Proc. Natl. Acad. Sci. USA Vol. 96, pp. 2031–2035, March 1999 Biophysics Stretching lattice models of protein folding i NICHOLAS D. SOCCI§¶ ,JOSE´ NELSON ONUCHIC**, AND PETER G. WOLYNES†† §Bell Laboratories, Lucent Technologies, 700 Mountain Avenue, Murray Hill, NJ 07974; ¶Center for Studies in Physics and Biology, The Rockefeller University, New York, NY 10021; **Department of Physics-0319, University of California at San Diego, La Jolla, CA 92093-0319; and ††School of Chemical Sciences, University of Illinois, Urbana, IL 61801 Contributed by Peter G. Wolynes, December 17, 1998 ABSTRACT A new class of experiments that probe folding A Minimalist Approach to Stretch Experiments of individual protein domains uses mechanical stretching to cause the transition. We show how stretching forces can be According to the energy landscape theory, folding mechanisms incorporated in lattice models of folding. For fast folding can be classified into several scenarios depending on the shape proteins, the analysis suggests a complex relation between the and character of the free energy surface. In the most common force dependence and the reaction coordinate for folding. regime, folding is described by an exponential process whose rate is limited by the time it takes to reach an entropic bottleneck, Several experimental groups recently have succeeded in moni- which consists of an ensemble of many structures. In this regime toring the unfolding of individual protein domains (1–3), by the rate is a smooth function of the protein stability. For a measuring the forces exerted when stretching titin, a giant mul- restricted range of stabilities the relationship between activation tidomain protein molecule. Single molecule experiments hold free energy and stability is linear. Nevertheless when the protein unique promise to give information about the mechanisms of is very stable, folding can occur by a downhill mechanism or biomolecular reactions. They can, in principle, confirm the co- conversely when very unstable, unfolding can be a downhill operative character of some conformational transitions, which near-barrierless process. Deviations from exponential kinetics are often is only indirectly inferred by observations of ensembles of expected in the downhill scenario. Also, when the process is not molecules. Also they can yield information about the intermit- dominated by a single bottleneck, the ruggedness of the landscape tency of transitions arising from the ruggedness of the energy is more apparent because events are much faster. landscapes of proteins, which has been predicted recently (4). This framework suggests that the way in which the force Stretching experiments also provide direct information about the affects stability is paramount. Because the force is a vector, mechanical forces and therefore the free energies involved in stability is a quadratic function of it for small magnitudes. In protein folding. Indeed the authors of those experimental papers this regime, models that treat the reaction coordinate as a made several intriguing inferences about the folding mechanism length can be misleading. Once the chain is polarized, the and energy landscape of the domains on the basis of their results. stability of the molecule will change more linearly with the In this paper we want to further discuss the interpretation of such additional force, and models that use length as the reaction single molecule mechanochemical experiments by using the en- coordinate may become more appropriate, although as we ergy landscape theory of protein folding and lattice model shall see some care is needed in interpreting them too literally simulations. as giving geometrical information about the transition state. There are two aspects of the mechanochemical experiments For very large extension forces, unfolding can become down- that require special attention. The first of these is that from the hill in character and again will vary slowly with tension. In this point of view of a protein chemist the stretching experiment regime, undoing specific traps can be the rate-limiting step. In perturbs the energy landscape for folding in an unusual way. addition, in the downhill regime, the force dependence of even The dominant energies that guide a protein to fold quickly are the elementary events of local conformation isomerization correlated strongly to the specific structure of the protein, that may become rate controlling (7). is, the energy landscape is a funnel (5, 6). The reaction We now illustrate these ideas with the help of a lattice coordinate for folding is a collective coordinate measuring, to simulation. The system used is a minimally frustrated three- a first approximation, the fraction of interactions that are letter code 27-mer on a cubic lattice (8–10). The energy native-like. This reaction coordinate is a scalar, i.e., it is landscape of this model system in a renormalized sense rotationally invariant. Most perturbations used to probe the corresponds to a fast-folding (millisecond time scale) small folding energy landscape therefore are also scalars. In contrast, helical (around 60 aa) protein (5). Without any stretching force the stretching force is a vector. Until the molecule is suffi- the energy is given by a sum of contact terms‡‡. The proper ciently polarized by stretching, the stretching force acts very incorporation of the stretching force into the model requires ineffectually on the reaction coordinate. Within the linear some subtlety because of the inherent rotational anisotropy of response regime, the Curie principle, in fact, would require any lattice. In reality, a configuration with any given end-to- there to be no effect of stretching on folding. On the other end length can be oriented in any direction. If the rotational hand, the single molecule stretching experiments generally motions of the chain are taken to be rapid, a contribution to were performed under extreme tension, which greatly distorts the energy landscape, favoring highly expanded configura- iTo whom reprint requests should be sent at present address: Center tions. This feature leads to the second unusual aspect of the for Studies in Physics and Biology, The Rockefeller University, New perturbation—it is very large on the scale of folding energies. York, NY 10021. e-mail: [email protected]. ‡‡ Energy landscape theory allows us to estimate the effects of A detailed description of the model can be found in refs. 8–10. The energy function used is a contact energy in which monomers that are these strong perturbations on the free energy surface that nearest neighbors on the lattice are considered in contact. There are controls the folding rate. two terms, one of which is the number of like contacts, Nl, (i.e., contacts between monomers of the same type) and the other of The publication costs of this article were defrayed in part by page charge which is the number of unlike contacts, Nu. The contact part of the energy is then: E 5 N E 1 N E . E and E set the strengths of payment. This article must therefore be hereby marked ‘‘advertisement’’ in cont l l u u l u like and unlike contacts, respectively. In this work they are 23 and accordance with 18 U.S.C. §1734 solely to indicate this fact. 21, respectively. The sequence used was ABABBBCBACBAB- PNAS is available online at www.pnas.org. ABACACBACAACAB. 2031 Downloaded by guest on September 25, 2021 2032 Biophysics: Socci et al. Proc. Natl. Acad. Sci. USA 96 (1999) the potential of mean force caused by the tension can be obtained by averaging over all orientations. This rotational averaging gives a contribution to the free energy of the chain given by 21 4p ^~ ! 5 F ~b !G lend b log b sinh Flend . [1] Flend F is the force on the ends, lend is the distance between the ends of the chain, and b is the inverse temperature. (The unit of distance is the separation of points in the lattice. Therefore two neighboring beads in the chain are a distance 1 apart.) This free energy contribution was first obtained by Wall in his theory of rubber elasticity (11). Lattice simulations were carried out with Monte Carlo kinetics (8, 9) using the sum of the internal (contact) energies and the stretch free energy contribution given by Eq. 1. The simulations were run at a temperature for which 76% of the time the system was in the ground state with no applied force. (In our simulation units, this condition represents a temper- ature of 1.4. Energies are measured in units of temperature, 5 i.e., kb 1. Again, the unit of length is the distance between neighboring residues in the chain.) Both folding and unfolding runs were performed. In addition, the multiple histogram method was used to evaluate the protein stability as well as free energy profiles as a function of the external force. In Fig. 1 we exhibit the unfolding times of the model protein as a function of the tensile force. The three regimes described in the beginning of these sections are clearly evident. For small and large forces, the logarithm of the unfolding time is roughly independent of the force, whereas for intermediate values it has a linear dependence on force. In Fig. 2 the stability of the lattice protein is plotted. Folding rates are related to the FIG. 2. Stability vs. stretching force. (Upper) The probability of stability through an extrathermodynamic relationship, which is being in the native state Pnat as a function of the stretching force. We illustrated in Fig. 3. The location of the transition state in the can define an equilibrium unfolding force, the force at which Pnat 5 Q coordinate is given by the slope of this curve (shown in Fig. 0.5, which is approximately 2.7. (Lower) The free-energy difference 3). Q is a measure of the number of native contacts for any (DG0) as a function of force.