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

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5.1 Introduction

105

FIGURE

5.1.

Characterizations of

phase separation,

that of the inverse temperature transition, for the

elastic-contractile model protein, (GVGVP)25i.(A) A

series of tubes show a clear solution of (GVGVP)25i

in water at temperatures below 25°C (left), a solu-

tion that became cloudy on raising the temperature

above 25°C (middle), and the phase separation that

occurs on standing (right). (B) Temperature profile

for the onset of turbidity. Starting below 25°C with

a clear solution containing at least 40mg/ml of

(GVGVP)25i in water placed in a spectrophotometer,

the temperature dependence for the onset of turbid-

ity is accurately

measured.

The

temperature at which

half-maximal turbidity occurs is defined as Tj. (C)

Differential scanning calorimetry curve of 40mg/ml

of (GVGVP)25i in water. The trace shows the heat

absorbed by the sample as the temperature is raised

through the temperature interval of the inverse tem-

perature transition. The area contained within the

peak gives the heat of the transition, and the heat

divided by the temperature, integrated over the tran-

sition, gives the entropy of the transition. (Adapted

with permission from Urry."^)

clear

solution

cloudy

solution

glue-like

phase

separates

out

(GVGVP)25i in 40 mg/ml of H2O

20 40 60 80 100

Temperature, °C

A quantitative measure of the decrease in

order for each of these phase transitions is the

entropy for the transition, ASt, which is simply

the heat of the transition, AHt, divided by the

temperature in degrees Kelvin (K) at which

ice ASt(ice -^ water) = 1,440 cal/mole/273°K =

-1-5.3

cal/mole-degree (EU), and for the vapor-

ization of water ASt(water -> vapor) =

9,720

cal/mole/373° K =

-i-26

cal/mole-degree =

-1-26

EU where

cal/mole-degree K is 1 entropy

the transition occurred. For the melting of unit (EU).

106

Consilient Mechanisms for Diverse Protein-based Machines

Schematic Representation of Transitions of the

poly(GVGVP)-water system

/u-^

60-

50-

40-

30-

20-

10-

-2

3 ASt(soln-^hase separated) = AH/Tt

E +1.2/303%+4.0 EU; Factor 409/18 = 23

(DO 1 ,_

-1 .i

A « 1

°-- IS 2

Bo 2 <D

2 -a 3 ^1

1 >3 -c

/ c ^

1 -

1 k_

•D

.5"

Jl :: ^Jq

T 1 1 1 1 r

0 20 40 60 80 IOC

)

FIGURE 5.2. The four phase transitions of

the model protein (GVGVP)25i in water

over the temperature range from -20° to

120°C. The familiar transition of the

melting of ice and the vaporization of

water are shown with the relative magni-

tudes of the heats of these transitions to

those of protein heat denaturation and to

the innocuous looking inverse temperature

transition near 30°C that we believe to be

the basis of the function of protein-based

machines of Life. See text for discussion.

Temperature °C

5.1.2.1.2 An Increase in Entropy Is a

Decrease in Order

During the melting of ice, water molecules go

from being precisely positioned, one molecule

with respect to the other, in the well-known

crystal structure of ice, to still being geometri-

cally constrained to associate but able to shift

positions in the liquid state. Melting of ice con-

stitutes a decrease in order that is measured

quantitatively as an increase in entropy, that is,

+5.3

cal/mole-degree.

The decrease in order is about five times

greater, however, for vaporization than for

melting; 26/5.3 ~ 5. This is understandable

because, in water, though quite mobile and able

to change positions, the molecules are still con-

strained to associate using geometries dictated

by their shapes and hydrogen bonding of one

water molecule to surrounding water mole-

cules.

On the other hand, once vaporized and in

the gas phase, individual water molecules are

no longer in contact. Instead, they are free to

fly around completely independent of each

other. These changes from states of more order

to states of less order (ice to liquid water and

liquid water to water molecules in the gas

phase) are positive entropy changes (increases

in entropy). A negative entropy change (a

decrease in entropy) on going from one state to

another indicates a change from a state of less

order to a state of more order.

Thus,

the reverse

processes (of water molecules in the air con-

densing on a cold window pane, or, if it is really

cold outside, of water vapor forming frost [ice

crystals] on the window pane) represent

decreases in entropy.

These phase changes, very familiar to our

daily experience, make real the thermodynamic

quantities, AH and AS, that are fundamental to

understanding how protein-based machines

function. With knowledge of heat changes, AH,

and entropy changes, AS, we unmask how

model protein-based machines access energies

in the environment and convert them to those

energies essential for biological function. The

phase transitions of the melting of ice and

the vaporization of liquid water in terms of the

thermodynamic quantities of heat change (also

called change in heat content or enthalpy) and

entropy change bring to life the energy conver-

sions that sustain Life.

5.1 Introduction 107

5.1.2.2 Phase Transitions of Proteins

5.1.2.2.1 Heat Denaturation of Protein

Along the temperature scale shown in Figure

5.2, sandwiched between commonplace phase

changes of the melting of ice at low tempera-

ture and the vaporization of liquid water at

high temperature, are transitions exhibited by

proteins in water. The higher temperature

transition, shown near 70° C, represents the

denaturation of proteins. Below the denatura-

tion transition temperature, the protein has a

regular structure, where each amino acid

residue of the protein chain occurs in a defined

space. Above the denaturation transition tem-

perature, the protein can be described as a dis-

ordered, irregular, and randomized tangle of

chains. The location in space of a given amino

acid residue in one chain with respect to other

amino acids in the chain is different for each

protein chain. During cooking (heating) of an

egg, the proteins of the clear gel, surrounding

the yellow yolk, become disordered, more

sohdified, and white on heat denaturation.

5.1.2.2.2 Low Heat of the Intermediate

Transition at Lower Temperature

The lower temperature transition, occurring

near 30° C for our basic elastic-contractile

model protein, is difficult to detect, as plotted,

because of the relatively small amount of heat

involved in this transition. The representations

in Figure 5.2 provide a visual bridge from the

more familiar transitions of water, considered

above, to the less familiar transitions central

to the function of our model protein-based

machines. Figure 5.2 is a composite of the tran-

sitions for pure or nearly pure water with the

curve for poly(GVGVP) with the relative areas

of the curves representing the circumstance for

1 gram of protein dissolved in 1 gram of water.

The relative areas are plotted for equal weights

of water and model protein, yet the differences

in the areas of the peaks, which measure the

relative heats of the transitions, are dramatic.

This seemingly innocuous transition, shown

here peaking near

30"^

C for poly(GVGVP), is

the source of the consilient mechanism for

energy conversion. We believe this inconspicu-

ous transition represents the primary physical

process enabling the energy conversions that

sustain Life.

5.1.2.2.3 Heat Maturation of Protein

For our model contractile protein,

poly(GVGVP), AHt = -hl.2kcal/mole-pentamer

and ASt(solution -^ phase separated) = +1.2/303

kcal/mole-degree K ~ +4.0cal/mole-pentamer-

degree (EU).'''' Because 1 mole of (GVGVP)

is 409 grams and 1 mole of water is 18 grams,

on a per gram basis (4.0/409 = 0.0098 and

5.3/18

= 0.29) the entropy change for the melting of

ice is 30 times greater (0.29/0.0098) than that of

the transition for poly (GVGVP).

Despite the absorption of heat for the tran-

sition and the overall increase in entropy of

+4.0

EU for the water plus protein, the protein

component actually increases in order on

raising the temperature. As unambiguously

demonstrated by crystallization of a cyclic

analog (see Figure 2.7), in this case the protein

component of the water plus protein system

becomes more ordered as the temperature is

raised. For this and additional reasons,

noted below in section 5.1.3, we call this tran-

sition exhibited by our model protein, poly

(GVGVP), an inverse temperature transition.

5.1.2.2.4 The Inconspicuous Inverse

Temperature Transition Is Equivalent to Cold

Denaturation of Protein

In the early 1900s, it was recognized that low-

ering the temperature could cause enzymes to

lose their catalytic activity due to an unfold-

ing and/or disassembly of proteins and their

subunits.^^ This is called cold denaturation.^^

Similarly with our model protein, lowering the

temperature from above to below the tempera-

ture interval of the inverse temperature transi-

tion causes the model protein to unfold and

disassemble, that is, to dissolve. In the case

of our model protein, the reverse process of

raising the temperature from below to above

the temperature of the inverse temperature

transition causes the massive, visible changes

in solubility represented in Figure 5.1A. The

phase diagram, discussed immediately below,

provides systematic characterization of this

108

Consilient Mechanisms for Diverse Protein-based Machines

phase separation. The phase separation analy-

sis,

however, accommodates many other

energy inputs in addition to thermal energy,

because many other energy inputs drive phase

separation.

5.1.3 Phase Diagram for Inverse

Temperature Transitions and

Related Analyses

5.13.1 The Inverse Temperature

Transition of Model Proteins

The model proteins of interest here are poly-

mers of a distinctive family of repeating peptide

sequences where the repeating unit may be

as few as two residues or as many as several

hundred residues. These elastic-contractile

model proteins exhibit remarkably functional

and distinctive reversible phase transitional

behavior to increased polypeptide order on

raising the temperature and on this basis was

called an inverse temperature transition.^^ This

inverse order-disorder transition demonstrable

by elastic-contractile model proteins is the basis

for many different energy conversions, includ-

ing, we believe, the energy conversions existing

in biology."^'^

5.1.3.2 Inverted Phase Transitional

Behavior of Inverse Temperature

Transitions

In general, solubility in a solvent, of whatever

molecular species, increases with heating. The

solubility of petroleum-based polymers in

organic solvents and most petroleum-based

polymers in water represent such cases. In

general, solubility of the polymers increases on

raising the temperature. For the case of the

amount of solute within a solvent, phase dia-

grams graphically represent the change in sol-

ubility as a function of temperature. In the case

of the phase diagram, the boundary between

insolubiUty and solubiUty is called the binodal

or coexistence

line.^^

At the temperature of the

coexistence line soluble and insoluble states

coexist. In general, insolubility occurs at lower

temperatures, and solubility takes place on in-

creasing the temperature.

5.1.3.2.1 Our Model Proteins Become

Insoluble on Increasing the Temperature

As demonstrated in Figure 5.3 for several

model proteins, essentially unUmited solubiUty

occurs at low temperature, and phase separa-

tion (insolubility) occurs as the temperature is

raised. Also, for our model protein composi-

tions,^^'^^

the curvature of the coexistence line is

inverted, having the shape of a valley instead of

a smooth mountain peak. Because of this we

call the phase transition, exhibited by elastic-

contractile model proteins, an inverse tempera-

ture transition. Even more compelling reasons

exist for the inverse temperature transition label.

5.1.3.2.2 Our Model Proteins Increase Order

on Raising the Temperature

A vital property of these model proteins is that

they are more ordered above the transition

temperature defined by the binodal or coexis-

tence line in Figure 5.3. The polymer compo-

nent of this water-polypeptide system becomes

more ordered or structured on increased tem-

perature from below to above the transition.

This behavior is the inverse of that observed for

most systems, as discussed above. In particular,

we developed the term inverse temperature

transition when the precursor protein and

chemical fragmentation products of the mam-

malian elastic fiber changed from a dissolved

state,

and therefore when molecules were ran-

domly dispersed in solution, to a state of paral-

lel-aligned twisted filaments as the temperature

was raised from below to above the phase

transition.^^'^^'^^

Chemically synthesized high molecular

weight polymers of repeating sequences found

within the elastic fiber similarly form parallel-

ahgned twisted filaments (see Figure 5.4C)

when the temperature is raised from below to

above the coexistence line. Most definitively,

cyclic analogues of the repeating peptide

sequences crystallize when the temperature is

raised above the transition temperature (see,

e.g.. Figure 2.5 and Figure 5.4A,B) and dissolve

when the temperature is lowered below the

transition.^^'^^

At first thought, this appears to contradict

the second law of thermodynamics, which states

5.1 Introduction 109

FIGURE 5.3. Phase diagram for several

elastic-contractile model proteins,

showing an inverted curvature to the

binodal or coexistence line (when com-

pared with petroleum-based polymers)

that is equivalent to the Tfdivide, with

the value of Tt determined as noted in

Figure 5.IB. Solubility is also inverted

with insolubility above and solubility

below the binodal line, that is, solubil-

ity is lost on raising the temperature

whereas solubility is achieved by

raising the temperature of most petro-

leum-based polymers in their solvents.

Note that addition of a CH2 group

lowers the Tfdivide and removal of

the CH2 group raises the Tfdivide.

For these and the additional reason of

increased ordering on increasing the

temperature, the phase transitions of

elastic-contractile model proteins are

called inverse temperature transitions.

(The curve for poly[GVGVP] is

adapted with permission from Manno

et al.^^ and Sciortino et al.^°).

°C

Operating-

temperature

poly[0.8(GVGVP),0.2(GEGVP)]

in PBS

4 COO"/100 residues

less hydrophobic

(hydration)

more hydrophobic

(hydration)

-I

I I I I I

binodal or

coexistence

line = Tt-divide

1 I I I

poly(GAGVP)

(-2C1^)

poly(GVGVP)

poly(GVGIP)

(+CH2)

.2 .4

volume fraction

that the order of a system must decrease as the

temperature is as raised, with the melting and

vaporization transitions of water shown in

Figure

5.2.

This seeming violation of the second

law of thermodynamics disappears on recog-

nizing the presence of ordered water that sur-

rounds oil-like side chains of the dissolved, low

temperature state.^^'^^ When the temperature is

raised, this more-ordered water surrounding

oil-like groups becomes less-ordered liquid or

bulk water, and thereby provides the measured

increase in entropy (see section 5.1.2.2) that

drives the thermally induced phase separation.

5.1.3.3 A Glimpse at the Underlying

Process Involving Water around

Oil-like Groups

5.1.3.3.1 Effect of Adding CH2 Units on

Solubility in Water

Oil is composed mostly of -CH2 and -CH3

groups. When the small single oil-like group is

attached to an OH, as in methanol, CH3-OH,

the methanol is infinitely soluble in water. For

the series of CH3-OH (methanol),

CH3-CH2-OH (ethanol, the alcohol of the

beverage variety), CH3-CH2-CH2-OH (1-

propanol), CH3-CH2-CH2-CH2-OH (1-

butanol), CH3-CH2-CH2-CH2-CH2-OH

(1-pentanol), the amount of the alcohol dis-

solvable in water decreases as each oil-like CH2

unit is added. In fact,

1-octanol,

CH3-CH2-

CH2-CH2-CH2-CH2-CH2-CH2-OH, is insolu-

ble in water, and the long chain,

CH3-CH2-CH2-CH2-CH2-CH2-CH2-CH2-, is

properly called water fearing or hydrophobic.

Until the loss of solubility comes about,

however, there is water surrounding the oil-like

groups. For example, water surrounds the

soluble oil-like butyl grouping, CH3-CH2-

CH2-CH2- when it is in water as

1-butanol.

Although it may seem to be a contradiction in

terms,

this water that surrounds oil-like groups

dissolved in water is called hydrophobic hydra-

tion.

Significantly, the thermodynamic quanti-

ties of AH and AS describe changes in

hydrophobic hydration. As discussed below,

the formation of hydrophobic hydration occurs

FIGURE

5.4. The products of the inverse temperature

transition of ehistic-contractile model proteins and

their analogues are self-assembled (ordered) struc-

tures.

(A) The cyclic analogue cyclo(GVGVAP)2

crystallizes on raising the temperature through the

inverse temperature and dissolves on lowering the

temperature below the transition. (Adapted with

permission from Urry et al.^^) (B) The crystal struc-

ture of the cyclic analogue cyclo(GVGVP)3, which

also aggregates on raising the temperature and dis-

solves on lowering the temperature. (Reproduced

with permission from Cook et al.^"^) (C) Electron

micrograph of incipient aggregates of poly(GVGVP)

after negative staining with uranyl acetate/oxalic

acid showing parallel aligned,

nm diameter, twisted

filaments that give the optical diffraction pattern

shown in the inset. The twisted filaments are associ-

ated into a fibril of about ().15)im in cross section.

(Reproduced with permission from Volpin et al.""^)

(D) Light micrograph, without any staining or fixa-

tive,

showing aggregated state of poly(GVGVP) that

has been chemically cross-linked above the temper-

ature of the inverse temperature traansition due to

the occasional inclusion of Lys and Glu residues.

Fibers are seen to be composed of fibrils of a cross

section similar to the fibril of part C. (Reproduced

with permission from Urry.^^^) (E,F) Scanning elec-

tron micrographs at two different magnifications of

a single fiber splaying out into parallel aligned fibrils

and re-coalesced back into the same-sized fiber.

(Reproduced with permission from Urry et aV^^)

5.1 Introduction

111

Contracted 100 —

>•

Relaxed 0 —

lower poorly

responsive range

unfolded and/or disassembled

(swollen) state

temperature

Interval

upper poorly

responsive range

folded and/or assembled

(contracted) state

Temperature

ON 100 H

OFF

lower poorly

responsive zone

unfolded and/or disassembled

(swollen) state

transition

zone

upper poorly

responsive zone

folded and/or assembled

(contracted) state

Independent Variable

(temperature, chemical potential, electrochemical potential, pressure, electromagnetic radiation, etc.)

FIGURE

5.5.

Transitions,

plotted as independent vari-

able versus dependent variable, showing a response

limited to a particular range of independent variable.

(A) Representation of the thermally driven contrac-

tion for an elastic-contractile model protein, such

as the cross-linked poly(GVGVP), plotted as the

percent contraction (dependent variable) versus

temperature (independent

variable).

The plot shows

a poorly responsive range below the onset of the

transition, the temperature interval of the inverse

temperature transition for hydrophobic association,

and another poorly responsive region above the tem-

perature range of the transition. (B) Generalized

representation of transitions where any of the inten-

sive (independent) variables of energies inter-

converted by designed elastic-contractile model

proteins, such as temperature for thermal energy,

chemical potential for chemical energy, electro-

chemical potential for electrical energy, and mechan-

ical force for mechanical work, are plotted versus

the normalized dependent variable. Again, there are

the responsive

transition

zone and poorly responsive

ranges above and below the transition zone.

with a negative AH and a negative

AS.

The signs

of AH and AS for the formation of hydropho-

bic hydration and the way they enter into the

expression governing solubility, of course,

determine solubility or insolubility of oil-like

groups and, thereby, provide the basis for

energy conversions by the consilient

mechanism.

112

Consilient Mechanisms for Diverse Protein-based Machines

5.1.3.3.2 Thermodynamic Expression for

Solubility: The Gibbs Free Energy

(AG = AH - TAS)

The thermodynamic quantity that determines

solubiHty is the Gibbs free energy, AG It is

made up of two terms, AH, the change in heat

on dissolution, and -TAS where T is the tem-

perature in degrees Kelvin (K) and AS quanti-

fies the change in order attending dissolution.

(These are the same thermodynamic quantities

discussed above in relation to the phase

changes exhibited by water and by proteins in

water.) Accordingly, AG(solubility) = AH -TAS.

When AG(solubility) for dissolution is negative,

solubility occurs. As AG(solubility) becomes

positive, solubility decreases.

In general, substances that form ions on dis-

solving in water evolve much heat; AH is nega-

tive.

An example is caustic soda, NaOH. When

water is carefully added to a test tube contain-

ing pellets of NaOH, they dissolve as Na^ and

OH" ions with the production of much heat.

The test tube becomes hot to the touch. In

general, hydration of ions is an exothermic

reaction. The same happens for glacial acetic

acid (the major component of vinegar,

CH3-COOH, at a very high concentration).

Glacial acetic acid must be added slowly and

very carefully to water, because it dissolves as

H^ and CH3-COO" with the release of so much

heat. AH for dissolution is large and negative.

The full hydration of ions in water is a very

exothermic, that is, very favorable, process.

5.1.3.3.3 The Thermodynamic Properties of

Hydrophobic Hydration

Remarkably, in the above series of alcohols,

addition of an oil-Uke CH2 unit increases the

release of heat when dissolved in water. Dis-

solving CH3-CH2-CH2-CH2-OH in water

releases more heat per mole added than does

dissolving CH3-CH2-CH2-OH in water. The

dissolution of these oil-like CH2 units in water is

exothermic. Therefore, formation of hydropho-

bic hydration is exothermic. Nonetheless, solu-

bility decreases as the number of oil-like CH2

units increases, despite a continuing favorable

release of heat on dissolution. Now we must

remember that AG(solubility) = AH - TAS is

the expression that governs solubility; and

solubility occurs when AG is negative and

decreases as AG becomes positive. Therefore,

the loss of solubility, as the number of oil-like

CH2 units increases, is due to the sign and mag-

nitude of the (-TAS) term. Based on the data

of Butler,^^ the average change in the AH and

(-TAS) terms for each of the four CH2 units

added on going from CH3-OH to

CH3-CH2-CH2-CH2-CH2-OH is a favorable

-1.4kcal/mole-CH2 for AH and an unfavorable

-hl.7kcal/mole-CH2 for (-TAS).Thus, insolubil-

ity ultimately happens when enough CH2 units

are added due to the larger and positive (-TAS)

term. It measures the contribution to the Gibbs

free energy due to an increase in order of water

as it goes from being bulk water to being

hydrophobic hydration. This ordering effect is

so pronounced that Frank and Evans^^ referred

to hydrophobic hydration as "icebergs" sur-

rounding oil-like groups.

5.1.3.3.4 The Inverse Temperature

Transition: A Hydrophobic Folding and

Assembly Transition

The protein-based polymer is soluble in water

at temperatures below its coexistence line

where the hydrophobic residues are sur-

rounded by hydrophobic hydration. As the

positive (-TAS) term due to hydrophobic

hydration becomes larger than the negative AH

term, simply due to increasing the value of T,

solubility of a protein-based polymer is lost,

and it hydrophobically folds and assembles. The

inverse temperature transition is a hydrophobic

association transition.

5.1.3.4 Capacity to Move the T^divide

as a Measure of Energy Available to the

Consilient Mechanism (That Is, Moving

the Temperature Interval)

5.1.3.4.1 Defining the Tt-divide

At the coexistence line of a phase transition,

such as that plotted in Figure 5.3, soluble and

5.1 Introduction 113

insoluble states coexist. Heat added at this tem-

perature, for an ideal case like ice dissolving in

water, converts one state to the other rather

than increases the temperature. At the temper-

ature of the phase transition when there are

two phases, the Gibbs free energy per mole of

the molecules in the two phases are the same,

that is, the chemical potential,

|Li

= AG/mole, is

the same for the molecules in solution and for

the molecules in the phase-separated state. For

the inverse temperature transition, the chemi-

cal potential for molecules in solution, that is,

for molecules in the hydrophobically dissoci-

ated state,

\iHD,

is the same as the chemical

potential of the molecules in the separated,

hydrophobically associated phase, IIHA, that is,

\iHD

\IHA'

This means for the transition that

AGt = 0 = AHt - TtASt, where the subscript t

stands for the phase transition. Therefore, AHt

= TtASt, and Tt = AHt/ASt, where Tt is the tem-

perature for the transition. Accordingly, the

coexistence line may also be called the Tr

divide between soluble and insoluble states (see

Figure 5.3).^^ Importantly for the transition,

AGt = 0 = AHt - TfASf at the Tf-divide and

AHt = TASt.

5.1.3.4.2 Dependence of Tt on Chain Length

and on Concentration

Meaningful comparison of the different means

whereby the value of Tt may be moved requires

definition of a common polymer chain length

and concentration to function as a reference

state.^^'^^'^^'^^

Definition of the reference condi-

tions occurred nearly two decades ago with

chemically synthesized polymers,^^ before the

following systematic comparisons began. In

particular, chemically synthesized polymers

dialyzed against 50,000 Da cut-off dialysis

membranes^^ at 4°C (i.e., a temperature sub-

stantially below Tt) exhibit a lowering of Tt on

increasing the concentration until a concentra-

tion of 40mg/ml was reached. Increasing the

concentration further had little effect on the

value of Tt such that this became the concen-

tration for comparison. This was extensively

substantiated in the process of developing the

phase diagrams for poly(VPGVG),^^'^^'^^ as the

phase diagram is a plot of temperature versus

concentration.

The chain length dependence was also deter-

mined on chemically synthesized polymers

using dialysis membranes. ^^ Subsequently,

protein-based polymers using recombinant

DNA technology were prepared with 41, 141,

and

251

repeats of the pentamer.^^ Again, above

100,000 Da or over 200 repeats of the pentamer,

the chain length dependence of Tt becomes

limited. Accordingly using polymers of approx-

imately 100,000 Da at concentrations of

40mg/ml have become the reference conditions

for the comparisons noted below.

5.1.3.4.3 Moving the Tt-divide by Adding a

CH2 Unit, That Is, Replacing Val by He (an

Approximation to the Change in Gibbs Free

Energy for Hydrophobic Association,

AGHA)

From the data of Butler,^^ we can predict how

the Tt-divide will change as CH2 units are added

or removed from the polymers plotted in

Figure 5.3. If we add a CH2 group to the R-

group of the second Val residue of (GVGVP)n,

it becomes (GVGIP)n, that is, the R-group for

the V residue, -CH(CH3)2, becomes

-CH(CH3)-CH2-CH3 for an I residue. Because

AHt = TtASt for both compositions at their

respective Tt-divides, the relevant equations

become

AHt(GVGVP) = Tt(GVGVP)ASt(GVGVP)

(5.1)

and

AH,(GVGIP) = T,(GVGIP)AS,(GVGIP).

(5.2)

Due to the experimental work of Butler and for

reasons of the additive nature of the compo-

nents within AH and of the components that

make up

AS,

we separate the effect of the addi-

tion of a single CH2 group and write that

AHt(GVGIP) = AHt(GVGVP) +AHt(CH2)

(5.3)

and

114

Consilient Mechanisms

for

Diverse Protein-based Machines

(GVGIP)

AS,

(GVGIP)

(GVGIP)[

AS,

(GVGVP) +

AS,(CH2)]

(5.4)

In general, therefore,

AGHA

(X)

= [T. (ref)

(z)]AS,

(ref)

(5.8)

Substitution

Equations (5.3)

and

(5.4) into

Equation

(5.2) and

subtracting Equation

(5.1)

gives

AH,(CH2)-T,(GVGIP)AS,(CH2)

[T,

(GVGIP)

(GVGVP)]

AS,

(GVGVP)

(5.5)

Again,

if the

data

Butler^^

for the

effect

addition

of a CH2

group

solubiUty

are

apphcable,

the

hydrophobic association

of the

phase separation would

such that

the

mag-

nitude

of the

Tt(GVGIP)ASt(CH2) term would

be larger than that

of the

AHt(CH2) term,

and

for hydrophobic association, that

is, for

insolu-

bilization, ASt(GVGVP)

positive

ordered hydrophobic hydration becomes less

ordered bulk water

hydrophobic associa-

tion. This means that Tt(GVGIP)ASt(CH2)

AHt(CH2), such that

(GVGVP) >

(GVGIP) (5.6)

This

found experimentally

for the

phase sep-

aration

these model proteins. Addition

of a

CH2 group lowers

the

value

Tt, that

is,

lowers

the Tfdivide

for

(GVGIP)n

shown

Figure

5.3.

The

experimental estimates

of Tt are

nominally 25°

C for

(GVGVP)^

and

10°

C for

(GVGIP)n

keeping with Equation

(5.6).

Also,

removal

two CH2 units,

replacing

the

first

V residue

(GVGVP) with

alanine residue,

A, having

R-group

-CH3, results

in a Tt

value

45°

As expected from Equation (5.6)

and from

the

data

Butler, Tt(GVGVP)

Tt(GAGVP). This simple relationship between

hydrophobicity

and Tt

provides insight into

the

Tfbased hydrophobicity scale

for

protein func-

tion

and

engineering, developed

section

5.3

below.

Returning

Equation (5.5),

define^^

-AGHA(CH2)

AHt(CH2)

Tt(GVGIP)ASt(CH2)

and rewrite Equation

(5.5) to

give

AGHA

(CH2)

[Tt

(GVGVP)

(GVGIP)]

ASt

(GVGVP)

(5.7)

AGHA

(X)

- - ATt

(x)ASt

(ref) (5.9)

where

is any variable that moves

the

Tt-divide

from that

of the

reference

(ref)

model protein,

and AGHA(Z) is

the

change

Gibbs free energy

for forming

the

hydrophobically associated

state

of the

model protein resulting from that

variable. (Note that

the

change

Gibbs free

energy

written here

for

hydrophobic associa-

tion, that

is, for

insolubilization, which

has the

opposite sign

AG(solubility),

as the

latter

considers

the

reverse process

solubilization

or solubility discussed

section 5.1.3.3.)

The

approximation

Equation

(5.9) is

demon-

strated below

Figure

5.10 in

terms

of the

diagonal straight fine through

the

sigmoid

data points. Within

the

approximation

Equation

(5.9)

ASt(ref) would

30cal/

mole-(GXGVP)-deg. That

is the

approxima-

tion

of the

practical Tj-based hydrophobicity

scale.

Equation (5.9) provides

initial insight into

the relationship between moving

the

Tt-divide

and changes

free energy

hydrophobic

association;

it is of

use when

the

value

ASt(x),

or AHt(x) from which

it is

derived,

is not

avail-

able.

Nonetheless,

the

limitations

Equation

(5.9) become apparent when both ASt(z)

^^^

ASt(ref)

are

known, because reversal

roles

the reference

and

altered state should simply

reverse

the

sign

AGHA(X),

^^^

this

is not the

case

due to the

approximation

Equations

(5.3)

and

(5.4).

As will

shown below

analysis

of the

data

Figure 5.10,

plot

Tt versus AGHA

has

a sigmoid appearance. This means that

the

derivative dTt/dGnA gives

bell-shaped curve

equal

(dSt)"^

and not a

straight line

imphed

Equation

(5.9) and

assumed

by the

Tfbased hydrophobicity scale. Equation (5.9),

however, does appear

to be

meaningful when

staying within

the

proper family

molecular

systems,

for

example,

defined

Figure

5.10

by lines

b (for the

aliphatic side chains),

c (for

the aromatic side chains),

and d (for the

effect

of charged species).