10% ionization,

charge-charge repulsion is no longer present

and the chain randomizes to give full contrac-

tion, (c) When the chemical potential of proton

in the bathing medium allows 20% ionization,

contraction will have occurred to the extent

allowed by charge-charge repulsion. On

stretching of the matrix at this point, there is an

increase in distance between charges, and for

the unchanged proton chemical potential in the

bathing medium there can be a release of

protons into the medium until the chemical

potential of the fiber again matches that of the

medium, (d) Therefore, stretching results in a

release of protons for the charge-charge repul-

sion mechanism, which is to say that

(3|a/Df)n=a=o.5 > 0 for the charge-charge repulsion

mechanism, whereas for the ATt-mechanism it

is exactly the opposite, (3|i/3f)n=a=o.5 < 0. (e) Con-

clusion: A mechanism other than charge-charge

repulsion is responsible for the stretch-induced

pKa shift exhibited by Tf-based molecular

machines.

Relationship of cooperativity to tspKa'.

Furthermore, the increase in Gibbs free energy

of hydroxyl ion required to remove the proton

from COOH, indicated by the pKa shift of 0.85,

is AG2,i = A|x(hydroxyl ion) = 2.3 RT ApKa =

1.16kcal/mole. In association with the pKa

shift, there is an increase in steepness of the

acid-base titration curve on stretching, indicat-

ing an increase in positive cooperativity. For

small pKa shifts this may be estimated using the

Wyman equation^^ AG = RT(1 - l/n)/a(l - a),

where n (the Hill coefficient)^^ is the slope of

the curve at the pKa and a is the degree of ion-

ization, which at the pKa is 0.5. Under zero

load, n was 1.07; on loading to 1 gram, n became

2.21.

For the zero load case, AGi is 0.15

kcal/mole, whereas for the 1 gram load, AG2 is

1.30, such that the difference, AGi,2, is 1.15

kcal/mole. Thus, both the pKa shift and the

increase in positive cooperativity represent

dif-

ferent expressions of the same energy of inter-

action. Conclusion: Positive cooperativity is

another expression of the competition for hydra-

tion between apolar and polar groups that gave

rise to the stretch-induced pKa shift.

Nonlinear stretch-induced pKa shift: The

stress/strain curve of the cross-linked pro-

tein-based polymer X''-poly[0.82(GVGIP),

0.18(GEGIP)] is shown in the inset of Figure

5.23A to be linear such that the work done on

the elastomeric band is directly proportional to

the apphed force. On the other hand, the pKa

shifts resulting from linear increases in load are

very nonlinear (Figure 5.23B).^^ This carries

significant implication with regard to the effi-

ciency of the conversion of mechanical energy

into the chemical work of pumping protons.

In addition, the acid-base titration curve

becomes steeper, that is, exhibits an increase in

positive cooperativity as the elastomeric band

is further extended (Figure 5.23B). This effect

was explicitly calculated above. Because the

change in chemical potential times the change

in number of moles involved to go from the

relaxed state to the contracted state is the

required chemical energy, the positive co-

operativity means less chemical energy will be

required to perform a given amount of mechan-

ical work. Not only is there a different mecha-

nism from that of the charge-charge repulsion

mechanism, but the new mechanism is more

efficient, as was briefly discussed in Chapter 2

and shown in Figure

2.16.

This will be discussed

more extensively below.

5.7 What Is the Physical Basis for the Consihent Mechanism of Energy Conversion?

187

5.7.8.3 Hydrophobic-inducedpKa Shifts

Four studies based on different model protein

systems and experimental variables have been

used to vary hydrophobicity systematically and

thereby to demonstrate the basis of hydropho-

bic-induced pKa shifts. The four studies are (1)

dilution of a recurrent ionizable residue in a

poly(5-mer) from 1 ionizable group every 5

residues to 1.2 per 100 residues by stepwise

replacement of a polar residue with an apolar

(hydrophobic) Val (V) residue, (2) maintenance

of the composition of a poly(30-mer) but

rearranging the proximity of more hydro-

phobic residues to the ionizable residue in a

poly(30-mer), (3) increasing systematically the

hydrophobic residues within a poly(30-mer)

while holding the ionizable residue of the

poly(30-mer) constant at 1 per 30 residues, and

(as discussed in section 5.7.8.2) (4) stretching

incrementally a hydrophobically folded cross-

linked elastic matrix containing ionizable

residues (in this case the polymer composition

was kept constant but the hydrophobicity was

increased by application of an extending force

that increasingly exposed hydrophobic groups

to water in sufficient proximity to the ionizable

function.

5.7.8.3.1 Increasing Hydrophobicity by

Stepwise Replacement of the Functional

Group with a More Hydrophobic Residue

The model system, poly[/v(GVGIP),

/x(GXGIP)],

is used where /x and f are mole

fractions and X is defined as follows for each of

three studies: (1) X = Glu(E),/x = 1.0 to 0.06;

(2) X = Asp(D), /x = 1.0 to 0.06; and (3) X =

Lys(K),/x = 1.0 to 0.06. In each of these studies,

the starting polymer is poly(GXGIP), and then

the residue X is stepwise replaced by the more

hydrophobic valine (Val, V) until the X residue

is very dilute in the polymer, 1.2 residues per

100 residues. As shown in Figure 5.30A,B

for starting compositions of poly(GEGIP)^^

and poly(GKGIP),^^ respectively, the charge

density is high enough such that charge-charge

repulsion is observed in the absence of

salt.

This

charge-charge repulsion is entirely removed by

0.15 N NaCl for X = E and D (data not shown)'^

and significantly relieved for X = K, and the

essentially normal pKa is obtained for these

ionizable functions. In the absence of salt, the

charge-charge repulsion disappears for

= 0.7 and for

= 0.75. The remarkable feature

of these studies is that, as the dilution of

charged species continues, the pKa shift

resumes with a vengeance, instead of decreas-

ing as it must if charge-charge repulsion were

relevant. The largest pKa shifts are found at the

highest dilutions examined with 1.2 ionizable

residues per 100 residues of polymer.

5.7.8.3.1.1 Poly[fv(GVGIP)Jx(GXGlP)] with

X = Glu(E) and fx Varied from 1.0 to

0.06.^^

Specifically for the Glu(E) residue, 10 polypep-

tides were synthesized, each with a different

value of ffi. The value for fn = 1 in pure water

is 4.7, and on addition of 0.15 N NaCl the

value drops to 4.35. The magnitude of the

charge-charge repulsion is about 0.49kcal/mole

as estimated from the ApKa of 0.35. A pKa of

4.35 is the lowest value obtained in the absence

of salt at fE= 0.7. On further dilution, that is, for

ffi < 0.7, the increase in pKa resumes until at fe

= 0.06 the pKa becomes 6.6, which relaxes to 6.1

on the addition of physiological saUne.Thus, the

mechanism operative in these Tt-type protein-

based polymers, which is not charge-charge

repulsion, has brought to bear a repulsive free

energy of interaction of over 3kcal/mole. As

will be discussed later, this new mechanism can

give rise to repulsive free energies of interac-

tion of 8 to lOkcal/mole.

Calculated curves using the charge-charge

repulsion argument are included in Figure

5.30A, and the means of calculating is given

in the following section on poly[/v(GVGIP),

/K(GKGIP)].The minimal experimental dielec-

tric constant for poly(GVGIP) at 36° is obtain-

able from Table 5.6, where the high-frequency

component (£°° of 23) and the

MHz compo-

nent (Ae of 42) sum to give a minimal value of

65.

With this value, only a few tenths of a pH

unit shift is calculated (see Figure 5.30A). Even

when taking the unrealistic value of 5 for the

dielectric constant in the absence of any ioniz-

188

Consilient Mechanisms for Diverse Protein-based Machines

able residue, the experimental curves are not

reasonably calculated by the Glu(E) residue.

With the charge-charge repulsion approach,

less than 40% of the shift observed for fE = 0.06

is calculated. The same calculation is carried out

below in more detail for the lysine series.

5.7.8.3.1.2 Poly[fv(GVGIP),fx(GXGIP)J with

X = Asp(D) and fo Varied from 1.0 to

0.06.^^

Specifically for the Asp(D) residue, 10 polypep-

tides were synthesized, each with a different

value of

fo.

The detailed information of the syn-

thesized compositions and the pKa shifts with

and without salt should be sought in Urry

et al.^^ The magnitude of the charge-charge

repulsion seen in the ApKa for an/o of

was 0.8

pH units, representing a charge-charge repul-

sion of l.lkcal/mole with a charged aspartate

every fifth residue. The repulsion was entirely

removed by the addition of 0.15 N NaCl. Thus,

1.00

0.80

0.60

0.40

0.20

0.00

4.0

1.00

0.75 h

0.50 h

0.25 h

0.00

7.0

• o poly[f^(IPGVG)4(IPGEG)] electrostatic

domain

•o

o •

4«a • O

• 4 0

« 4 O

hydrophobic

domain

• O.lSNNaCl

O Type

H2O

• 6(75,65)

4 e(75,25)

D e(75,5)

5.0

6.0

7.0

1 poly[^^(IPGVG)4(IPGKG)]

1 AG(SE) = ^ £kll£i = 2.3RTApKa

•...-.• 6j5 =75.

£Q

=65 (experimental)

I ---e,5 =75.80=25

1 -.-8,5=75.60=5

"•- experimental data in water

"cn:^ 1

i_i 1 1 1

7.5

8.0

8.5

9.0

9.5

10.0

FIGURE

5.30. (A) Hydrophobic-induced pKa shifts

as Glu is replaced by

Val

in the family of model pro-

teins,

poly[fv(GVGIP),fE(GEGIP)], demonstrate a

limitation of the electrostatic argument. (Repro-

duced with permission from Urry et al.^^) (B) Using

the protein-based polymer poly[fv(GVGIP),

fK(GKGIP)],

hydrophobic-induced pKa shifts as Lys

is replaced by

Val,

as with the data in A, also demon-

strates limitation of electrostatic argument. (Repro-

duced with permission from Urry et al.^^)

5.7 What Is the Physical Basis for the Consilient Mechanism of Energy Conversion?

189

TABLE

5.6. Dielectric relaxation of poly(GVGIP) in water as a function of temperature.^

Temp (°C)

(ns) a (mS/m)

6°-

100

MHz ± 0.5

±3

±4

±2

34 ±2

±3

±2

±3

±4

139

±6

87 ±10

49 ±3

44 ±3

±3

±2

36 ±2

±2

±3

±2

0.36 ± 0.08

0.3 ± 0.1

0.11 ± 0.03

0.08 ± 0.02

0.08 ± 0.07

0.08 ± 0.02

0.09 ± 0.02

0.10 ± 0.02

0.11 ± 0.03

1.2 ± 0.2

1.8 ± 0.6

0.5 ± 0.1

0.6 ± 0.1

0.7 ± 0.1

0.7 ± 0.2

0.9 ± 0.2

1.0 ± 0.2

32.3

30.8

26.3

24.7

23.7

23.2

22.7

22.6

22.4

^ The minimal dielectric constant at 37° C is 65, that is, 23 for the high frequency (100 MHz) limit plus 42 for the

MHz

band. Values of 5 are not correct.

^ The dielectric parameters of the samples at 0° and 6° C are only suggestive because the relevant dielectric relaxation is

not yet very well resolved. (Data from Table 1 of Buchet et al.^^)

under physiological conditions with an aspartic

carboxylate every fifth residue, there is no

charge-charge repulsion in 0.15N NaCl. On the

other hand, on dilution to 1.2 residues/100

residues,

the ApKa

was

2.1

pH units in water and

1.5 pH units in

0.15

NaCl. Significantly, even

in this domain where charge-charge repulsion

is ruled out, there is yet an effect of salt to

relax the repulsion between hydrophobic and

charged

groups.

This region of the hydrophobic

domain will be calculated explicitly below for

the lysine NHa^ example.

5.7.8.3.1.3

Poly[fv(GVGIP)Jx(GXGIP)] with

X= Lys(K) andfK Varied from

1.0

0.06.^^

For

carboxylates, the high pH species is the charged

-COO"

species, whereas for amino functions

the low pH species is the charged,

-NHs^,

group.

Accordingly, the pKa values of lysine (Lys, K)

residues will reverse direction and shift to

lower values. Specifically, to examine the pKa

shifts of the Lys(K) residue, 12 polypeptides

were synthesized, each with a different value of

fx.

The data in Figure

5.SOB

are for water. The

highest pKa value is 9.6, and the value for fx =

1 is 9.2 in Type I (low conductivity) water and

9.4 in the presence of normal

saline.

The relax-

ation of charge-charge repulsion appears to be

less effective when the CI" anion is the shield-

ing species rather than

Na"^

cation. As the value

of fx approaches 0, the pKa decreases in pure

water to 8.2 and to 8.6 in normal saline. Inter-

estingly, the shielding by the CI" counter-ion

appears to be more effective in the hydropho-

bic domain rather than in the charge-charge

repulsion (electrostatic) domain.

We now focus on an attempt to use the elec-

trostatic mechanism to calculate the large pKa

shifts in the range for lower values of fx (<0.75).

The calculation uses the expression of solvation

energy (SE) due to Born, SE = [(ze)V2r](l -

1/e), where z is the charge on the ion, e is the

unit electron charge,

is the dielectric constant

of the surrounding medium, and r is the radius

of the ion. For the change in solvation energy,

ASE, on going from

to 8i_i would be

ASE = Q[8i_i-ei)/ei_iEi]

where Q is substituted for the coefficient

[(zcf/2r].

The four steps of the calculation are (1) the

choice of dielectric constants for boundary con-

ditions of fx=0.75 (at the highest pKa value) and

0, when there is no K present; (2) the change in

dielectric constant for each replacement of a

polar K residue by the more hydrophobic V

residue; (3) use of an experimental value of

ApKa to evaluate Q; and (4) calculation of the

ApKa for each ionizable residue replaced by Val

per 100 residues.The results for

Lys(K)

are given

in Figure

5.30B

and for Glu(E) in Figure

5.30A.

At the temperature of the experiment, the

dielectric constant of pure water is

75;

this is the

highest value possible. The dielectric constant

for fK = 0 has been experimentally determined

190

Consilient Mechanisms

for

Diverse Protein-based Machines

be 65 (See the

36°

data

Table 5.6).^^ The

use

these values

for the

boundary conditions

results

in a

maximal ApKa

only 0.3

units,

whereas

the

true ApKa

five times larger. Only

when

fictitious dielectric constant

used

for the phase-separated state can the electrosta-

tic calculations begin

approach

the

experi-

mentally observed shifts.

One might argue that

the

lysine side chain

buried

in a

local dielectric constant

of 5

even

though

the

macroscopic dielectric constant

65.

Tliis

is not

reasonable because

the

barrier

to backbone mobility

poly(GVGIP)

only

l.lkcal/mole.^^

barrier many times greater

would

required

hold

lysine side chain

a local dielectric constant

for a

viable elec-

trostatic argument.

fact

the

magnitude

of the

pKa shifts

can be as

large

as 8 to

lOkcal/mole

for

an Asp

residue

in a

more hydrophobic

Tt-type protein-based polymer. ^^^

Conclusion: Neither charge-charge repulsion

nor being buried

in a

medium

of low

dielectric

constant provides

satisfactory explanation

for

the increasing

pKa

shift attending dilution

ionizable side chains

in a

moderately hydropho-

bic protein-based polymer

5.7.8.3.2 Carboxylate-Containing

Polytricosapeptides

Constant Composition

but with Rearrangement

Hydrophobic Residues: The Power

of Sequence!

5.7.8.3.2.1 Effect

Different Pentamer

Sequence Arrangements

of the

Same 30-mer

Composition. Chemically synthesized polytri-

cosapeptides, poly (30-mers), were prepared

with compositions

of 1

aspartic acid residue

(Asp,

D) and 5

more-hydrophobic phenylala-

nine (Phe,

residues replacing valine (Val,

residues

per

repeat

of 30

residues,

but

with

dif-

ferent relative locations

of D and F

residues.

These compositions

are

written:

Polymer

XI:

Poly(GFGFP GVGVP

GDG

VP GFGFP GFGVP GVGVP)

Polymer

Poly(GVGVP GVGFP

GDG

FP GVGVP GVGFP GFGFP)

The

pKa for

Polymer

V was

found

to be 6.7,

whereas that

for

Polymer

under identical

conditions

was

determined

to be 10 (see

Figure

5.32, below)."^^

The

normal

pKa for

aspartic

this polymer without

any F

residues having

replaced

residues

(see

Polymer

in Table

5.5)

is 3.8.

The

polymers were designed with

the (5-

spiral structure

as the

basis. (This structure

was

developed

section

5.2.)

Given

the

p-spiral

structure as the design

basis,

Polymer XI had

the

more-hydrophobic

residues more distal from

the

residue, whereas

Polymer

V the

more-

hydrophobic

residues were more proximal

the

residue,

shown

Figure 5.32, below).

Accordingly, without changing

the

mean com-

position

of the

polytricosapeptide,

but by

designing structures with different proximities

the

ionizable function,

the pKa

shift changed

from 6.7

3.8 = 2.9

3.8

6.2.'^ The Varied

hydrophobic proximity

the functional groups,

resulting from

the

primary structure, causes

dif-

ferences

hydrophobic-induced

pKa

shifts.

5.7.8.3.2.2 Comparison

a Fixed Sequence

Six Pentamers with

Random

Mix of

the Same

Pentamers.^^^

The

importance

sequence

control

also understood through

two

addi-

tional comparisons.

The

polytricosapeptide

Polymers

X and XII in

Table 5.5, with specified

sequences, exhibited

pKa

values

8.1

and 7.8,

respectively, whereas

random mixture

the same combination

pentamers, namely,

poly[(GEGFP),2(GVGVP),2(GVGFP),

(GFGFP)],

gave

a pKa of 5.2.

Similarly,

the

specified sequence poly(GEGFP GVGVP

GVGVP GVGVP GFGFP GFGFP) gave

pKa

of 7.7,

whereas

the

same pentamers

in random mixture, namely, poly[(GEGFP,

3(GVGVP),2(GFGFP)] gave

a pKa of

4.7

(see

Table

1 in

Urry

al.).^^^

The pKa

shift with

specified sequence

four times larger than

for

random polymers

of the

same mean pentamer

composition.

terms

energies,

the

average

pKa shift

of the

polymers composed

random

pentamers,

0.65,

represents

repulsive free

energy

less than Ikcal/mole, whereas

the

average

pKa

shift

of the

fixed sequence

protein-based polymers

(3.6)

represents

repulsive free energy

about

kcal/mole.

For

energy conversions required

living organ-

isms,

sequence matters.

part, this

is why

biology puts

much energy into synthesis

protein,

discussed

Chapter

5.7 What Is the Physical Basis for the ConsiUent Mechanism of Energy Conversion?

191

5.7.8.3.3 Increasing Hydrophobicity While

the Composition of the Functional Group

Remains Constant

In the polytricosapeptides of this study the

aspartic acid residue in one series and the glu-

tamic acid residue in a second series are kept

constant at 1 per 30 residues, and the number

of valine residues replaced by the more

hydrophobic phenylalanine residues systemati-

cally increases from 0 to 2 to 3 to 4 and to 5, as

foUow^s:

Polymer

II:

III:

IV:

VI:

VII:

VIII:

IX:

Poly(GVGVP GVGVP

Poly(GVGVP GVGFP

Poly(GVGVP GVGVP

Poly(GVGVP GVGFP

Poly(GVGVP GVGVP

Poly(GVGVP GVGFP

Poly(GVGVP GVGVP

Poly(GVGVP GVGFP

GDGVP GVGVP GVGVP GVGVP) D/OF

GDGFP GVGVP GVGVP GVGVP) D/2F

GDGVP GVGVP GVGFP GFGFP) D/3F

GDGFP GVGVP GVGFP GVGFP) D/4F

GDGFP GVGVP GVGFP GFGFP) D/5F

GEGVP GVGVP GVGVP GVGVP) E/OF

GEGFP GVGVP GVGVP GVGVP) E/2F

GEGVP GVGVP GVGFP GFGFP) E/3F

GEGFP GVGVP GVGFP GVGFP) E/4F

GEGFP GVGVP GVGFP GFGFP) E/5F

The magnitude of the pKa shifts are given

under the representations of the structures in

Figure 5.32 (see below) with the reference

value being 4.0 for aspartic acid and 4.4 for glu-

tamic acid in the polymers without F residues."^^

Observed is a remarkably nonhnear pKa shift

with increasing numbers of more hydrophobic

Phe residues, which is analogous to that of the

stretch-induced pKa shift discussed above and

even similar to the form of the pKa shifts

observed on replacing a V by E, D, or K, dis-

cussed immediately above.

Conclusion: There is a dramatic nonlinear

pKa shift with increasing number of hydropho-

bic Phe residues and an associated increase in

positive cooperativity that is quite analogous to

the stretch-induced pKa shift (see Figure 5.23B)

and with other means of systematically increas-

ing hydrophobicity to achieve pKa shifts.

A similar series of polytricosapeptide model

proteins containing the lysine residue has been

prepared biosynthetically and character-

i^^d

67,102,103

namely:

Model Protein i': (GVGVP GVGVP GKGVP GVGVP GVGVP GVGVP)22(GVGVP) K/OF

Model Protein ii': (GVGVP GVGFP GKGFP GVGVP GVGVP GVGVP)22(GVGVP) K/2F

Model Protein iii': (GVGVP GVGVP GKGVP GVGVP GVGFP GFGFP)22(GVGVP) K/3F

Model Protein iv': (GVGVP GVGFP GKGFP GVGVP GVGFP GVGFP)2i(GVGVP) K/4F

Model Protein v': (GVGVP GVGFP GKGFP GVGVP GVGFP GFGFP)2i(GVGVP) K/5F

Lys (K) residues exhibit similar nonhnear pKa

shifts,

in this case to lower pKa values, on

increasing hydrophobicity by increasing the

number of phenylalanine residues per 30-mer,

but the magnitudes of the shifts are less than

for glutamic and aspartic acids."^^"^^

TTie above profusion of studies demonstrate

the breadth and depth of the foundation for the

physical basis of the consilient mechanism of

the inverse temperature transition. The studies

discussed in section 5.7.8 preceded the

microwave dielectric relaxation studies and

were quite conclusive in their own right, but

nothing is so convincing as identifying the

hydrophobic hydration and directly observing

the loss of hydrophobic hydration as the

charged species form. They also point to more

explicit descriptions and subtleties such as the

basis for efficiency and cooperativity. We begin

exploring these below.

192

Consilient Mechanisms for Diverse Protein-based Machines

5.7.9 Primary Source of pKa Shifts in

Model Proteins i Through v:

Competition for Hydration Between

Hydrophobic and Charged Groups

In section 5.1.3.4, we derived expressions for

the Gibbs free energy of hydrophobic associa-

tion, AGHA- In this section 5.7, hydrophobic-

induced pKa shifts were shown to arise from a

competition for hydration between the polar

(ionized) and apolar (hydrophobic) groups.

Accordingly, the pKa shifts provide an oppor-

tunity to express the Gibbs free energy repre-

sented by this competition for hydration, AGap

and to compare it to the

AGHA

determined

under similar circumstances and to Monod's

"second secret of life" embodied in allostery-

based positive cooperativity.^^'* In our case,

the allostery involves the two conformational

states,

hydrophobically associated and hydro-

phobically dissociated.

5.7.9.7

Generally Considered Physical

Bases for pKa Shifts in Polymers

The three generally considered sources for pKa

shifts are (1) the electrostatic-based charge-

charge repulsion, (2) the ionizable group being

forced into an environment of low dielectric

constant where it cannot ionize, and (3) the per-

spective developed by the systematic series of

model proteins considered here of a competi-

tion for hydration between hydrophobic and

charged groups constrained by sequence to

coexist along a chain molecule such as a protein.

5.7.9.1.1 Electrostatic Charge-Charge

Repulsion

As seen for the series poly[/v(GVGIP),

/E(GEGIP)] in Figure 5.30, when fn = 1, a small

charge-charge repulsion occurs with four

residues separating the ionizable carboxylate-

containing side chain of the Glu (E) residues.

The addition of 0.15 N NaCl removes that

repulsion. When f^ = 0.17, as in Model Proteins

i through v of Table 5.5, with 29 residues sepa-

rating charged residues, charge-charge repul-

sion is no longer a tenable explanation. This is

verified by the observed cooperativity, which is

negative for charge-charge repulsion, as shown

in Figure 5.35 below for poly(methacrylic acid)

with a carboxyl on every other backbone atom.

On the other hand. Model Proteins i through v

exhibit a positive cooperativity that increases

with increased replacement of less hydrophobic

Val (V) residues with more hydrophobic Phe

(F) residues.

5.7.9.1.2 Local Environment of Low

Dielectric Constant

The classic argument for pKa shifts in proteins

is the claim that the energy of a folded state is

so favorable that it can force the ionizable

species into the energetically unfavorable cir-

cumstance of a local environment of low die-

lectric constant. This issue was considered in

some detail and rejected by Urry et al.^^ in

connection with the series poly[/v(GVGIP),

/K(GKGIP)] for which the data are given in

Figure 5.30.

5.7.9.1.3 Competition for Hydration Between

Charged and Hydrophobic Groups

The conclusion that this is the mechanism

for the hydrophobic-induced pKa shifts of

the model proteins of consideration here was

extensively developed earlier in this section 5.7.

In what follows, this mechanism is explored

further.

5.7.9.2 Derivation of

AGap,

the Change

in Gibbs Free Energy Due to the

Apolar-Polar Repulsive Free Energy

of Hydration

Derivation of AGap, the change in Gibbs free

energy due to an apolar-polar repulsive free

energy of hydration, begins with the expression

for chemical potential of acid (proton, H^),

|IH

RTln(a), where a is the activity of proton. Under

the normal low concentrations for acid, the

activity coefficient is 1, and a can be replaced by

the concentration of the acid, [ff], to give

|IH

RTln[H^].

Because acid-base titrations are

plotted using a log scale,

|LIH

= 2.3RTlog[H^] and

as pH = -log[H'^], the chemical potential of

proton becomes,

|IH

= -2.3RTpH. The chemical

potential for the value at which the pH = pKa is

simply |IH = -2.3RTpKa. Now the change in

5.7 What Is the Physical Basis for the Consilient Mechanism of Energy Conversion?

193

chemical potential, A|LI, resulting from the

hydrophobic-induced pKa shift can be written

as A|i2,i = ^2 - 1^1 = -2.3RT(pKa2 - pKai) =

-2.3RTApKa. Finally, we note that the ApKa

arises from the competition for hydration

between charged (polar) and hydrophobic

(apolar) groups, that

is,

competition resulting in

repulsion arises on formation of the carboxylate,

-COO",

which parallels formation of hydroxyl

ion. As we would like to express the change in

Gibbs free energy per mole resulting from the

apolar-polar repulsion as each species reaches

out for water unperturbed by the other, this has

the effect of sign reversal, such that we can write

A|IH

= -A|LioH = -AGap, that is.

AGap = 2.3RT ApKa

(5.13)

where the subscript ap stands for apolar-polar

repulsion. It will be of interest to see how this

quantity relates to

AGRA

and to positive coop-

erativity and charge-charge repulsion in the

treatment of acid-base titration data that con-

sider both interactions in section 5.8. Before

doing so, however, we note another experi-

mental demonstration of AGap, but in this case

there occurs delineation between the compo-

nent that results from sequence without the

capacity to form a conformation to reUeve

apolar-polar repulsion and that component

resulting from lowering repulsion by hydropho-

bic association and charge neutralization.

5.7.9.3 Hill Plots Delineate AGap Due

Solely to Sequence and That Associated

with Allosteric Effects (Due to Formation

of a Hydrophobically Associated State)

Analysis of the experimental data in Figure 5.31

allows fundamental delineation between

hydrophobic-induced pKa shifts in the absence

of the ordering due to hydrophobic association

and hydrophobic-induced pKa shifts associated

with positive cooperativity and the presence of

a more-structured hydrophobically associated

state.

The analysis is consistent throughout the

series of Model Protein I through v in Table 5.5.

Sufficient for our analysis here is consideration

of the acid-base titrations of two model pro-

teins,

namely,

A 100

1.0

0.1

0.01

B 100

1.0

0.1

0.01

1 ' '

1 pKa for

r weakest

binding

1—1 1 1—7

/' / n=l -

a' /

(no

cooperativity)

/ I /titration slope

/ 1 1

pKa for

tightest J

binding

1 1

6 7

pKa for

weakest

binding

O/'

/ P''

T 1 ry— —1

/'/ n=l J

o / (no cooperativity)

'/ /

/'//titration slope

/i^O-1n=1.9)

1 pKg for

] tightest -J

1 binding

1 1 1 1 1 1

FIGURE 5.31. Hill plots, log[a(l - a)] vs. pH, for

Model Proteins iii and v. The normal pK for

E(COOH) is ~4. Model Proteins iii and v exhibit

their first COOH pKa values at 5.53 and 7.0, respec-

tively, and their last COOH pKa values at 4.84 and

5.7 for the 39th and 42nd COOH pKa values,

respectively. The pKa values of completely unfolded

model proteins exhibit residual shifts due to compe-

tition for hydration between oil-like and vinegar-

like groups. Repulsions of 1.2 (=0.84 x 2.3RT) and

2.4 (=1.7 X 2.3RT) kcal/mole carboxyl remain in

the completely hydrophobically unfolded model

protein.

194

Consilient Mechanisms for Diverse Protein-based Machines

2 Phc/30 mcr

(proximal)

3 Phc/30 mcr

(proximal)

4 Phc/30 mcr

(proximal)

5 Phc/30 mcr

(proximal)

5 Phc/30 mcr

(distal)

Asp

C31u

Pblymcrll

ApKa=0.4

PblymcrVn

ApKa=0.3

Pblymerm Polymer IV

ApKa=0.7 ApKa=1.4

Polymer VEI Polymer IX

ApKa=0.6 ApKa=l.l

Q Aspartic Acid (D) or Glutamic Acid (E)

Polymer V

ApKa=6.1

Polymer X

ApKa=3.8

^ Phcn^dalatiinc (F)

Polymer XI

ApKa=2.9

PblymcrXn

ApKa=3.5

FIGURE 5.32. As indicated below the schematic (3-

spiral representations of the structures, stepwise

increases in hydrophobicity as Phe replaces Val

result in suprahnear increases in carboxyl pKa values

in studies of chemically synthesized polymers in

analogy to the hydrophobic-induced pKa shifts in

Figures 5.29, 5.30, and 5.31. (Reproduced with per-

mission from Urry et

al.'^^).

Model Protein iii: (GVGVP GVGVP GEGVP GVGVP GVGFP GFGFP)39(GVGVP) E/3F

Model Protein v: (GVGVP GVGFP GEGFP GVGVP GVGFP GFGFP)32(GVGVP) E/5F

The Hill plots in Figure 5.31 provide remark-

able insight into the hydrophobic-induced pKa

shifts.

Starting in the hydrophobically associ-

ated state for Model Protein v, on decreasing

acid concentration, that is, raising the pH, the

pKa of the first carboxyl to form carboxylate is

7.0. In Figure 5.31 A, this is the pKa for the most

tightly bound proton. As more carboxylates

form, further ionization becomes easier, and

finally the last carboxyl to ionize does so with a

pKa of 5.7. Remarkably, the last carboxyl to

ionize does so with a pKa shifted 1.7 pH units

from that of the unperturbed Glu pKa of about

Even in the completely unfolded state there

remains an apolar-polar repulsion of 2.4 (= 1.7

2.3RT) kcal/mole-Glu.^^^

Those who cannot abandon the concept of

the carboxyls being bound within a local envi-

ronment of low dielectric constant may still

wish to argue that the carboxyl is held within a

low dielectric constant provided by the most

proximal Phe (F) side chains. Thus it becomes

of interest to remove the two most sequence

proximal Phe (F) residues. This is achieved with

Model Protein iii, and the experimental results

are shown in Figure 5.31B. Although the mag-

nitude of the pKa shift is reduced, the conun-

drum remains. The last carboxyl to ionize does

so with a pKa shifted 0.84 pH units from that

of the unperturbed Glu pKa of about 4. Here

there remains a repulsion of 1.2 (= 0.84 x

2.3RT) kcal/mole-Glu even when the more

hydrophobic Phe (F) residues are at a distance

of some 11 residues along the sequence. It

seems inescapable that apolar-polar repulsion,

the effort of the charged and hydrophobic

residues to reach water unperturbed by the

other, reaches out several nanometers. More

detailed scenarios follow.

5.7,9.4 Scenario for pKa Shifts Coupled

with Positive Cooperativity: Starting from

the Hydrophobically Associated State

During an opening fluctuation of a pair of asso-

ciated hydrophobic domains, the formation of