Proc. Nat. Acad. Sci. USAVol. 71, No. 5, pp. 1977-1981, May 1974

Denaturation: An Example of a Catastrophe(protein conformation/optical rotatory dispersion)

JOHN J. KOZAK* AND CRAIG J. BENHAMt* Department of Chemistry, and t Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

Communicated by Stuart A. Rice, December 10, 1973

ABSTRACT We show that a "phase diagram" for theconformational states of proteins can be constructedusing the mathematical theory of singularities of func-tions developed recently by Rent Thom. For proteinssubjected to the disruptive influences of temperature and/or concentration of denaturing agent, this theory can beused as a predictive tool to account for the shape, sense,and changes in the curves that result when the opticalrotatory dispersion is plotted as a function of these twoconstraints. The mathematical model is found to be inaccord with known experimental data on collagen, lyso-zyme, and ribonuclease.

Recently, Rene Thom has classified the singularities associatedwith certain classes of functions (1). The main theoreticalsignificance of Thom's classification is that it allows one todetermine the stable equilibria of gradient systems subjectto a small number of constraints, and to describe how theseequilibria change as the constraints are changed. The use-fulness of Thom's theory goes far beyond pure mathematics,however; in fact, his methods have been applied to the analysisof problems in disciplines as diverse as biology, sociology,economics, and engineering (2-4). In this report, we showthe relevance of Thom's theory to a problem of contemporaryinterest in biochemistry-protein denaturation.Many phenomena may be thought of as governed by a

potential function of some form. The stable states of thesystem, i.e., those states observed to occur, may then beregarded as states for which some (potential) function isminimized. If this function has multiple minima, then morethan one stable state may be accessible to the system. Chang-ing the controlled parameters in an experiment may alterthe form of the governing potential in such a way as to changethe positions, relative heights, or total number of local minima.Thus, the stable states accessible to the system may changein a discontinuous way as the controls are changed smoothly.Observed discontinuous changes in state have been called"catastrophes" (2-4).In most situations, the precise form of the potential func-

tion is not known. However, there are only a few generictypes of potentials. These have been classified completelyby Thom (1) for one or two observed variables and up tofour control variables. All potentials of a given type lead toqualitatively similar sorts of discontinuous behavior. It isbecause of these factors that Thom's theory is of such greatgenerality and usefulness.The intention of this report is to apply Thom's ideas to the

analysis of observed changes in the conformation of a pro-tein when such a system is subjected to the combined in-fluence of controlled temperature and denaturant concentra-tion. This amounts to considering a control space of dimen-sion two, a situation for which there are two generic types

1977

of catastrophes-the fold and the Riemann-Hugoniot cusp.In the next section, we review those aspects of the mathe-matical theory that are relevant to our study of denaturation.Then, in section II we apply the theory of Thom, placingparticular emphasis on the proteins collagen, lysozyme, andribonuclease. Finally, in section III we comment on the gen-eral applicability of the theory, and indicate directions offurther research.The main conclusion of our analysis is that Thom's theory

provides a mathematical framework within which experi-mental data on protein denaturation can be organized andcorrelated. It is to be emphasized at the very outset that theapproach taken here is independent of thermodynamics, andin several important respects (see section III) goes beyondthermodynamics in being able to describe situations notaccessible to thermodynamic analysis and/or measurement.The price one pays for this generality is that, in contrast tothermodynamics, one loses the possibility of providing, ulti-mately, a molecular description of the processes involvedin protein denaturation. Taken together, these remarks sug-gest that further progress in understanding denaturationmight come from a marriage of catastrophe theory withthermodynamics, a situation which could be realized if thepotential function mentioned above were identified with athermodynamic potential. We believe that this identificationmight follow from a correlation of the predictions of catas-trophe theory with the observed behavior on denaturationof a protein for which both spectroscopic and thermodynamicdata are available, e.g., ribonuclease. Before such a studyis undertaken, however, it is necessary to demonstrate that,for the problem at hand, Thom's theory "works," and it isthis task that is the objective of the present paper.

I. MATHEMATICAL PRELIMINARIES

The generic case of the Riemann-Hugoniot catastrophearises from a potential of the form

V(x) = x4/4 + ax2/2 + bx.

The minima of V(x) are found when

XI + ax + b = 0

3x2 + a > O.

[1]

[2a]

[2b]

Assume that a system is governed by the potential V(x),and identify a and b as controlled variables to be set at will.Three situations may arise depending upon the sign of thenumber

n = b2/4 + a0/27.

1978 Biophysics: Kozak and Benham

0

.o~l CW/onstraints

m~~~II/P,1I~~~~~P

P2 P3.FIG. 1. Generic form of the Riemann-Hugoniot catastrophe

(case I).

If n > 0, then V(x) has one minimum. If n = 0, V(x) has oneminimum and a critical point that is neither a minimum nora maximum. If n < 0, there are two distinct local minimaand one local maximum. The minima are plotted in Fig. 1as the upper and lower sheets of the folded surface. The maxi-mum is the middle sheet inside the fold.

If one holds the parameter a fixed (at a = ao < 0) andchanges the parameter b, then for

b < - [-4ao8/27]'12there is one minimum, seen as the upper sheet in Fig. 1. Onthe other hand, if

b > + [-4aos/271 /2

the one available minimum is the lower sheet. Thus, as onechanges the parameter b through the interval

(- [-4ao3/27]"/2, + [-4ao'/27]'/2)

the system must at some point b = bo jump from the upperto the lower sheet. The set of all such points of discontinuityis called the bifurcation set of the system. This set of pointsmust lie within the cusp

b2/4 - a3/27 ( 0.

Similarly, if one fixes b = bo and varies the parameter a, thenas one crosses the curve

a = [-27b02/4]P/3the number of local minima changes from two to one. Thus,it may happen that as one changes a, one again observesbifurcation.

In general there will be two bifurcation sets-the set ofpoints where the system jumps from the lower sheet to theupper one, Bl,,, and the set of points where the jump is fromthe upper sheet to the lower one, B,,. If these sets happen tocoincide, the system is said to obey a Maxwell convention.

However, there are two other possible relative dispositionsof B., and BLu, best illustrated by their respective extremecases. If the system changes sheets as soon as another mini-mum becomes accessible (that is, if B,, is the branch of thecusp with b < 0 and B I. is the branch with b > 0 in Fig. 1),the system is said to obey the saturation convention. On theother hand, if the system jumps only when the state it is inbecomes inaccessible, it obeys the delay convention. In thiscase B,4 will be the right branch of the cusp and B I. will bethe left branch, as drawn in Fig. 1. The dashed arrow in Fig. 1shows where the catastrophe B. I will occur in a system obeyingthe delay convention.Assume an experiment is performed to measure a quantity

x' for different values of the constraints a',b'. It may happenthat the value of x' changes drastically from one value toanother near certain values of (a',b'), while at other valuesof these constraints the experimentally observed x' holdsnearly constant or changes slowly. Such a situation may bemodelled by the Riemann-Hugoniot catastrophe. The values(a',b') for which the change is sudden and marked correspondto the bifurcation sets; the relatively static values of x' cor-respond to points on the upper and lower sheets of the fold.By plotting the bifurcation sets B,, ,B1, [that is, by observingthe changes in x' in both directions as (a',b') change], onedetermines which catastrophe convention applies. If one findseither the delay or the saturation convention, then the plottedbifurcation sets B,, and B,. correspond to the two edges ofthe cusp in (a',b')-space. If the Maxwell convention applies,then the bifurcation set B = B. I = B,. may correspond to themidline of the cusp in (a',b')-space. In practice the observedbifurcation sets will not be precisely sharp, especially instochastic problems where large populations are involved.Once the above determinations have been made, one may

plot the observed data in (x',a',b')-space. In general, it isfound that the generic form (given by Eqs. 1 and 2) doesnot fit the data exactly. However, the mathematical theorystates that there is a smooth transformation (diffeomorphism)

[3]

that will map the generic catastrophe to the observed data.It must be emphasized that this transformation may have awide variety of forms. One may visualize the possibilitiesby imagining the surface M and its cusp in (a,b)-space plottedin Fig. 1 to be made of rubber. Then any distortion of thesurface or the cusp is permitted as long as it does not resultin creases, sharp points, cuts, or additional folds. The flex-ibility permitted by the many possible transformations (Eq.3) gives the theory its great applicability. Once one has plottedthe observed data (x',a',b'), one may construct the trans-formation appropriate to that case. This permits one to gainunderstanding both of the underlying potential functioninvolved and of the general behavior of the system subjectto the constraints (a',b').Although in this paper we confine ourselves to situations

where a single variable x' is observed as two controls (a',b')are changed, more general situations can be considered eventu-ally. As mentioned earlier, for systems governed by a po-tential function, the generic catastrophes have been classifiedcompletely for one or two observed variables and up to fourconstraints. Thus, it is expected that the approach suggestedin the following section can be extended to deal with manyexperimental aspects of denaturation.

Proc. Nat. Acad. Sci. USA 71 (1974)

f : (xab) --.>. (x',a',b)

Denaturation: An Example of a Catastrophe 1979

II. APPLICATION TO DENATURATION

To show the relevance of Thom's theory to the phenomenonof denaturation, we consider the response of a protein tochanges in (i) the temperature T of the system, and (ii) theconcentration C of denaturing agent. To monitor the be-havior space, we choose as an appropriate variable the opticalrotatory dispersion. Whereas experimental plots of opticalrotation as a function of temperature or concentration areusually reported in terms of - [a]x, it is more convenienthere to choose: x = + [aIx. The motivation for introducingthis sign change. is that the interrelationship between opticalrotation, temperature, and concentration is most clearlydisplayed if the 2-fold curve in Fig. 1 opens outward towardthe reader, rather than into the plane of the paper. As regardsthe experimental results, the effect of this sign-change is sim-ply to reverse the sense of the published transition curves.The control variables are identified as follows:

a = C = C-COb = T= T-To.

Here, To represents the midpoint of the thermal denaturationcurve for zero concentration of denaturant. Then, with re-spect to some fixed reference temperature, CO (>O) representsthe midpoint of the transition that results upon addition ofdenaturant. Note that the point (ToCo) need not correspondto the origin of the coordinate system (a = b = 0); in mostsituations, the point (To,Co) will lie just inside the cusp.

Laboratory experiments on the effect of denaturants onprotein conformation are usually performed at a certain pHand in a buffer. In terms of the present analysis, these addi-tional constraints could be represented by two additionalcontrol variables (see previous section). Apart from the ob-vious fact that the many-dimensional surfaces that arisewhen one considers constraints >3 are difficult to schematicize,we point out that the usefulness of Thom's theory can beillustrated for the nontrivial case of two controlled variables.Moreover, at constant pH and constant buffer concentration,the projection of the many-dimensional surface onto the space(xab) is precisely that depicted in Fig. 1.As our first application of the theory, we seek to demon-

strate how the existing experimental data on a protein sub-jected to the combined disruptive influences of temperatureand calcium chloride concentration can be modelled in termsof the Riemann-Hugoniot catastrophe and its associatedbifurcation set. For the morphology displayed in Fig. 1, itmust be observed that as one changes the controls througha part of the bifurcation set, the value of x = + [ax] mustchange abruptly. In particular, consider a slice perpendicularto the plane (a,b) and containing Path 1 (PI) in Fig. 1. Theintersection of that slice with the upper and lower sheets ofthe surface M represents the accessible states for the systemas P is changed with C = constant. The system proceedsalong the upper sheet of the fold until at some point (dashedline in M1) it drops to the lower sheet of the surface Mi. Thispoint is in the bifurcation set of the system. As is easily seen,the path followed generates a sigmoid curve. That is, theanalysis predicts that a plot of + [aix versus T at constantCaCl2 concentration should be sigmoid in shape. A similaranalysis for a path such as P3 (i.e., increasing CaCl2 concen-tration at constant temperature) predicts that a plot of + [aIxversus C for constant temperature should also be sigmoidin shape and of the same sense. If one follows the evolution

FIG. 2. Diffeomorphic distortion of the generic form of theReimann-Hugoniot catastrophe displayed in Fig. 1 (case II).

of the melting line for successively higher concentrations ofsalt, the splitting between the upper and lower sheets of themanifold M (i.e., the height of the step in [aix from nativeto denatured state) is seen to decrease. This splitting can bevisualized in Fig. 1 by looking into the fold, and consideringsuccessive, constant concentration slices receding towardthe origin of the coordinate system in the manifold M. Wenote further that, given the morphology and catastropheconvention depicted in Fig. 1, it follows that Tm, the midpointof the thermal denaturation curve, should move to lowertemperatures as the concentration of CaCl2 is increased. Thatis, in terms of Fig. 1, the bifurcation set must lie betweenthe right wing of the cusp and the line b = 0.As emphasized earlier, the surface M in Fig. 1 corresponds

to the simplest possible morphology-that correspondingto the generic form of the Riemann-Hugoniot catastrophe.It seems, however, that for the system ribonuclease subjectedto the combined disruptive influences of temperature andCaCl2, nature follows exactly this pattern, as can be verifiedby examining the experimental data cited in Fig. 5 of ref. 5.We remark that the morphology displayed in Fig. 1 is alsothe appropriate model for ichthyocol collagen subjected tothe combined influences of temperature and concentrationof CaCl2 (see Fig. 2 of ref. 5; note that the experimental curvescan be analyzed as displayed).The importance of considering diffeomorphic distortions

of the generic surface M displayed in Fig. 1 was emphasizedin the preceding section. The simplest distortion of the sur-face lI in Fig. 1 is that schematicized in Fig. 2. This surfacemay be realized by imagining that one's thumb and firstfinger are inserted at the point in Fig. 1 indicated by the foldcurve arrow, then, by moving the thumb and first fingerapart, the "rubber sheet" M is stretched. This diffeomorphismgenerates a surface that models existing data on the denatura-tion of ribonuclease by pH or guanidine hydrochloride (withtemperature the other constraint), or lysozyme by guanidinehydrochloride (with T the other constraint).

Proc. Nat. Acad. Sci. USA 71 (1974)

1980 Biophysics: Kozak and Benham

To take the simplest case first, suppose we model the re-sponse of ribonuclease to the combined constraints of pHand temperature by the manifold M of Fig. 2. Here we choosethe direction of increasing-C to be the direction of decreasingpH. We note that the intersection of M with a plane perpen-dicular to the control space (a,b), and containing the path P1(i.e., a constant pH slice), generates a sigmoid curve. Pro-ceeding along the path traced out by the intersection of themanifold M with a plane perpendicular to the control space(a,b) and containing the line P3 (i.e., a constant temperatureslice), we find that a sigmoid curve is also generated havingexactly the same sense as the one generated above. Noticethat as one considers constant pH slices closer and closerto the origin of the coordinate system (i.e., as one recedesfrom the plane of the paper), then, given the catastropheconvention of Fig. 2, the melting temperature Tm moves tolower temperatures, a behavior similar to that observed forthe denaturant calcium chloride. In terms of the control space(a,b), this means that the bifurcation set for ribonucleasefor this set of constraints (T, pH) lies between the right wingof the cusp and the line b = 0. In contrast to the case of CaCl2,however, the splitting between the native and denaturedbranches of the curve, [aix versus T, becomes larger as onedecreases the pH (i.e., as one moves toward the origin of thecoordinate system in M), given the morphology of Fig. 2.These observations based on Fig. 2 are consistent with theexperimental data on ribonuclease compiled in Fig. 11 of ref. 6.

It is interesting to note that at room temperature, someproteins are exceptionally stable to a reduction in pH. Forexample, Tanford (6) points out that for ribonuclease atmoderately high ionic strengths (and for lysozme at an ionicstrength of 0.1), the pH can be reduced to below pH 2 with-out any sign of conformational change. That this behavior isconsistent with the morphology depicted in Fig. 2 becomesclear if one considers a path to the left of path P2 (outsidethe bifurcation set). This would be a path for which one wouldnot expect to find any catastrophic behavior upon loweringthe pH; the system would simply remain on the upper (mini-mum) sheet of the fold.Without considering each case in thumping detail, we point

out that ribonuclease subjected to the combined constraintsof temperature and guanidine hydrochloride concentration(guanidine- HCl concentration increasing in the positivesdirection) can be modelled by the morphology depicted inFig. 2; that this is indeed the case can be verified by examiningthe data presented in Fig. 7 of ref. 5. As a third example of thismorphology, we cite the protein lysozyme subjected to theconstraints (T, guanidine - HCl); this can be checked byexamining the data in ref. 7. If we term the generic morphologywith catastrophe convention displayed in Fig. 1 as Case I,and the morphology displayed in Fig. 2 as Case II, then a,classification scheme for denaturing agents acting on par-ticular proteins can be constructed (see Table 1).

In all Case I and Case II situations considered so far, ithas been noticed that the observed bifurcation set lies betweenthe right wing of the cusp and the line b = 0. The questionnaturally arises whether there might exist a protein/de-naturant system whose bifurcation set lies between the leftwing of the cusp and the line b = 0. As is clear from an ex-amination of Figs. 1 and 2, in this situation one would expectthe melting temperature Tm to increase with increasing salt

other situations have been determined experimentally. Wecite the comprehensive studies of von Hippel reported in Figs.6 and 8 of ref. 5. An examination of these data reveals thatthe bifurcation sets for most pairs of controls lie between theright wing of the cusp and the line b = 0 (i.e., Tm decreaseswith increasing salt concentration). However, for a few pairsof controls, it is seen that the melting temperature does in-crease with increasing salt concentration, so apparently inthese cases the bifurcation set does lie between the left wingof the cusp and the line b = 0. As a specific example, we citeribonuclease denaturation under the combined influences oftemperature and guanidine sulfate concentration.

It was emphasized in section I that the bifurcation setsB., and B,. (i.e., the observed catastrophe points for thetransition from the upper sheet to the lower one and fromthe lower sheet to the upper one) need not coincide. In termsof protein transconformation studies, this means that thebifurcation set for renaturation need not be the same as thebifurcation set for denaturation. In other words, the theorypredicts that hysteresis effects may be observed in some situa-tions. As an example of this possibility, the sensitive studiesof Jardetzky (8) on the unfolding and refolding of staphylococ-cal nuclease at high pH reveal a slight hysteresis; the midpointof the unfolding transition lies at a slightly higher pH thanthe midpoint of the refolding transition.Two final remarks should be made. First, it must be cau-

tioned that for a given experiment only a portion of the sur-faces M in Figs. 1 and 2 may be accessible. That is, it maynot be possible to alter the constraints (a,b) in such a way asto sample broad regions of the manifold M. Hence, it may bedifficult to decide which morphology best describes a partic-ular protein/denaturant system. Second, we note that, forpurposes of simplicity, we have not considered explicitly theprobable multiplicity of states that arise when a native pro-tein is subjected to denaturation. Even in terms of broadclassification, Tanford (6) has pointed out that heat-de-natured lysozyme can undergo a further denaturation to arandom coil; the heat-denatured and random coil regionsin his "phase diagram" for lysozyme represent many possibleconformational states. However, we emphasize that thetechniques outlined here may be used to model any proteintransconformation to any denatured state. That is, any ob-served change in conformation may be regarded as a changefrom one stable state to another, and hence modelled usingcatastrophe theory.

III. CONCLUDING REMARKS

The successful correlations realized in the preceding sectionsuggest that one should be able to use the approach laid downin this paper to generate "phase diagrams" for protein/de-

TABLE 1. Classification scheme for denaturing agents acting onparticular proteins*

Protein Constraint 1 Constraint 2 Case

Collagen Temperature CaCl2 ILysozyme Temperature Guanidine HCl IIRibonuclease Temperature CaCl2 IRibonuclease Temperature Guanidine HC1 IIRibonuclease Temperature pH II

concentration. As it happens, the bifurcation sets for many * See Figs. 1 and 2.

Proc. Nat. Acad. Sci. USA 71 (1974)

Denaturation: An Example of a Catastrophe 1981

naturant systems. More specifically, for two constraints only(in our case, temperature and concentration of denaturant),the mathematical theory states that if a change of morphol-ogy from one state to another is observed, then this changemust be characterized by a catastrophe of the Riemann-Hugoniot type, as schematicized in Fig. 1 (or equivalently,Fig. 2). Therefore, if one has, say, two or three representativepoints from the bifurcation set of a given denaturation experi-ment, it should be possible to construct the surface M, andthereby to predict the behavior of the system for values ofthe constraints not studied experimentally. More generally,it is conceivable that a systematic study of the manifoldsgenerated for different classes of denaturing agents mightresult in the elucidation of a "phase rule" similar in spiritto the Gibbsian phase rule of thermodynamics. Pursuant tothis remark, it may be noted that the Gibbs phase rule itselfcan not be applied to the transitions between conformationalstates of proteins, since these transitions are not infinitelysharp, and since there is no limit to the number of confor-mations that can coexist away from the native state of theprotein. Operationally, then, the methods of this paper mightbe viewed as an alternative to thermodynamics for organizingexperimental data on biochemical systems subjected to an

arbitrary number of constraints.In conclusion, we note that once we have modelled the

catastrophe and bifurcation set for -a given system subjectedto a given set of constraints, it becomes possible to estimatethe behavior of the system as the constraints change si-multaneously in any manner. For example, in an experimentalstudy, it may happen that two variables change simulta-

neously as certain constraints are changed. To cite a specificcase, a property specific to membrane protein and a secondproperty specific to the phospholipid bilayer may both changeas a membrane is subjected to changes in pH, temperature,and salt concentration. In this case, by using experimentaldata to construct the morphology of the resulting 5-dimen-sional manifold, it may be possible to ascertain interrelation-ships between observables and constraints in other situationsnot readily accessible to experimental control. The primarygoal of our further research will be to establish a correlationbetween thermodynamics and the theory elaborated in thepresent paper.

1. Thom, R. (1972) Stabilit6 Structurelle et Morphogenese (W. A.Benjamin, Inc., Reading).

2. (a) Thom, R. (1969) Topology 8, 313-335; (b) Thom, R.(1972) "Phase transitions as catastrophes," in StatisticalMechanics, New Concepts, New Problems, New Applications,eds. Rice, S. A., Freed, K. F.-& Light, J. C. (The Universityof Chicago Press, Chicago), pp. 93-107.

3. Zeeman, E. C. (1973) Applications of Catastrophe Theory(preprint).

4. Abraham, R. (1973) Introduction to Morphogenesis (preprint).5. von Hippel, P. H. & Schleich, T. (1969) "The effects of

neutral salts on the structure and conformational stabilityof macromolecules in solution," in Structure and Stability ofBiological Macromolecules, eds. Timasheff, S. N. & Fasman,G. D. (Marcel Dekker, Inc., New York), pp. 417-574.

-6. Tanford, C. (1968) Advan. Protein Chem. 23, 121-282.7. Hamaguchi, K. & Sakai, H. (1965) J. Biochem. 57, 721-732.8. Jardetzky, O., Thielmann, H., Arata, Y., Markley, J. L. &

Williams, M. N. (1971) Cold Spring Harbor Symp. Quant.Biol. 36, 257-261.

Proc. Nat. Acad. Sci. USA 71 (1974)