J. Phys. Chem. 1993,97, 13831-13840
13831
Analysis of Global Shape Fluctuations and Configurational Transitions in Chain Molecules, Simulated Thermal Behavior of Dodecane Gustavo A. Arteca Dgpartement de Chimie et Biochimie, Laurentian University, Ramsey Luke Road, Sudbury (Ontario), Canada P3E 2C6 Received: July 20, 1993; In Final Form: September 29, 1993’
A procedure is developed and applied to characterize the fluctuations in global shape and folding features, in simple molecular chains, as a function of the temperature (2‘). The approach uses the novel concept of probability of the number of overcrossings associated with all rigid placements of a given configuration as a tool to describe the compactness of a chain, as well as the topology and distribution of its entanglements. A number of shape descriptors are derived from this notion. One of the descriptors provides an absolute characterization of molecular shape, using a value of 1 for stretched, linear chains and a value of 0 for compact and entangled ones, regardless of their size and anisometry. The averages of these descriptors along molecular dynamics trajectories (or over configurational space) can be studied as a function of T. Fluctuations in these averages serve to assess the flexibility of a molecular configuration regarding its deformation into a new fold. The procedure is applied to two models of simulated ‘melting” of an initially linear, all-trans configuration of dodecane. A broad transition is recognized in terms of the shape descriptors, from the stretched chains at low T to the folded chains at high T. The results provide a quantitative expression of the view that melting of some crystalline, linear polymer chains can be interpreted, in a first approximation, in terms of the molecular rigidity and the dominant types of entanglements. A similar transition is also found when studying the shape descriptors as a function of excluded volume in the simulation of polymer swelling with a simple necklace model. Our results suggest a new viewpoint to relate configurational transitions with fluctuations in the macromolecular shape.
I. Introduction The Occurrenceof a large number of degrees of freedom makes some propertiesof macromolecules resemble those of macroscopic systems. A single macromolecule may be regarded as an ensemble of microstates, defined by local configurational energy minima very close to each other in energy. As is the case for other macroscopic systems, macromolecular configurations exhibit a marked cooperative behavior as a function of external factors, such as temperature and solvation. It is possible to recognize the Occurrenceof configurationaltransitions (or dynamic transitions) in single macromolecules.’ For some single-molecule models, equilibrium statistical thermodynamics is sufficient to show rigorously theexistence of a phase transition (e.g.,a discontinuity, as a function of temperature, in the rate with which “folds” appear for an infinitely long macromolecule).’ Well-known phenomena, such as the transition from helix to a random coil in many polymers, can be described adequately by this single-molecule model (see refs 1-5 and others therein). Protein unfolding, except in some particular cases,5appearsto follow a different mechanism than that of helix-coil transitions,”lI but yet some of its essential features can be captured in single-molecule simulations. Other phenomena, such as the disappearance of the catalytic properties in some biomolecules at temperatures below 220 K, can also be interpreted in terms of single-moleculeconfigurational transition^.'^-^^ Finally, the traditional view that melting in crystalline polymers is a phenomenon dependent mostly on the flexibility properties of isolated polymeric chain^^.^,'^ can still be considered a good first approximation, despite some controversy.1.w In the configurational ‘phase” transitions mentioned above, two regimes can be recognized in the nuclear motions. On the one hand, one has the quasi-harmonic,localized motions Occurring in conformations accessible at low temperature.6-20,2iOn the other hand, large-amplitude collective modes appear at high tem~erature.12-I~These latter motions, characterized by wave-
* Abstract
published in Aduance ACS Abstracts, December 1, 1993.
0022-3654 /93/2091-13831S04.00/0
numbers of 30-120 cm-1,22,23 are the more relevant towards the understanding of biological function. It is assumed that these two regimes correlate with changes in the flexibility of the molecular backbone. (These flexibility changes may be an intrinsic molecular property at a given temperature, or they may be partly due to changes in the thermal motions transmitted by thecoupling to the ~ o l v e n t . ’ ~ Changes ~ ~ ’ ~ ~ ~in)other experimental properties, such as heat capacities2’ and average deviations in atomic positions (e.g.,deduced from neutron scattering,2O M&sbauer spectro~copy,2~ and nuclear magnetic resonancezs), can also be interpreted from the temperature dependence of the molecular rigidity. In summary, configurational transitions and other observables of some large molecules may, in principle, be rationalized in terms of the flexibility of single chains. In this work, we deal with the analysis of global molecular flexibility as a tool to recognize and monitor configurational transitions. Intuitively, the small-amplitude motions should be dominant in the case of stiff, rigid molecules, whereas large nuclear geometry distortions should be characteristic of flexible molecules. A number of approaches have been proposed in the literature to quantify molecular flexibility. The results depend on what is understood by flexibility. From the conceptual point of view, we shall follow here an approach connected to the notion of persistence length,2s4,5,26,27 used in polymer theory. The persistence length ( a ) measures the extent of the correlation in the motion of monomers, or single segments, in the polymer chain. In a rigid molecule, the motions are correlated along the chain for many units, whereas flexibility is maximized in chains with very small persistence length. (For typicalvaluesofa, see ref 28.) Therefore, a simple measurement of flexibility in a chain can be given in terms of the dimensionless parameter x = a / L , where L is the contour length of the p o l ~ m e r . ~Even . ~ ~ though x can be determined from the geometry of the chain and from the parameters defining the model, it is not a very satisfactory measure since it does not discriminate on how much a backbone maintains a given fold. Note that molecules with large x (i.e., rigid 0 1993 American Chemical Society
13832 The Journal of Physical Chemistry, Vol. 97, No. 51, 1993 molecules) can still deform to new configurations,but their shape may remain essentially unchanged. The notion of persistence length provides a global approach to flexibility, since the characterizationdoes not depend very strongly on the details of the geometry. In this work, we characterize the deformability of a chain using topological notions of molecular shape. In our approach, the degree offlexibility is measured in terms of the amplitude offluctuations in the topological shape. Other methods, based on local and global geometrical properties, have been used in the literat~re.~%3~ Geometricmethods, however, do not provide a characterization which is sufficiently discriminating so as to recognize large-scale folding features. The description they provide is more related to size and anisometry measures than to the type of entanglements in the polymer. Ultimately, as stated already by Yamakawa in his closing remark, “there is no unique quantitative definition of chain flexibility; it depends on the molecular model on the basis of which a theory is developed”.Z7 In this work, we use the persistence of global folding features as a method to assess the effect of temperature on the flexibilty. The approach provides a new viewpoint to the analysis of folding reorganizations in finite molecular systems. Macromolecules exhibit features characterized by different scaling lengths.26 Some properties depend on local 2D features of the molecular surface, as is the case of docking and molecular recognition by complementarity (e.g., refs 38-40). We are concerned only with the large-scale 3D features, which are more relevant for configuration1 transitions, including polymer melting4*5J9and protein transitions (namely, helix-t~-coil,~-~ nativeto-denatured state unfolding,6J0 liquid-to-glass in hydration shell^,^^.^^ and swelling in good solvent^^^^^^^^^). The study of flexibility in large molecules combines shape analysis with statistical mechanical techniques. Molecular dynamics (MD) trajectories provide geometrical information, from which one must describe the molecular shape. Note, however, that “shape”is not merely a geometricalproperty. Small changes in the nuclear geometry do not necessarily modify propertieswhich depend on large-scale shape; therefore, a global, nongeometricalapproach to molecular shape seems~referable.4~~” In this work, we consider compactness and degree of entanglements in the macromolecular fold as essential global shape characteristics. These features are incorporated into the shape descriptors. Fluctuations in the shape descriptors along MD trajectories (or by a random sampling of configurational space) provide a characterization of flexibility. The persistence over time of a given fold (stability) can then be assessed. The work is organized as follows. In section 11, the global shape descriptors for the backbone’s fold are discussed in detail. Section I11 deals with the evaluation of the average and fluctuations in the descriptors over MD trajectories or configurational sampling. Section IV applies the previous methodology tostudy the shape fluctuationsalong MD trajectoriesof dodecane, as a function of the temperature of a thermal bath. We monitor the changes in molecular shape, starting with an initial stretched all-trans structure. The results obtained are discussed in terms of their relevance to simulated melting of rigid chains. Section V contrasts the results in section IV by using a simplified model of dodecane. The configurational space of a necklace model (random chain with excluded volume) of dodecane is analyzed, and the fluctuations in shape are compared to those obtained along MD trajectories and various temperatures. Conclusions and further comments are found in Section VI. 11. Shape Descriptors for Molecular Chains
We deal with the simplest representation of chain molecules, namely, their backbone. In the present approach, the analysis is restricted to backbones defined by sequences of straight-line segments (C-C bonds), which are essentially open strings with
Arteca no branching, bridging, or self-intersections. The molecular shape is characterized by some global features of the folding of the backbone. Our analysis is based on the use of topological invariants to characterize the shape of a molecular space curve. A number of these techniqueshave been introduced in the These approachesusually do not take into account importantgeometrical aspects of the shape, such as the compactness and the spatial distribution of entanglements. In contrast, methods proposed to characterize compactness (e.g., refs 31, 55, and 56) do not discriminate well among global shape features. Topological descriptors are global, and their values do not depend strongly on the precise atomic positions. An improved descriptionof the backbone’s shape can be achieved by using the molecular geometry to computethe actual topological descriptors. In this work, we use a recently introducedconcept to study flexible polymers: the probability of overcrossings in rigid threedimensionalplacementsof a backbone. Some algorithmicaspects of the methodology are described elsewhere;57d0for the sake of consistency, we shall discuss here the essential ideas, adapted to the present needs. A related procedure has been applied to the classification of secondary structural motifs in protein^^^.^^ as well as to the study of configurational minima in open and cyclic c h a i n ~and ~ ~some , ~ ~a-helices$I at constant temperature. Here, we apply a version of this method to monitor folding in a realistic model chain (dodecane)over a whole range of temperaturevalues. The basic idea is as follows. Consider an isolated backbone with n atoms in a configuration K. This configuration is an equivalenceclass of three-dimensional (3D) rigid placements of the backbone, equivalent under rigid translation and rotation.62 We shall characterize the shape of all rotationally equivalent 3D placements. When projected along a given direction in space, the actual instantaneous placement of a freely rotating backbone will appear as a planar curve, exhibiting crossings where the original 3D bonds cross over each other. Theseprojected crossings are referred to as over crossing^.^^^^^ A given placement may have N overcrossings. The case N = 0 corresponds to no overcrossings. As the molecule rotates, instant snapshots projected along the same direction in space will produce planar curveswith, in general, different numbers of overcrossings. Let A d n ) be the probability of observing the backbone with configuration Kin a 3D rigid placement which exhibits, by projection, Novercrossings. From the point of view of counting the number of overcrossings, all rigid placements which have the same number N are considered as “equivalent”. Clearly
where AN@) = 0 for N > max N = (n- 2)(n - 3)/2. Note that a backbone with n = 3 has no overcrossings, Le., Ao(3) = 1, since it is always planar. The probability distribution (A&)) is a continuousglobal shape descriptorof the molecular conformation K. The descriptor is built from the knowledge of the nuclear geometry, since the latter is needed to determine the Occurrence of overcrossings. However, the characterization is not local, since small changes in the geometry do not necessarily lead to dramatic changes in the overcrossing probabilities. The histogram of the probabilities (A&)) (the “overcrossing spectrum”) can be used to compare folding features in proteins.60 The description provided by (A&)) complements that of other current techniques, in the sense that the present shape characterization describes not the Occurrence of local motifs but the overall entanglement pattern of the backbone. Some essential information contained in the spectrum can be conveyed by its maximum, the probability A*, associated with the most probable
The Journal of Physical Chemistry, Vol. 97, No. 51, 1993 13833
Analysis of Shape Fluctuations
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Figure 1. Four 3D rigid placements of the carbon chain of dodecane in a foldedconformationK(?*). Theplaccmentsaredcrivedfromoneanother by performing 90° rotations. The number below the diagrams indicates the number of overcrossings in the projection. (The conformation corresponds to one particular snapshot of the molecular dynamics trajectory of dodecane a t T = lo00 K. Note that a small distortion of the placement with N = 5 may lead to N = 7 overcrossings.)
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A similar magnitude has recently been proposed, independently, to characterize complexity in self-avoiding random walks in lattices.63@ The minimum number of crossings has also been used to characterize self-entanglements in ring polymer^.^^.^^ The computation of the overcrossing probabilitiesis performed automatically. The algorithm used to determine of the number of overcrossings N, associated with a given 3D rigid placement, is discussed in ref 57. To evaluate the probabilities (Adn)],we repeat the calculation of N for a random distribution of points on the surface of the smallest sphere (centered at the geometrical center of the backbone) that completelyencloses the chain. (The radius of this sphere is R,and one an use it as a crude descriptor of molecular size, since it is proportional to the hydrodynamic radius of the molecular chain?) Each of these random points defines a new projection of the backbone (or, it defines a new placement of thesameconformation K obtained by rigid rotation). The final evaluation of ( A d n ) }is done accurately by repeating several of the above calculations and averaging. The choice of the randomization procedure is important to achieve accurate results. In this work, we use the algorithm discussed in ref 67. (The overcrossing probabilities can also be interpreted as fractional areas on the sphere with radius R,as done in refs 58 and 59.) As an illustration, we provide here an examplewhich is relevant to the discussion in the next sections. Figure 1 shows the projectionsof four 3D rigid placementsof the same configuration of dodecane (n = 12). This configuration is quite bent, and it
Figure 2. Overcrossing spectrum of the configuration K(r*) displayed in Figure 1. The results shown superimpose several calculations with 8 OOO, 10 OOO,and20 000randomprojections. Themostprobablenumber of overcrossings is P = 0 (with A* = 0.70).
corresponds to a snapshot of a structure encountered along a molecular dynamics trajectory of the foldingat high temperature of dodecane, initially at a linear, all-trans configuration. The snapshot corresponds to a time f * , and the overcrossing probabilities for this structure are indicated by (AN(12,t*)]. Figure 1 displays only the carbon backbone; underneath each projection, the number of overcrossings N is given. When a large number of similar projections is studied, the overcrossing probabilities are evaluated. Figure 2 shows the results of several repeated computations of the overcrossing spectrum associated with this configuration. The figure superimposes six spectra, computed with two random series of 8 000, 10 000, and 20 000 points each on the smaller sphere containing the backbone. Even allowing for some variation in the values of A2(12,t*), the differences found between the various randomizationsare minor. From these results, we can accurately estimate for this configuration that P = 0 and A* = 0.696 f 0.003. In contrast, note that a linear, all-trans configuration of decane will have P = 0 and A* = 1. Therefore, the shape of the fold can be recognized by the values of A*, even if the conformations show the same number of most probable number of overcrossings. (The mean overcrossing number, of these two conformations will of course be different.) A similar procedure is applied below for the characterization of all other molecular configurations.
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III. Quantitative Assessment of Shape Fluctuations The values A* and P are used to characterize the folding shape of molecular chains. Each value of these two descriptors is associated with a configuration K ; we shall use the notation A*(K) and P(K)to indicate this dependence when needed. Depending on external conditions,the values of A*(K) and P( K ) may oscillate along a reaction path or folding trajectory. If the flexibility of a molecular chain is interpreted as its capacity to move away from an initial shape due to collective nuclear motions, then these oscillations in shape descriptors can be used to quantify the degree of flexibility of a given initial configuration.59.60 We shall refer to these oscillations in A* and P along a trajectory as shapefluctuations. The amplitude of these fluctuations is related, in principle, to the molecular f l e ~ i b i l i t y . ~ ~ The configurations encountered along a molecular dynamics (MD) trajectory are parametrized in terms of the parameter time, t, Le., K ( t ) . The shape descriptor A*(K(t))changes along the trajectory as a function of time, and its trajectory time average can be defined as follows: 1 (A*) = lim - ~ f ' + ' * ' A * ( K ( t )dt, ) Af > 0
(4) sAt where the result is independent of t'. When using MD, t ' > 0 P-
Arteca
13834 The Journal of Physical Chemistry, Vol. 97, No. 51, 1993 can be taken as the equilibration time and the interval At as the time step used in the integration of the equations of motion in the form of s discrete steps (in our case, At = 1 fs). The time-average ( A * ) characterizes the folding type of the dominant type of conformations occurring along the trajectory. In order to monitor flexibility, one may choose to start the simulation from a (reference) configurational minimum, for instance, the linear (all-trans) structure of a chain, where A* = 1. The more rigid the chain, the closer ( A * ) will remain to this initial value. The extent of the shape fluctuations is described by the standard deviation of the shape descriptor along the trajectory: UA
= ( ( A * * )- (A*)2)"2
The role of the external factors in modifying both the molecular shape and the molecular flexibility can be assessed then by monitoring the changes in values of ( A * ) and uA. If the MD trajectories are sufficiently long, the time averages above will coincide with the phase-space (or ensemble) averages over configurations:6*
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TW) Figure 3. Average value and fluctuations in the torsional contributions to the potential energy (Em) along the MD constant-temperature trajectories of dodecane, as a function of the simulation temperature T. (Both the average and the standard deviation (fluctuations)grow almost linearly with T. Em is in kcal/mol.)
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direct coupling of the molecule to a thermal bath. The system equilibrates within the first picosecond; the largest fluctuation in the total energy is found to be below 596, with a standard error where Pi is the probability of the molecule to appear in a of less than 0.1 kcal/mol. The trajectories can thus be considered configuration('microstate") Ki, among a number M (sufficiently as quasi-microcanonical. During the simulated dynamicsof this large) of random configurations. Since the present MD simusmall chain, the configurations found show mostly no overcrossings lations are quasi-microcanonical (Le., the "allowed" configurations (W = 0); therefore, in this case we find A* ( = A N I )= Ao, as in M are those with energy values within the energy shell (E the problems studied in ref 59. AE/2, E AE/2)),all configurations are equally probable (a As expected, the torsional motion accounts for most of the priori), and eq 6 becomes fluctuations in the potential energy. Figure 3 shows the results for the average and standard deviation in the torsional potential energy Em found along the MD trajectories, at each temperature. (7) Note that both the average and the deviation grow linearly with the temperature. No remarkable structural change as a function The standard deviation uA can also be written in terms of of the temperature can be assessed by studyingonly these energetic microcanonicalensembleaverages. The sum involves a sampling fluctuations. of thecomplete phase (or configurational)space, within the energy Figures 4 and 5 describe the interplay of energy and shape shell. This approach is used below to compare the findingsin the fluctuations along the trajectories. These figures represent the simulated thermal behavior of a molecular chain with those MD trajectories as a sequenceof snapshots, each one characterized correspondingto configurational fluctuations of simple polymer by its position in a 2D shape vs potential energy diagram. The models. "shape axis" indicates the maximum probabilityA*(t)associated with the instant configuration K ( t ) (snapshot), and the "energy" IV. Temperature Dependence of Shape Fluctuations in axis indicates the instantaneous potential energy. The arrow on Simulated Dynamia of Dodecane the diagram gives the orientation of the walk, starting from the first equilibrated configuration. Figure 4 contains the results for Dodecane has been taken as the working example to explore the trajectories at low temperatures and Figure 5 the results systematicallytheeffect of temperature on the shape fluctuations and molecular flexibility of small chains. The thermal behavior corresponding to high temperatures. In Figure 4, the average value of A* decreases with the temperature, indicating that the has been studied by simulated MD, using standard molecular mechanics potentials and constant-temperature trajectorie~.69*~0 chain spends most of the trajectory in structures folded away from the linear shape. This effect appears accompanied by an The results shown below have been derived by using the Dreiding increase in the fluctuations in the energy. In contrast, Figure 5 force field,71as implemented in the program Biograf. We have as well repeated some of the trajectories using the MM2 shows that at larger temperatures a smaller increase in the potential,'Z as implemented in the molecular modeling package potential energy fluctuations can lead to larger oscillations in the HyperChem. The final results obtained with both types of shape descriptor. In other words, at larger temperature, the potentials are very similar. Since hydrocarbon chains can be molecule has access to many more deformation modes leading described accurately with most potential~,s.~3.~4 the fluctuations to changes in the folding pattern. encountered along the trajectories should simulate reasonably The results in Figures 4 and 5 show that the shape fluctuations well the intramolecular effects on flexibility. appear to depend markedly on the temperature. Figure 6 makes The constant-temperatureMD trajectories have been computed this apparent by representing theconfigurationallyaveraged shape at temperatures ranging from T = 50 to 1000 K. In each case, descriptor ( A * ) as a function of the temperature. The figure we have computed 30-ps trajectories, starting from the same initial indicates the result of averaging A*(t) over the MD trajectory minimum energy configuration,namely, the completelystretched, and the amplitudes in its oscillations. For further clarity, these linear all-trans-dodecane. The sequenceof C-C bonds is planar amplitudes (mean square deviations) appear in Figure 7 as a at this configuration (Le., W = 0 and A* = 1). The trajectories function of T. The error bars in Figure 7 correspond to our are short, but they provide a reasonable survey of accessible estimate in the accuracy of the fluctuations. This error is tied configurationsat each temperature. The constant-temperature to the accuracy in the computation of overcrossing probability, MD trajectories are computed with rescaling in velocities, not by which is estimated at most with a +0.005 error (A*(t) < 1).
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The Journal of Physical Chemistry, Vol. 97, No. 51. 1993 13835
Analysis of Shape Fluctuations l r
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on the diagram representsone in a seriesof regular snapshots taken every 0.2 p along the trajectory. Each snapshot (a configuration K ( t ) ) is characterizedby theA* shapedescriptor ofthechain and itscorresponding potential energy value U(K(t)). Snapshots are joined sequentially by straight line segments. The arrow indicates the walking direction along the trajectory, starting from the initial structure (an almost linear chain obtained after equilibration).
Figures 6 and 7 provide a clear characterization of the types of folding features accessible to the isolated dodecane chain in a thermal bath. At temperatures lower than ca. 200 K,the folding features resemble those of the initial structure, since the descriptor ( A * ) is close to 1 (i.e., no overcrossings) and its fluctuation is lower than 0.5%. However, when the temperature is close to 300 K there is a rapid decrease in the value of ( A * ) and an increase in the amplitude of shape fluctuations. Between 300 and 400 K, these two changes are remarkable. Figure 6 and 7 show clearly the existence of two inflection points in both (A*) and U A as a function of T. Below the first inflection point, the chain is completely linear. Between the inflection points, there is a recognizable change in the accessible shapes, as the chain begins to fold. As shown in Figure 7, for larger temperatures, the oscillation in folding features dampens (as a function of 7‘) as the whole range of folds becomes accessible. The behaviors exhibited in Figures 6 and 7 resemble closely the broad bimodal change observed in many properties of finite systems during configurational phase transitions.12J4 For comparison, the experimental melting and boiling points of dodecane (263.4 and 489.3 K,re~pectively~~) are indicated by arrows in Figures 6 and 7. These two values are surprisingly close to the inflection points found in the shape descriptors. Even though phase transitions are collective phenomena where the intermolecular interactions play an essential role, polymer melting can
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Figure 5. Shapeand potentialenergy fluctuationsalong MD trajectories of dodecane at high temperatures, T = 425,650, and lo00 K. Each point on the diagram representsone in a series of regular snapshots taken every 0.2 ps along the trajectory. (See further explanation in Figure 4.)
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approximately be correlated to the configurational flexibility of single molecule~.4.~J~ Our findings for the shape descriptors of dodecane conform to this view, even though the closeness in the
13836 The Journal of Physical Chemistry, Vol. 97, No. 51, 1993 7.5T
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agreementwith themelting point is probably accidentaland should be taken with caution. In general, melting in crystalline polymers is accompanied by a rapid change in theaccessibleconfigurations. In the solid phase of crystalline polymers, molecules are deemed to be confined to essentially linear configurations (see, though, discussion in ref 16). As a condition accompanying melting in linear crystalline polymers (and, perhaps, polymers with modest branching), the isolated molecules would be expected to oscillate into configurations where their folding shape is markedly different. The configurational change accompanying melting can be assessed by studying shape fluctuations in single molecules. In a traditional approach, melting points of crystalline polymers can be estimated from the “flexibility” of a single ~ h a i n , 2 * ~the .~J~ latter being given in terms of the number of energy of the accessible torsions. Our results indicate that such an assessmentcan indeed be made in terms of flexibility, but where the latter is described in more familiar terms as a capacity to move away from the fold characteristic of the molecule in the crystal. This result is in line with the latest improvementson the mean-field theory of polymer melting, where a first-order transition is observed from a phase with stretched molecules at low temperature to a ‘liquid” phase at larger T.I9 Although we have illustrated our procedure for dodecane, the results should be also valid for larger hydrocarbon chainsand polymethylenes,since the melting point does not depend strongly on the number of carbon atoms.4 However, assessing the full significance of the agreement found between the change in shape descriptors and the phase transition requires further studies of other chains and polymer topologies. Complementing the above discussion, Figure 8 shows the results for a measure of the chain’s size as a function of the temperature. The parameter R correspondsto the radius of the smallest sphere (centered at the geometriccenter of the backbone) which encloses completely the molecular chain. The configurational or time average of R, (R), satisfies the same scaling laws as the averaged radius of gyration and the end-to-end di~tance.4J*~~,~* In addition, ( R ) can also be estimated experimentally, since it is related to the hydrodynamic radius and the equivalent spherical radius of a molecule,76both of which can be obtained from diffusion and molecular volume measurements. Figure 8 gives (R) and uR over the MD trajectories as a function of temperature (values are in A). The temperature dependence of molecular size shows a qualitative correlation with the behavior observed in the shape descriptors. For temperatures above 300 IC,the averaged radius
decreases (and its fluctuations increase), thus revealing that the chain adopts more compact structures. The fact that the transitions in the geometrical descriptor ( R ) and in the shape descriptor (A*) take place in the same range of temperatures indicates that both are describing the same phenomenon. However, the information they provide is not the same. From Figures 6 8 , we can compare the two descriptors: (i) At temperatures T < 200 K, the average molecular size is constant and its fluctuations are negligible. The results for the shape descriptor show that (A*) decreases slowly in the same range, thus indicatingthat although the size changeslittle, the molecular chain is coiling away from linearity. (ii) The contrast is also clear at high temperatures. The average (R)and its fluctuations changealsolittlefor T > 500 K. In contrast, the shapedescriptors (Figures 6 and 7) indicate that the molecule continues to coil as the number of overcrossings increases. In other words, the shape characterization at high temperatures shows that the complexity of entanglements may still increase while maintaining the molecular size essentially constant. The results above provide an illustration of the complementary information provided by the purely geometrical and the topological shapedescriptors. The key findings can be summarizedas follows: (a) The present shape descriptors can be more discriminating than a geometrical measure such as (R) since they convey the degree of entanglements of the molecular fold, as well as its compactness. In comparison,the molecular radius provides only informationabout the latter. Features which remain hidden while following (R) along a MD trajectory can be recovered by analyzing also ( A + )or ( W ) . (b) The fact that bimodal behavior and configurational transitions can be monitored by means of geometrical measures (such as the radius) is well-known (e.g.,refs 5 , 14, 30 and 33). Here, we have shown for the first time that geometrical and topological descriptors of global molecular shape can also characterizethesame phenomenon. Thisensuresthat it is possible to analyze other molecular properties and yet convey the same physical picture. (c) A sole analysis of macromolecular size is not enough to recognize the type of fold present. The shape descriptors permit one to establish the type of dominant folding features occurring during the phase transition and thus separate neatly size and shape effects. This is important toward a detailed study of folding mechanisms, where a differential characterization between folds with similar sizes is essential. The approach may be valuable to
The Journal of Physical Chemistry,Vol. 97,No. 51, 1993 13837
Analysis of Shape Fluctuations recognize and assess the formation of molten globula, secondary structure, or other specific macromolecular features believed to occur in many folding-unfolding processes.”Il
V. Shape Fluctuations and Configurational Transitions in a Necklace Model of Dodecane
High T regime
t
I Low T regime I
\
In the previous section, we discussed the fluctuations in the 0.6 shape of a molecular mechanics model of dodecane along MD trajectories. In this section, we consider whether the continuous, 0.4 broad transition found in theshape descriptors can also be retrieved I from the configurational fluctuations of a much simpler model. A simple model of a 12-unit molecular chain with excluded volume is chosen. The configurationally-averaged shape descriptors for this chain are evaluated as a function of the excluded 0 1 2 3 volume parameter. Since the role of the temperature and that of the excluded volume are related,27.42*77 these results can be contrasted qualitatively with those in Section IV. The model used is similar to the necklace polymer ~ h a i n . ~ ~ - S ~Figure 9. Changes in the configurational averages of the global shape descriptors for the self-avoiding walk model of dodecane as a function The chain described here is essentially a Pearson random walk of the radius rexof the excluded volume interaction. The curves shown with constant-length steps including excluded volume. Attached correspond to the average of the most probable number of overcrossings to each ‘nucleus”, a sphere of radius rex is considered. The in the chain, (hn);its corresponding configurationallyaveraged overconfigurations permitted are those in which no other “nucleus” crossing probability, ( A * ) ;and the fluctuation UA in the latter. The value rex = 3.06 A corresponds to a completely stretched, rigid linear (a node in the walk) can penetrate inside any of the spheres. The chain. (Two regimes can be recognized, separated by a dashed vertical macromolecule is therefore an off-lattice “stick-bead” chain.s1 line in the figure. First, for re, > 2.5 A, the average ( A * )grows linearly Similar models have been used recently to simulate polymer towards unity (linear chain),its fluctuationdecays linearly, and the most swelling in various solvents,77to compute molecular excluded probable number of crossings (hn)is almost zero. The linear behaviors volumes,8’ and to study the behavior of rigid and flexible fibers are represented by dashed lines. The second regime corresponds to r, < 2.5 A, where ( A * ) increases slowly from its minimum value, the in flow fields.82 Self-avoidingrandom walk (off-lattice) models fluctuation UA is almost constant at its maximum value, and (hn)decays provide a more realistic representation of the dynamic behavior rapidly. The first regime, corresponds to the rigid rod, is identified with of chains than the lattice models of configurational transitions the low-temperaturecase of the simulation of dodecane. Similarly, the in polymers and proteins.4,’,11,55~56,77~83-86 Nevertheless, both low r,,valuesareidentified with the flexiblechain,Le.,a high-temperature approaches should produce the same scaling laws for configuregime.) rationally-averaged proper tie^.^^^^^ ature” P,in such a way that rex 21 for P = 0, and rex 0 We consider a constant internuclear distance (C-C singlefor T* -: bond distance, I = 1.53 A) and a randomly chosen bond angle, subject to the constraint set by a given excluded volume radius. Note that in this approach rexcan take a maximum value of 21. At this limit value, the chain is forced to adopt a linear structure. where b is some constant. A similar dependence with the For smaller value of rex,the chain can adopt infinitely many temperature can be argued for the persistencelength.26 By using configurations,although the range of configurationswith distinct eq 8 for a given b, one can pass from the rexscale to a P scale. folding features will increase markedly only when rexis quite For each rexvalue, we have constructed molecular configusmall. In the limit of rex 0, this range is maximized. This rations of the 12-bead (n = 12) polymer by random walks, with change in the nature of the chain configurations as a function of a constant step length I = 1.53 A (Kuhn’s length). For each of the excluded volume radius re, can be compared qualitatively to these configurations of the ‘dodecane” chain, we have computed the effectoftemperat~re.~~.~~,~~ At high T,the (initially-stretched) the overcrossing spectrum and evaluated W , A*, R, and the chain can fold and thus adopt the type of configurations which radius of gyration (RG). A series of random configurations was are characteristic of a *compact” p 0 1 y m e r . l ~ Conversely, ~~~ a generated until the configurational averages ( W ) ,( A * ) , (R), rigid, elongated chain at low T will appear in configurations and (RG) (cf. eq 7) and the mean-square-root deviations U A and characteristic of swollen p o l y m e r ~ . l ~Therefore, -~~ the transition OR (cf. eq 5 ) were stable within a desired accuracy. It is found between these two regimes caused by the change in excluded that at least some 500 accepted chain configurations are needed volume about each atom should roughly correlate with the to reach an accuracy of more than two significant figures in the temperature effect on the chain rigidity discussed in Section averaged shape descriptors. The actual computation becomes, IV.5*10,87 Note, however, that in the present model, with only of course, longer for larger rex,since more random walks are repulsive interaction between beads, the change between the two rejected for failing to meet the excluded volume criterion. regimes does not represent a true “phase” transition. The change Figure 9 shows, in the same scale, the changes in averaged represents rather a “crossover phenomenon” between dominant shapedescriptors ( W )and (A*),aswellas thestandarddeviation configurations (stretched vs folded), which is characteristic of UA. The largest value of rexused is 3.03 A. The diagram includes though the exact result in the limit rex 21 = 3.06 A: polymers with finite si~e.*~,~O To quantitate the relation between temperature and excluded (W>-O, ( A * > - l , UA’0 (9) volume, one can describe the effect of the temperature on the effective radius for the atomic collisions (which is taken as rex). representing the fact that there are no overcrossings in the linear chain. The averaged most probable number of overcrossings, Intuitively, this radius will decrease with increasing T,since the ( W ) ,shows the most remarkable change as a function of rex. atomic spheres become ‘softer” and they exclude less volume (Note that, in this model, A* does not always correspond to A0 about them. At T = 0 K, the radius should be a maximum, whereas re, should be a minimum at T -. For the sake of for a given configuration.) For values rex> 2.5 A, the chain is essentially linear ( ( N * ) = 0). The value ( W ) grows rapidly illustration, we shall assume here that re, adopts a simple Boltzmann form as a function of a virtual, “equivalent temperwhen re, decreases, finally reaching a maximum of ( W )= 1.3
-
-
-
-
-
-
Arteca
13838 The Journal of Physical Chemistry, Vol. 97, No. 51, 1993 High T
9 1 8 7
,
LOW
T
-'
-.
:::I/;
6 -. 5 -. 4
-.
,
0.2 1
0.1
0
0 0
2
1
3
OA
0
rex(A)
250
500
1000
750
F (equivalent tempcraturc, K) T*=-250/ln(l-rcxlU)
Figure 10. Changes in the configurationalaveraged size parameters for the self-avoiding walk model of dodecane as a function of the radius rex
of the excluded volume interaction. The curves shown correspond to the effectiveradius (R), its fluctuation UR, and the configurationalaverage of the radius of gyration ( R o ) . (The behavior of (R)and that of (RG) are coincident,as expected. Low- and high-Tregimes can be defined for these parameters in a manner similar to that done in Figure 9.) in the limit of no-excluded volume about the beads (freely-jointed chain). In this limit, the chain is definitely folded. Based on our results for the rigid configurations of small chains and protein motifs,5*"' a value of (iV) = 1.3 should correspond to a chain resembling a @-turnor loop. The transition in ( W )as a function of rexcorrelates with the changes in the other shape descriptors. For small excluded volumes, the average probability of the most probable number of overcrossings, (A*), changes slowly with rex, whereas its standard deviation remains almost constant. In contrast, for rex> 2.5 A, (A*) grows linearly with rex,whereas oAdecreases linearly in the same interval. Therefore, two regimes can be recognized, separated by rex 2.5 A. Above this "critical" rex value, the chain is virtually rigid; below it, the chain is sufficiently flexible to fold away from linearity. For this reason, the two regimes are identified in Figure 9 as "low-T and "highT" regimes, respectively. Figure 10 shows the results for the size descriptors. The configurational averages of the radius R and the radius of gyration RGbehave similarly. Both exhibit an initially slow increase with the excluded volume radius rexand a faster linear growth for re, > 2.5 A. The standard deviation U R shows also a two-regime behavior as a function of rex,remaining almost constant for rex < 2.5 A and decreasing to zero for larger rexvalues. The behavior of ( R ) in Figure 10 is similar to that of (A*) in Figure 9. This indicates that the averaged overcrossing probabilities describe both entanglements and compactness (size) features. Similar conclusions were obtained for the "realistic" dodecane in Section IV, for temperatures lower than 500 K. The analogy between the results derived with the two models can be made more explicit by transforming the excluded volume to a T* scale. Figure 1 1 shows the results obtained for the average (A*), and its fluctuation U A , for the 12-bead necklace model with two possible "equivalent" temperature scales. The upper diagram shows theresultswith thechoiceb = 250K,andthelowerdiagram corresponds to b = 750 K. Both figures describe the same behavior, although the change is more gradual for larger bvalues. In the 7" scale, the descriptors exhibit a bimodal behavior, with an inflection point separating two temperature regimes (dotted line). For T+ values below this inflection point, thechains remain in stretched and rigid configurations ((A*) = 1 and uA 0). For equivalent temperatures above the inflection point, the chains fold away from the linear structure. The shape fluctuations become stationary for sufficiently large temperatures (cf. upper diagram), indicating that the polymer can access to all possible folded conformations ('maximum disorder"), The same behavior
-
\-I
0.9
0.6 0.5
0.2 0.3
I =A
0.1 0
I
0
250
500
750
1000
T+(equivalent temperature, K) T*=-750fln(l-;xlU)
Figure 11. Dependence of the averaged shape descriptor ( A * ) and its fluctuation uA, with an equivalent temperaturescale P,for the necklace model of dcdecane with excluded volume interaction. The equivalent temperatureis defined so that the excluded volume is maximum at 0 K and zero at infinite temperature. (The two diagrams are constructed with the results from Figure 9, displayed for two possible choices of the parameters defining the equivalent temperatureP.Both results display a very similarbehavior to that found in Figures 6 and 7 for the molecular dynamics simulations of the actual dcdecane. The line indicatts the transition from the low- T,regime, dominated by small fluctuations, to the high-T regime with large fluctuations. The actual position of the inflection points in ( A * )and u,, (and the boundaryvertical line) depends on the choice of the equivalent temperature scale.)
is observed in all T+ scales; a different choice of b changes only the location of the inflection point. The results in Figure 11 for the necklace dodecane agree with those in Figures 6 and 7 for (A*) and UA in the real dodecane, respectively. This suggests that some essential features of the shape fluctuationsaccompanying the folding of realistic polymers can be approximated by studying self-avoiding random walk models, where the atoms behave as soft spheres with temperaturedependent radii. Moreover, the results provide a quantitative expression to the relation between the configurationaltransitions observed in the melting and the swelling of polymers. The results in this section conform also with the conclusions in Section IV regarding the contrast between the geometric descriptor (such as ( R ) or (R G ) )and the shape descriptors (such as ( A * ) or ( W ) ) .In both cases, it is possible to recognize the occurrence of a configurational transition. The information conveyed by the two types of descriptors is complementary. The analysis of overcrossings allows one to assess the type of folding (shape) features associated with the configurations at a given excluded volume, whereas this information is blurred when
Analysis of Shape Fluctuations
The Journal of Physical Chemistry, Vol. 97, No. 51, 1993 13839
studying only size and compactness. Furthermore, the results in Figure 9 show that (P)is a sensitive descriptor of the configurational transition, allowing a neat division between the regimes dominated by unfolded ((P ) 0) and folded (( P ) E 1) dodecane chains.
de Dtmarrage 1993 (Laurentian University), and by computing time from the Mathematical Chemistry Research Unit of the University of Saskatchewan (directed by. P. G. Mezey).
VI. Further Comments
(1) Wiegel, F. W. In Phase Transitions and Critical Phenomena; Domb, C., Lebowitz, J. L., Eds.; Academic: London, 1983; Vol. 7, pp 101-149. (2) Birshtein, T. M.; Ptitsyn, 0. B. Conformotions of Macromolecules; Interscience: New York, 1966. (3) Poland, D.; Scheraga, H. D. Theory of Helix-Coil Transitions in Biopolymers; Academic: New York, 1970. (4) Volkenstein, M. V. Configurational Statistics of Polymeric Chains; Interscience: New York, 1963. (5) Flory, P. J. Statistical Mechanics of Chain Molecules; Interscience: New York. 1969. (6) Jaenicke, R. Prog. Biophys. Molec. Biol. 1987, 49, 117. (7) Bradshaw, R. A.; Purton, M. (Eds.) Proteins: Form and Function; Elsevier: Cambridge, 1990. (8) Freire, E.; Murphy, K. P. J. Mol. Biol. 1991, 222, 687. (9) Chothia, C.; Finkelstein, A. V. Annu. Rev. Biochem. 1990,59, 1007. (10) Dill, K. A.; Shortle, D. Annu. Rev. Biochem. 1991, 60, 795. (11) Skolnick, J.; Kolinski, A. J. Mol. Biol. 1991. 221, 499. (12) Doster, W.; Cusack, S.;Petry, W. Nature 1989, 337, 754. (13) Kuczera, K.; Kuriyan, J.; Karplus, M. J . Mol. Biol. 1990,213, 351. (14) Rasmussen, B. F.; Stock, A. M.; Ringe, D.; Petsko, G. A. Nature
In this work, we have developed and illustrated an approach that links molecular shape fluctuations to dynamic transitions in simple chain molecules. The fluctuations are described in terms of global shape descriptors. These descriptorsconvey the folding type of a chain in terms of the compactness and complexity of its entanglements. In this work, we have employed (A*) and (P ) to extract structural information from computer-generated dynamic trajectories. (A descriptor such as ( E )can be used for the same purpose.) In principle, it is conceivable that some of these shape descriptors could also be measured experimentally. To date, however, I cannot give any answer as to which experimental approach would provide a measure of (P) or (N). As we have shown, when the fluctuations are followed as a function of external factors, such as the temperature, it is possible to recognizea regime where a linear chain is rigid. The transition found in (A*) and ( N * ) as a function of Tcan be compared to "melting" in a single molecule. As the results suggest, the shape descriptors (and their fluctuations) allow one to retrieve some structural information which sometimes may not be available from energy or molecular size analyses. Moreover, we have verified that the study of configurationsof simple polymer models as a function of the volume excluded by the atoms leads to a picture similar to that of the effect of temperature. The folding features and the rigidity found in chains with small atomic excluded volume are comparable to those found for isolated chain molecules at high temperatures (in melted configurations). In closing, we make some remarks regarding (A*). This global shape descriptor represents the configurational average of the probabilityof the most probablenumber of overcrossingsobserved in rigid 3D placements of the molecular backbone. This probability can be viewed also as a fractional area on the smallest sphere enclosing the backbone and centered at its geometrical center. This function is a maximum ((A*) 1) when an open chain is forced to adopt only a linear configuration. In contrast, it is minimized when the molecule can access all possible configurations (e.g., at high r). Moreover, the more entangled the accessible configurations, the smaller the value of (A*). Intuitively, (A*) would appear to be proportional to a reciprocal "entropy" of the chain (or a reciprocal free energy in the microcanonical case). The analogy is further supportedby results on configurational averages of necklace polymers for n >> 1,6OvS8 which indicate
-
-
-
(A*) an', with b -1 (10) Assuming that the number of configurations il of the polymer grows as1926
fl- g"ny (1 1) and that the microcanonical entropy So is proportional to In a, then it follows that (A*) also scales approximately as (So)-]for n >> 1. This suggested connection between thermodynamic functions of the chain and global shape descriptors of its entanglements is another indication of the possible relevance of molecular shape analysis to the study of configurationaltransitions in polymers. Acknowledgment. I thank R. Kari and D. Jobe (Laurentian University) for discussions. This work has been supported partly by an operating grant from the Natural Sciencesand Engineering Research Council (NSERC) of Canada, by the Fonds de Recherche de I'Universitt Laurentienne (FRUL) and the Fonds
References and Notes
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