References

1

E.-L. Florin, V. T. Moy, H. E. Gaub, Science 264, 415 (1994); V. T. Moy, E.-L. Florin, H. E. Gaub, Science 266, 257 (1995).

2

The figure was prepared with the use of MOLSCRIPT, P. J. Kraulis, J.\ Appl. Crystallogr. 24, 946 (1991).

3

G. U. Lee, D. A. Kidwell, R. J. Colton, Langmuir 10, 354 (1994).

4

N. M. Green, Adv. Protein Chem. 29, 85 (1975); O. Livnah, E. A. Bayer, M. Wilchek, J. L. Sussman, Proc. Natl. Acad. Sci. USA 90, 5076 (1993).

5

S. Miyamoto and P. A. Kollman, Proteins 16, 226 (1993).

6

Reviewed in R. Elber, in New developments in theoretical studies of proteins, R. Elber, Ed. (World Scientific, Tai Seng, Singapore, in press).

7

J. A. McCammon, B. R. Gelin, M. Karplus, Nature (London) 267, 585 (1977); W. F. van Gunsteren and H. J. C. Berendsen, Mol.\ Phys. 34, 1311 (1977).

8

For algorithmic advances, see C. Niedermeier and P. Tavan, J. Chem.\ Phys. 101, 734 (1994); M. Eichinger, H. Grubmüller, H. Heller, P. Tavan (in preparation); and H. Grubmüller, Phys. Rev. E 52, 2893 (1995) and references therein. For parallel computing, see [19] and H. Heller, H. Grubmüller, K. Schulten, Mol. Simul. 5, 133 (1990).

9

Alternatively, one may compute the free energy profile along the unbinding reaction path from the MD simulation and derive the rupture force from that profile. However, because free energies pertain to systems in equilibrium, this suggested equilibrium approach rests on the questionable assumption that the rupture is a quasistationary process. Furthermore, it requires computations of free energies that notoriously exhibit slow convergence. Thus, in the case at hand our non-equilibrium approach appears to be more appropriate and is computationally less expensive.

10

P. C. Weber, D. H. Ohlendorf, J. J. Wendoloski, F. R. Salemme, Science 243, 85 (1989). The x-ray structure contains only residues 13 through 133 of the total of 159 residues. However, this cleavage does not considerably affect the binding affinity. The structure suggests that the interaction between the loop containing TRP 120 of the neighboring streptavidin monomer and the biotin in the binding pocket is considerably weaker than that for avidin [4].

11

The simulations were performed with the program EGO [19], which uses the CHARMM force field [20]. New multi-scale approximations [8] enabled us to consider the full Coulomb interaction (with a dielectric constant equal to 1) --- that is, no cut-off [20] was applied. For biotin, the partial charges were taken from [5], and the force constants from [21]. All simulations were carried out with an integration step size of fs. The lengths of chemical bonds involving hydrogen atoms were fixed [7], non-polar hydrogen atoms were described through compound atoms [20], and no explicit term for the hydrogen bond energy was included. To quantify the strength of hydrogen bonds we used a combination of energetic (mainly electrostatics) and steric criteria, as suggested in [22]. The protein--ligand-complex was placed within a sphere of TIP3 [23] water molecules with a diameter of Å\ ( total atoms). All surface water molecules were subjected to deformable boundary forces [24] [approximated by a quartic polynomial [20]] and coupled to a heat bath of K [19] with a coupling coefficient () of . Additionally, all atoms were weakly coupled to the heat bath (). After minimization, the system was equilibrated for ps because it exhibited relaxations for more than ps. After relaxation the protein backbone atoms showed a root mean square deviation from the x-ray structure of Å. Closer inspection revealed that a large fraction of the observed deviation arose from two loops (residues 65 to 70 and 112 to 122) at the tetramer interface, both of which are flexible in our monomer model but fixed by intermolecular interactions in the tetramer. The geometry of the binding pocket in our model does not deviate significantly from that of the x-ray structure (root mean square deviation Å). Structures randomly extracted from a subsequent MD run of ps in duration were used for the simulations described in the text.

12

This choice allows thermal fluctuations for of roughly Å , a value commonly observed for unperturbed atomic fluctuations in proteins. Therefore, the thermal motion of the pulled atom is essentially unaffected by the spring potential.

13

We note that thermal fluctuations entail a logarithmic dependency of the rupture force on the pulling velocity [3]; however, for the large velocities studied in our simulations a linear dependency due to friction dominates. This friction heats the binding region by at most K in the case of our slowest simulation. That heat is then effectively transferred to the heat bath; see [11].

14

The computed force profiles exhibit fast fluctuations caused by (artificial) resonance of our spring. Accordingly, by simply taking the maximum force from these unprocessed profiles one overestimates the rupture force. Therefore, before computing rupture forces, we removed the fast fluctuations by smoothing the force profiles with a Gaussian distribution of ps width; that width has been estimated from the decay of the position autocorrelation function of atom `O2' (compare Fig. 1 B). This approach introduces some arbitrariness into the force computation, because the maximum force depends on the smoothing width. To quantify that uncertainty, we determined error bars by considering force profiles smoothened on the scale of ps (providing an upper bound) and ps (lower bound), respectively.

15

H. Frauenfelder, G. A. Petsko, D. Tsernoglou, Nature (London) 280, 558 (1979).

16

This simulation covered ns, within which the cantilever was moved by Å. Considering the structural heterogeneity mentioned above, one cannot expect the finest details of the simulation or of the force profile to be reproducible. Therefore, we restrict our discussion to those `typical' features, which appeared in several of our simulations. We emphasize, however, that many more interactions contribute to the binding forces than those discussed here and that the rupture process is more complex than our simplifying pictures may suggest.

17

All hydrogen bonds found after equilibration are also found in the x-ray structure [P. C. Weber et al., J. Am. Chem. Soc. 114, 3197 (1992)].

18

For subtle reasons simple integration of our force profile does not yield correct equilibrium binding free energies. The question of whether and how these can be obtained from our simulations is, therefore, left for future discussion.

19

M. Eichinger, H. Grubmüller, H. Heller, User Manual for EGO_VIII, Release 2.0, Theoretische Biophysik, Institut für Medizinische Optik, Universität München, Theresienstr. 37, D-80333 München, Germany (1995); electronic access: http://www.mpibpc.gwdg.de/abteilungen/071/ego.html or http://www.imo.physik.uni-muenchen.de/tavan/molgroup/moloverview.html.

20

B. R. Brooks et al., J. Comp. Chem. 4, 187 (1983).

21

A. T. Brünger, X-PLOR, The Howard Hughes Medical Institute and Department of Molecular Biophysics and Biochemistry, Yale University, 260 Whitney Avenue, P.O. Box 6666, New Haven, CT 06511 (1988), 1992.

22

A. Kitao, F. Hirata, N. Go, J. Phys. Chem. 97, 10223 (1993).

23

W.L. Jorgenson, J. Chandrasekhar, J.D. Madura, J. Chem. Phys. 79, 927 (1983).

24

C. L. Brooks III and M. Karplus, J. Chem. Phys. 79, 6312 (1983).

25

We thank H. Gaub for stimulating this work, V. Moy for explaining details of the AFM experiment, M. Eichinger and H. Heller for discussions and help with the program EGO. Supported by the Deutsche Forschungsgemeinschaft, grant SFB 143/C1.