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Energy landscapes and dynamics of ion translocation through membrane transporters: a meeting ground for physics, chemistry, and biology

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Abstract

The dynamics of ion translocation through membrane transporters is visualized from a comprehensive point of view by a Gibbs energy landscape approach. The ΔG calculations have been performed with the Kirkwood–Tanford–Warshel (KTW) electrostatic theory that properly takes into account the self-energies of the ions. The Gibbs energy landscapes for translocation of a single charge and an ion pair are calculated, compared, and contrasted as a function of the order parameter, and the characteristics of the frustrated system with bistability for the ion pair are described and quantified in considerable detail. These calculations have been compared with experimental data on the ΔG of ion pairs in proteins. It is shown that, under suitable conditions, the adverse Gibbs energy barrier can be almost completely compensated by the sum of the electrostatic energy of the charge–charge interactions and the solvation energy of the ion pair. The maxima in ΔGKTW with interionic distance in the bound H+ – A− charge pair on the enzyme is interpreted in thermodynamic and molecular mechanistic terms, and biological implications for molecular mechanisms of ATP synthesis are discussed. The timescale at which the order parameter moves between two stable states has been estimated by solving the dynamical equations of motion, and a wealth of novel insights into energy transduction during ATP synthesis by the membrane-bound FOF1-ATP synthase transporter is offered. In summary, a unifying analytical framework that integrates physics, chemistry, and biology has been developed for ion translocation by membrane transporters for the first time by means of a Gibbs energy landscape approach.

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Abbreviations

A− :

Anion

a :

Hydrated ion radius (m)

d :

Distance between c-rotor and a-stator in FO portion of ATP synthase (m)

E :

Local electrical field (Vm−1)

E :

Enzyme

E :

Young’s modulus (kg m−1 s−2)

ΔG :

Gibbs energy change (kJ mol−1)

ΔG sol :

Solvation Gibbs energy change (kJ mol−1)

ΔG desolvation :

Desolvation Gibbs energy change (kJ mol−1)

H+ :

Proton

H+ − A− :

Proton–anion charge pair

I :

Moment of inertia (kg m2)

k :

Boltzmann constant (= 1.38 × 10−23 J K−1)

k :

Torsional spring constant of γ-subunit in F1 (kg m2 s−2)

k":

Torsional spring constant of c-subunit α-helix in FO (kg m2 s−2)

L :

Characteristic length (m)

L :

Length of α-helix (m)

l :

Membrane thickness (m)

l :

Distance within the c-ring of FO (m)

m :

Mass (kg)

n :

Number of c-subunits in the c-ring of FO

P n :

Legendre polynomial of degree n

Δp :

“Protonmotive force” (kJ mol−1)

q :

Charge (C)

R :

Inter-ionic distance within H+ − A− charge pair (m)

R :

Radius of α-helix (m)

r :

Radial position (m)

r :

Radial distance of a single charged species from the center (m)

T :

Temperature (K)

t :

Time (s)

U :

Stored elastic energy (kJ mol−1)

z :

Distance along membrane access channel (m)

γ :

γ-Subunit of FOF1-ATP synthase

γ :

εW/εm

ε m :

Dielectric constant of membrane

ε w :

Dielectric constant of water

ζ :

Frictional coefficient (kg m2 s−1)

θ :

Angle subtended by the rotating c-subunit with the center of the c-ring in FO (°)

θ :

Angle swept by the imaginary line joining the trailing c-rotor residue and the upper a-stator residue in FO with respect to the equilibrium position (°)

θ″ :

Angle of rotation about the axis of the c-subunit (°)

λ :

Inter-ionic length scale (m)

λ D :

Debye length scale (m)

ρ :

Charge density (C m−3)

σ :

Poisson’s ratio

τ m,d :

Driving electrostatic motor torque in FO (kg m2 s−2)

τ m,r :

Resisting electrostatic motor torque in FO (kg m2 s−2)

τ m,net :

Net electrostatic motor torque in FO (kg m2 s−2)

Φ:

Electrical potential (V)

φ :

Delocalized electrical potential (V)

ψ :

Local electrical potential (V)

c:

Charges in water

D:

Debye

Eq:

Equivalent

m:

Membrane

max:

Maximum

s:

Solution

w:

Water

*:

High-energy or transition state of intermediate

Arg:

Arginine

Asp:

Aspartic acid

ATP:

Adenosine triphosphate

Glu:

Glutamic acid

His:

Histidine

DASS:

Divalent anion sodium symporter

K:

Kirkwood

KTW:

Kirkwood–Tanford–Warshel

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Acknowledgements

The author acknowledges the very thoughtful and constructive comments of both referees that have greatly contributed to improve overall readability and the presentation of many detailed aspects in the paper.

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  1. Department of Biochemical Engineering and Biotechnology, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016, India

    Sunil Nath

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Nath, S. Energy landscapes and dynamics of ion translocation through membrane transporters: a meeting ground for physics, chemistry, and biology. J Biol Phys 47, 401–433 (2021). https://doi.org/10.1007/s10867-021-09591-8

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