The dynamics of biological polymers, including proteins, RNA, and DNA, occur in very high-dimensional spaces. Many naturally occurring polymers can navigate a vast phase space and rapidly find their lowest free energy (folded) state. Thus, although the search process is stochastic, it is not completely random. Instead, it is best described in terms of diffusion along a downhill energy landscape. In this context, there have been many efforts to use simplified representations of the energetics, for which the potential energy is chosen to be a relatively smooth function with a global minimum that corresponds to the folded state. That is, instead of including every type of physical interaction, the broad characteristics of the landscape are encoded in approximate energy functions. We describe a particular class of models, called structure-based models, that can be used to explore the diffusive properties of biomolecular folding and conformational rearrangements. These energy functions may be regarded as the spherical cow for modeling molecular biophysics. We discuss the physical principles underlying these models and provide an entry-level tutorial, which may be adapted for use in curricula for physics and non-physics majors.

1.
J.
Harte
,
Consider a Spherical Cow: A Course in Environmental Problem Solving
(
University Science Books
,
Sausalito
,
1988
).
2.
C.
Clementi
,
H.
Nymeyer
, and
J.
Onuchic
, “
Topological and energetic factors: What determines the structural details of the transition state ensemble and ‘en-route’ intermediates for protein folding? An investigation for small globular proteins
,”
J. Mol. Biol.
298
(
5
),
937
953
(
2000
).
3.
P. C.
Whitford
,
J. K.
Noel
,
S.
Gosavi
,
A.
Schug
,
K. Y.
Sanbonmatsu
, and
J. N.
Onuchic
, “
An all-atom structure-based potential for proteins: Bridging minimal models with all-atom empirical forcefields
,”
Proteins
75
(
2
),
430
441
(
2009
).
4.
J. K.
Noel
,
M.
Levi
,
M.
Raghunathan
,
H.
Lammert
,
R. L.
Hayes
,
J. N.
Onuchic
, and
P. C.
Whitford
, “
SMOG 2: A versatile software package for generating structure-based models
,”
PLoS Comput. Biol.
12
(
3
),
e1004794
(
2016
).
5.
M.
Cheung
,
J.
Finke
,
B.
Callahan
, and
J.
Onuchic
, “
Exploring the interplay between topology and secondary structural formation in the protein folding problem
,”
J. Phys. Chem. B
107
(
40
),
11193
11200
(
2003
).
6.
L.
Chavez
,
J.
Onuchic
, and
C.
Clementi
, “
Quantifying the roughness on the free energy landscape: Entropic bottlenecks and protein folding rates
,”
J. Am. Chem. Soc.
126
(
27
),
8426
8432
(
2004
).
7.
S.
Yang
,
S.
Cho
,
Y.
Levy
,
M.
Cheung
,
H.
Levine
,
P.
Wolynes
, and
J.
Onuchic
, “
Domain swapping is a consequence of minimal frustration
,”
Proc. Natl. Acad. Sci. U. S. A.
101
(
38
),
13786
13791
(
2004
).
8.
J. K.
Noel
and
P. C.
Whitford
, “
How EF-Tu can contribute to efficient proofreading of aa-tRNA by the ribosome
,”
Nat. Commun.
7
,
13314
(
2016
).
9.
M.
Levi
,
J. K.
Noel
, and
P. C.
Whitford
, “
Studying ribosome dynamics with simplified models
,”
Methods
162–163
,
128
140
(
2019
).
10.
M.
Levi
,
K.
Walak
,
A.
Wang
,
U.
Mohanty
, and
P. C.
Whitford
, “
A steric gate controls P/E hybrid-state formation of tRNA on the ribosome
,”
Nat. Commun.
11
,
5706
(
2020
).
11.
C.
Levinthal
, “
How to fold graciously
,” in
Mossbauer Spectroscopy in Biological Systems
(
University of Illinois
,
Urbana
,
1969
), Vol.
67
, pp.
22
24
.
12.
P. E.
Leopold
,
M.
Montal
, and
J. N.
Onuchic
, “
Protein folding funnels: A kinetic approach to the sequence-structure relationship
,”
Proc. Natl. Acad. Sci. U. S. A.
89
(
18
),
8721
8725
(
1992
).
13.
J.
Bryngelson
and
P.
Wolynes
, “
Spin glasses and the statistical mechanics of protein folding
,”
Proc. Natl. Acad. Sci. U. S. A.
84
(
21
),
7524
7528
(
1987
).
14.
J. D.
Bryngelson
and
P. G.
Wolynes
, “
Intermediates and barrier crossing in a random energy-model (with applications to protein folding
,”
J. Phys. Chem.
93
(
19
),
6902
6915
(
1989
).
15.
J.
Bryngelson
and
P.
Wolynes
, “
A simple statistical field-theory of heteropolymer collapse with application to protein folding
,”
Biopolymers
30
(
1–2
),
177
188
(
1990
).
16.
J. D.
Bryngelson
,
J. N.
Onuchic
,
N. D.
Socci
, and
P. G.
Wolynes
, “
Funnels, pathways, and the energy landscape of protein-folding: A synthesis
,”
Proteins
21
(
3
),
167
195
(
1995
).
17.
H.
Gould
,
J.
Tobochnik
, and
W.
Christian
,
An Introduction to Computer Simulation Methods: Applications to Physical Systems
(
CreateSpace Independent Publishing Platform
,
Scotts Valley
,
2017
).
18.
L.
Verlet
, “
Computer ‘experiments’ on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules
,”
Phys. Rev.
159
,
98
103
(
1967
).
19.
M.
Levi
and
P. C.
Whitford
, “
Dissecting the energetics of subunit rotation in the ribosome
,”
J. Phys. Chem. B
123
,
2812
2923
(
2019
).
20.
D. L.
Mobley
,
J. D.
Chodera
, and
K. A.
Dill
, “
Confine-and-release method: Obtaining correct binding free energies in the presence of protein conformational change
,”
J. Chem. Theory Comput.
3
(
4
),
1231
1235
(
2007
).
21.
H.
Kim
,
S. C.
Abeysirigunawarden
,
K.
Chen
,
M.
Mayerle
,
K.
Ragunathan
,
Z.
Luthey-Schulten
,
T.
Ha
, and
S. A.
Woodson
, “
Protein-guided RNA dynamics during early ribosome assembly
,”
Nature
506
(
7488
),
334
338
(
2014
).
22.
M.
Eastwood
and
P.
Wolynes
, “
Role of explicitly cooperative interactions in protein folding funnels: A simulation study
,”
J. Chem. Phys.
114
(
10
),
4702
4716
(
2001
).
23.
K.
Lindorff-Larsen
,
S.
Piana
,
R.
Dror
, and
D.
Shaw
, “
How fast-folding proteins fold
,”
Science
334
(
6055
),
517
520
(
2011
).
24.
R.
Kubo
, “
The fluctuation-dissipation theorem
,”
Rep. Prog. Phys.
29
(
1
),
255
284
(
1966
).
25.
J. K.
Noel
,
P. C.
Whitford
, and
J. N.
Onuchic
, “
The shadow map: A general contact definition for capturing the dynamics of biomolecular folding and function
,”
J. Phys. Chem. B
116
(
29
),
8692
8702
(
2012
).
26.
H.
Lammert
,
P. G.
Wolynes
, and
J. N.
Onuchic
, “
The role of atomic level steric effects and attractive forces in protein folding
,”
Proteins
80
,
362
373
(
2012
).
27.
R.
Pathria
and
P. D.
Beale
, “
Fluctuations and nonequilibrium statistical mechanics
,” in
Statistical Mechanics
, 3rd ed., edited by
R.
Pathria
and
P. D.
Beale
(
Academic
,
Boston
,
2011
), pp.
583
635
.
28.
A.
Einstein
, “
On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat
,”
Ann. Phys.
17
,
549
560
(
1905
).
29.
A.
Einstein
, “
Investigations on the theory of the Brownian movement
,”
Ann. Phys.
19
,
371
381
(
1906
).
30.
T.
Tomé
and
M. J.
de Oliveira
,
Stochastic Dynamics and Irreversibility
(
Springer-Verlag GmbH
,
New York
,
2015
).
31.
M. A.
Islam
, “
Einstein–Smoluchowski diffusion equation: A discussion
,”
Phys. Scr.
70
(
2–3
),
120
125
(
2004
).
32.
H.
Kramers
, “
Brownian motion in a field of force and the diffusion model of chemical reactions
,”
Physica
7
,
284
304
(
1940
).
33.
H.
Brinkman
, “
Brownian motion in a field of force and the diffusion theory of chemical reactions
,”
Physica
22
(
1–5
),
29
34
(
1956
).
34.
H.
Brinkman
, “
Brownian motion in a field of force and the diffusion theory of chemical reactions. II
,”
Physica
22
(
1–5
),
149
155
(
1956
).
35.
N. D.
Socci
,
J. N.
Onuchic
, and
P. G.
Wolynes
, “
Diffusive dynamics of the reaction coordinate for protein folding funnels
,”
J. Chem. Phys.
104
(
15
),
5860
5868
(
1996
).
36.
S. S.
Cho
,
Y.
Levy
, and
P. G.
Wolynes
, “
P versus Q: Structural reaction coordinates capture protein folding on smooth landscapes
,”
Proc. Natl. Acad. Sci. U. S. A.
103
(
3
),
586
591
(
2006
).
37.
H.
Risken
,
The Fokker-Planck Equation
(
Springer Berlin Heidelberg
,
Berlin, Heidelberg
,
1996
).
38.
S.
Yang
,
J. N.
Onuchic
, and
H.
Levine
, “
Effective stochastic dynamics on a protein folding energy landscape
,”
J. Chem. Phys.
125
(
5
),
054910
(
2006
).
39.
R. J.
de Oliveira
, “
Stochastic diffusion framework determines the free-energy landscape and rate from single-molecule trajectory
,”
J. Chem. Phys.
149
(
23
),
234107
(
2018
).
40.
F. C.
Freitas
,
A. N.
Lima
,
V. D. G.
Contessoto
,
P. C.
Whitford
, and
R. J. D.
Oliveira
, “
Drift-diffusion (DrDiff) framework determines kinetics and thermodynamics of two-state folding trajectory and tunes diffusion models
,”
J. Chem. Phys.
151
(
11
),
114106
(
2019
).
41.
The associated computational tools are available for download at <
https://github.com/ronaldolab/DrDiff>.
42.
D. I.
Kopelevich
,
A. Z.
Panagiotopoulos
, and
I. G.
Kevrekidis
, “
Coarse-grained kinetic computations for rare events: Application to micelle formation
,”
J. Chem. Phys.
122
(
4
),
044908
(
2005
).
43.
A.
Szabo
,
K.
Schulten
, and
Z.
Schulten
, “
First passage time approach to diffusion controlled reactions
,”
J. Chem. Phys.
72
(
8
),
4350
4357
(
1980
).
44.
J. G.
Kirkwood
, “
Statistical mechanics of fluid mixtures
,”
J. Chem. Phys.
3
(
5
),
300
313
(
1935
).
45.
B.
Roux
, “
The calculation of the potential of mean force using computer-simulations
,”
Comput. Phys. Commun.
91
(
1–3
),
275
282
(
1995
).
46.
A.
Ferrenberg
and
R.
Swendsen
, “
New Monte-Carlo technique for studying phase-transitions
,”
Phys. Rev. Lett.
61
(
23
),
2635
2638
(
1988
).
47.
A.
Ferrenberg
and
R.
Swendsen
, “
Optimized Monte-Carlo data-analysis
,”
Phys. Rev. Lett.
63
(
12
),
1195
1198
(
1989
).
48.
S.
Kumar
,
D.
Bouzida
,
R.
Swendsen
,
P.
Kollman
, and
J.
Rosenberg
, “
The weighted histogram analysis method for free-energy calculations on biomolecules. 1. The method
,”
J. Comput. Chem.
13
(
8
),
1011
1021
(
1992
).
49.
C. A.
McPhalen
and
M. N. G.
James
, “
Crystal and molecular structure of the serine proteinase inhibitor CI-2 from barley seeds
,”
Biochemistry
26
(
1
),
261
269
(
1987
).
50.
See <https://github.com/smog-server/SMOG2_tutorial> for the repository with instructions on how to make the simulations and all the simulated data analyzed in this work.
51.
E.
Lindahl
,
B.
Hess
, and
D.
van der Spoel
, “
GROMACS 3.0: A package for molecular simulation and trajectory analysis
,”
J. Mol. Model.
7
(
8
),
306
317
(
2001
).
52.
M. J.
Abraham
,
T.
Murtola
,
R.
Schulz
,
S.
Páll
,
J. C.
Smith
,
B.
Hess
, and
E.
Lindahl
, “
GROMACS: High performance molecular simulations through multi-level parallelism from laptops to supercomputers
,”
SoftwareX
1–2
,
19
25
(
2015
).
53.
J.
Chahine
,
R. J.
Oliveira
,
V. B. P.
Leite
, and
J.
Wang
, “
Configuration-dependent diffusion can shift the kinetic transition state and barrier height of protein folding
,”
Proc. Natl. Acad. Sci. U. S. A.
104
(
37
),
14646
14651
(
2007
).
54.
R. J.
Oliveira
,
P. C.
Whitford
,
J.
Chahine
,
V. B. P.
Leite
, and
J.
Wang
, “
Coordinate and time-dependent diffusion dynamics in protein folding
,”
Methods
52
,
91
98
(
2010
).
55.
R. B.
Best
and
G.
Hummer
, “
Coordinate-dependent diffusion in protein folding
,”
Proc. Natl. Acad. Sci. U. S. A.
107
(
3
),
1088
1093
(
2010
).
56.
K.
Schulten
,
Z.
Schulten
, and
A.
Szabo
, “
Dynamics of reactions involving diffusive barrier crossing
,”
J. Chem. Phys.
74
(
8
),
4426
4432
(
1981
).
57.
M.
Gruebele
, “
The fast protein folding problem
,”
Annu. Rev. Phys. Chem.
50
(
1
),
485
516
(
1999
).
58.
V.
Munoz
and
W. A.
Eaton
, “
A simple model for calculating the kinetics of protein folding from three-dimensional structures
,”
Proc. Natl. Acad. Sci. U. S. A.
96
(
20
),
11311
11316
(
1999
).
59.
J.
Kubelka
,
J.
Hofrichter
, and
W. A.
Eaton
, “
The protein folding ‘speed limit’
,”
Curr. Opin. Struct. Biol.
14
(
1
),
76
88
(
2004
).
60.
G.
Hummer
, “
From transition paths to transition states and rate coefficients
,”
J. Chem. Phys.
120
(
2
),
516
523
(
2004
).
61.
S. V.
Krivov
and
M.
Karplus
, “
Diffusive reaction dynamics on invariant free energy profiles
,”
Proc. Natl. Acad. Sci. U. S. A.
105
(
37
),
13841
13846
(
2008
).
62.
S. V.
Krivov
, “
Is protein folding sub-diffusive?
,”
PLoS Comput. Biol.
6
(
9
),
e1000921
(
2010
).
63.
F. C.
Freitas
,
G.
Fuchs
,
R. J.
de Oliveira
, and
P. C.
Whitford
, “
The dynamics of subunit rotation in a eukaryotic ribosome
,”
Biophysica
1
(
2
),
204
221
(
2021
).
64.
W.
Humphrey
,
A.
Dalke
, and
K.
Schulten
, “
VMD: Visual molecular dynamics
,”
J. Mol. Graph.
14
(
1
),
33
38
(
1996
).
65.
J.
Ousterhout
,
TCL and the TK Toolkit
(
Addison-Wesley
,
Reading, MA
,
1994
).
66.
See <https://www.ks.uiuc.edu/Research/vmd/> for more information about the VMD software.
67.
The PDB Protein Data Bank is located at <https://www.rcsb.org/>.
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