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Indian Journal of Biochemistry & Biophysics
Vol. 51, April 2014, pp. 93-99
Kinetic modelling of coupled transport across biological membranes
Kalyani Korla and Chanchal K Mitra*
School of Life Sciences, University of Hyderabad, Hyderabad, India
Received 21 October 2013; revised 10 March 2014
In this report, we have modelled a secondary active co-transporter (symport and antiport), based on the classical
kinetics model. Michaelis-Menten model of enzyme kinetics for a single substrate, single intermediate enzyme catalyzed
reaction was proposed more than a hundred years ago. However, no single model for the kinetics of co-transport of
molecules across a membrane is available in the literature. We have made several simplifying assumptions and have
followed the basic Michaelis-Menten approach. The results have been simulated using GNU Octave. The results will be
useful in general kinetic simulations and modelling.
Keywords: Antiport, Symport, Simulation, Octave, Michaelis-Menten kinetics
Living cells maintain different concentrations of ions
and metabolites across biological membranes, so as to
maintain the functionality and compartmentalization
by accumulating high concentration of molecules that
the cell needs, such as ions, glucose and amino acids
etc. Such a strategy helps the cell to maintain
intracellular environment different from the
extracellular environment1. Cells have evolved several
transport systems, including co-transport systems,
which can transport molecules against their
concentration gradient to cater to the personalised
needs of various compartments. However, movement
of molecules against the concentration gradient
requires energy and is termed as active transport,
which can be further classified as primary and
secondary active transport.
Primary active transport mainly includes
transmembrane ATPases and uses ATP to transport
ions across the membrane, while other transporters
use redox energy or light energy to facilitate the
transport. In secondary active transport or coupled
transport (co-transport), the energy is derived from
electrochemical potential difference (free energy)
created by the ion pumps, instead of directly using
ATPs. The structural basis of secondary active
transport mechanism is widely studied and is
reviewed by Forrest et al2. Co-transport (or secondary
active transport) can be further classified into symport

*Author for correspondence
E-mail: [email protected]
Tel: +91 40 2313 4668
and antiport, depending on the ligands movement in
the same or opposite directions. Cells facilitate the
transport of some substances against a concentration
gradient by using energy already stored in molecular
concentration gradients across the membrane, such as
proton, sodium or other charge/concentration
gradients via membrane proteins called transporters3.
Even after hundred years since Michaelis-Menten
kinetics model was proposed, none could explain the
kinetics behind co-transport of molecules across a
membrane in a satisfactory, usable manner. Various
earlier approaches in this direction have resulted in
overly complicated models4,5.
Sometimes, the system facilitates the transport of a
species by transporting another species across the
membrane, where the two species may move in same
or opposite direction, termed as symport and antiport,
respectively (Fig. 1). The movement of one species
across the membrane along its concentration gradient
supplies the free energy for the movement of second
species, irrespective of its concentration gradient and,
therefore, the second species may or may not move
along its concentration gradient6. These transports are
usually tightly coupled, so that the transport is strictly
1:1 ratio of ligands across the membrane7.
In many cases, the substrates being transported are
electrically charged. For such molecules, any electric
field across the membrane shall cause a force acting
on the molecule and this will be in addition to the
force due to the concentration gradient. Depending on
the charge on the substrate molecule and the electric
94
INDIAN J. BIOCHEM. BIOPHYS., VOL. 51, APRIL 2014
Fig. 1—Diagrammatic representation of two types of coupled
transport: antiport and symport [A1, A2 represent the
concentrations of ligand A in sides 1 and 2 of the membrane,
respectively. Similarly, B1 and B2 represent the concentrations of
ligand B in sides 1 and 2 of the membrane, respectively. White
opaque circles indicate the ligand binding sites and it is expected
that at any given point of time two of these will be occupied,
giving rise to six such possibilities in both the cases. Of these six
possibilities, only two will result in transportation for a given set.
The arrow shown in the centre of the figure indicates the positive
direction of flux and has been fixed arbitrarily for the sake of
uniformity during scripting]
field, it is to be added to the concentration gradient
and the computations may be modified accordingly8.
Extending Michaelis-Menten equation to a process,
where no active biochemical reaction takes place, has
not been widely used. In the present study, the
secondary active transport across membrane is
considered as a rate process and the equivalent
biochemical reaction is the transport of the molecule
across the membrane. The present approach is
intuitive, simple and conceptually elegant.
Methodology
The possible intermediates for a system with a
transporter (protein) E and two species A and B
located on sides 1 and 2 of a membrane and labelled
as A1, A2, B1 and B2 are shown in Fig. 1.
The protein has two binding sites on both sides of
the membrane — one for A and B each, so that the
transporter can function in a reversible manner. The
possible intermediates are shown in Fig. 2 in a
simplified form. After binding the correct number of
ligands, for example, EA1B1/EA2B2 for symport or
EA1B2/EA2B1 for antiport, the protein undergoes a
conformational change, resulting in the transport of
ligands to the opposite side (Fig. 2).
All the binding sites of transporter are considered
independent and equivalent. Therefore, all the binding
constants (equilibrium constants in the above
equations, k) are considered equal. This simplifies the
Fig. 2—Intermediate reactions and the products formed during
transport [Reactions involved in co-transport are not perfect
bisubstrate reactions, but are essentially carried forward via two
substrate binding and the substrates can bind in any order. The
outline shown here represents possible combinations and thus the
possible intermediates and their formation route. Of the six
bisubstrate intermediates formed, two are suitable for symport and
two for antiport and the rest two will block one of the channels
and no transport can take place. After transport, the ligands enter
the other side and thus the changes in subscript (1→2 and 2→1)
indicate the respective change of side. The intermediates with
three and four ligands bound are not shown for the sake of clarity
because they do not give rise to effective transport and are
inhibitory in nature]
subsequent equations and is also a very reasonable
assumption. Looking at the symmetry of equations,
one can immediately write down the concentrations of
various intermediate species (Table 1). Flux is derived
using Michaelis-Menten approach9.
In case when one or both the ligands are
electrically charged, the membrane potential i.e.
potential difference existing between the two sides of
the transporter will also play an important role8,10.
However, in the present study, we have not included
the role of electric field on the transport to maintain
conceptual simplicity. The electric potential can be
added to the total free energy and the computations
can be carried out without loss of generality.
Following the Michaelis-Menten approach, we
calculate the total enzyme concentration:
 E 0  = [ E ] + [ EA1 ] + [ EA2 ] + [ EB1 ] + [ EB2 ] + [ EA1 A 2 ] +
[ EA1B1 ] + [ EA1B2 ] + [ EA2 B1 ] + [ EA2 B2 ] + [ EB1B2 ]
+ [ EA1A 2 B1 ] + [ EA1A 2 B2 ] + [ EA1B1B2 ] +
[ EA2 B1B2 ] + [ EA1A 2 B1B2 ]
where [E] is the free enzyme concentration and [E0] is
the total enzyme concentration. Substituting values
from the rate equation given above, we get
KORLA & MITRA: KINETIC MODELLING OF COUPLED TRANSPORT
95
Table 1—Intermediate reactions taking place during coupled transport and their rate equations [The states of transporter with three and
four ligands bound to it are not shown for sake of clarity. The complete set has been included in the equations]
[ EA1 ] EA = k E A
[ ] [ ][ 1 ]
[ E ][ A1 ] 1
[ EA2 ] EA = k E A
E + A2 ⇔ EA2 k =
[ ] [ ][ 2 ]
[ E ][ A 2 ] 2
[ EB1 ] EB = k E B
E + B1 ⇔ EB1 k =
[ ] [ ][ 1 ]
[ E ][ B1 ] 1
[ EB2 ] EB = k E B
E + B2 ⇔ EB2 k =
[ ] [ ][ 2 ]
[ E ][ B2 ] 2
[ EA1A 2 ] EA A = k EA A = k 2 E A A
EA1 + A2 ⇔ EA1A 2 k =
[
] [ 1 ][ 2 ] [ ][ 1 ][ 2 ]
[ EA1 ][ A 2 ] 1 2
[ EA1B1 ] EA B = k EA B = k 2 E A B
EA1 + B1 ⇔ EA1B1 k =
[
] [ 1 ][ 1 ] [ ][ 1 ][ 1 ]
[ EA1 ][ B1 ] 1 1
[ EA1B2 ] EA B = k 2 EA B = k 2 E A B
EA1 + B2 ⇔ EA1B2 k =
[
] [ 1 ][ 2 ] [ ][ 1 ][ 2 ]
[ EA1 ][ B2 ] 1 2
[ EA2 B1 ] EA B = k 2 EA B = k 2 E A B
EA2 + B1 ⇔ EA2 B1 k =
[
] [ 2 ][ 1 ] [ ][ 2 ][ 1 ]
[ EA2 ][ B1 ] 2 1
[ EA2 B2 ] EA B = k 2 EA B = k 2 E A B
EA2 + B2 ⇔ EA2 B2 k =
[
] [ 2 ][ 2 ] [ ][ 2 ][ 2 ]
[ EA2 ][ B2 ] 2 2
[ EB1B2 ] EB B = k 2 EB B = k 2 E B B
EB1 + B2 ⇔ EB1B2 k =
[
] [ 1 ][ 2 ] [ ][ 1 ][ 2 ]
[ EB1 ][ B2 ] 1 2
E + A1 ⇔ EA1 k =
 E 0  = [ E ] + k [ E ][ A1 ] + k [ E ][ A 2 ] + k [ E ][ B1 ] + k [ E ][ B2 ]
+k 2 [ E ][ A1 ][ A 2 ] + k 2 [ E ][ A1 ][ B1 ]
+k 2 [ E ][ A1 ][ B2 ]
+ k 2 [ E ][ A 2 ][ B1 ] + k 2 [ E ][ A 2 ][ B2 ] + k 2 [ E ][ B1 ][ B2 ]
+k 3 [ E ][ A1 ][ A 2 ][ B1 ]
+k 3 [ E ][ A1 ][ A 2 ][ B2 ]
+ k [ E ][ A1 ][ B1 ][ B2 ] + k [ E ][ A 2 ][ B1 ][ B2 ]
3
3
+k 4 [ E ][ A1 ][ A 2 ][ B1 ][ B2 ]
This equation can be reduced to give,
 E 0 
= 1+ k [ A1 ] + k [ A 2 ] + k [ B1 ] + k [ B2 ] + k 2 [ A1 ][ A 2 ]
[E]
+k 2 [ A1 ][ B1 ] + k 2 [ A1 ][ B2 ] + k 2 [ A 2 ][ B1 ]
+k 2 [ A 2 ][ B2 ]
+ k [ B1 ][ B2 ] + k [ A1 ][ A 2 ][ B1 ] + k [ A1 ][ A 2 ][ B2 ]
2
3
3
+k 3 [ A1 ][ B1 ][ B2 ] + k 3 [ A 2 ][ B1 ][ B2 ] + k 4 [ A1 ][ A 2 ][ B1 ][ B2 ]
 E 0 
= S , where S depends on A1, A2, B1 and B2, the
[E]
expression on the right hand side (RHS) in the
equation above.
The concentration of free enzyme, E can be given
as
[ E ] =  E 0  / S
This value of E has been further used for equation
of symport and antiport.
Symport
For symport, the rate of transport will be given by
symport flux = k' [ EA1B1 ] − k'' [ EA2 B2 ]
Since only A1B1 and A2B2 are the correct
conformations to facilitate symport (See Figs 1 and 2).
The fluxes in the two terms of the above equation are
in opposite directions.
INDIAN J. BIOCHEM. BIOPHYS., VOL. 51, APRIL 2014
96
The enzyme function must be symmetric and,
therefore, binding constants of the enzyme for A and
B are expected to be same on both sides of the
membrane. So, we assume, k' = k'' for all practical
purposes.
From Table 1 we have,
[ EA1 B1 ] = k 2 [ E ][ A1 ][ B1 ] and
[ EA2 B2 ] = k 2 [ E ][ A2 ][ B2 ]
substituting values of E, we get
[ EA1 B1 ] = (k 2 [ E 0 ][ A1 ][ B1 ] / S and
[ EA2 B2 ] = k 2 [ E 0 ][ A2 ][ B2 ] / S
substituting these values in the equation for symport
flux, we have
2
0
symport flux= (k ′k [ E ]([ A1 ][ B1 ] − [ A2 ][ B2 ] / S
Antiport
For antiport, the rate of transport will be given by
antiport flux = k' [EA1B2]- k''[EA2B1] (k' and k'' are
forward and reverse rate constants, respectively) since
only A1B2 and A2B1 are the correct conformations
to facilitate antiport.
The enzyme function must be symmetric and we
expect k' and k'' to be equal, i.e. k' = k''.
From Table 1 we have,
[ EA1 B2 ] = k 2 [ E ][ A1 ][ B2 ] and
[ EA2 B1 ] = k 2 [ E ][ A2 ][ B1 ]
substituting value of E, we get
[ EA1 B2 ] = (k 2 [ E 0 ][ A1 ][ B1 ] / S and
[ EA2 B1 ] = (k 2 [ E 0 ][ A2 ][ B1 ] / S
substituting these values in the equation for antiport
flux, we have
antiport flux = (k ′k 2 [ E 0 ]([ A1 ][ B2 ] − [ A2 ][ B1 ]) / S
We have used these equations for calculating the
flux of each ligand for both systems, viz. antiport and
symport.
GNU Octave is a high-level interpreted language,
primarily intended for numerical computations.
It provides the numerical solution of linear and
non-linear problems and performs other numerical
experiments. It also provides extensive graphics
capabilities for data visualization and manipulation11.
We have used Octave for simulation of the
transporters using ODE (ordinary differential
equation) solver. In our earlier works, we have used
similar approach to simulate Krebs cycle, oxidative
phosphorylation and mitochondrial translation12.
Octave scripts written for simulation of coupled
transport process are provided in Table 2. The initial
values can be varied and system behaviour can be
studied accordingly.
Table 2—Octave script used for simulation of symport and
antiport
Script for symport simulation:
1;
% solve symport
% Equations for symport
% the rate equations follow:
ds=zeros(4,1);
s(1) = 0.4;
s(2) = 0.3;
s(3) = 0.1;
s(4) = 0.6;
%a1
%a2
%b1
%b2
function ds=sp(s)
t1=1+s(1)+s(2)+s(3)+s(4)+s(1)*s(2)+s(1)*s(3)+s(1)*s(4)+s(2)*s(3)
+s(2)*s(4)+s(3)*s(4)+s(1)*s(2)*s(3)+s(1)*s(2)*s(4)+s(1)*s(3)*s(4)
+s(2)*s(3)*s(4)+s(1)*s(2)*s(3)*s(4);
alpha=(0.1*(s(1)*s(3)-s(2)*s(4)))/t1;
ds(1)= -alpha;
ds(2)= alpha;
ds(3)= -alpha;
ds(4)= alpha;
return
end
% use lsode to solve this set...
lsode_options("relative tolerance",1e-4);
lsode_options("absolute tolerance",1e-3);
t=0:0.1:50;
% supply the initial concentrations ...
s0=s;
[s,T,MSG]=lsode(@sp,s0,t);
T
MSG
plot(t,s(:,1),"linewidth",5,t,s(:,2),"linewidth",5,t,s(:,3),"linewidth",
5,t,s(:,4),"linewidth",5);
grid on;
set (gca,'FontSize', 20);
axis([0,50,-0.1,1.01]);
% title ("symport");
xlabel ("Time/arbitrary units");
ylabel ("Concentration/arbitrary units");
set (gca,'FontSize', 20)
set(get(gcf,'children'), 'linewidth', 4 )
legend("A1","A2","B1","B2", "location", "northeast");
(Contd.)
KORLA & MITRA: KINETIC MODELLING OF COUPLED TRANSPORT
97
Table 2—Octave script used for simulation of symport and
antiport(Contd.)
Script for antiport simulation:
1;
% solve antiport
% Equations for antiport
% the rate equations follow:
ds=zeros(4,1);
%a1
s(1) = 0.4;
s(2) = 0.3;
%a2
s(3) = 0.1;
%b1
s(4) = 0.6;
%b2
function ds=ap(s)
t1=1+s(1)+s(2)+s(3)+s(4)+s(1)*s(2)+s(1)*s(3)+s(1)*s(4)+s(2)*s(
3)+s(2)*s(4)+s(3)*s(4)+s(1)*s(2)*s(3)+s(1)*s(2)*s(4)+s(1)*s(3)*
s(4)+s(2)*s(3)*s(4)+s(1)*s(2)*s(3)*s(4);
alpha=(0.1*(s(1)*s(4)-s(2)*s(3)))/t1;
ds(1)= -alpha;
ds(2)= alpha;
ds(3)= alpha;
ds(4)= -alpha;
return
end
% use lsode to solve this set...
lsode_options("relative tolerance",1e-4);
lsode_options("absolute tolerance",1e-3);
t=0:0.1:50;
% supply the initial concentrations ...
s0=s;
[s,T,MSG]=lsode(@ap,s0,t);
T
MSG
plot(t,s(:,1),"linewidth",5,t,s(:,2),"linewidth",5,t,s(:,3),"linewidth",
5,t,s(:,4),"linewidth",5);
grid on;
set (gca,'FontSize', 20);
axis([0,50,-0.1,1.01]);
% title ("antiport");
xlabel ("Time/arbitrary units");
ylabel ("Concentration/arbitrary units");
set (gca,'FontSize', 20)
set(get(gcf,'children'), 'linewidth', 4 )
legend("A1","A2","B1","B2", "location", "northeast");
The simulations have been carried out with GNU
Octave and the graphs have been made using gnuplot.
Results and Discussion
We plotted a curve for [EA1] vs. [A1] to see the
binding behaviour of one of the ligand, when the
Fig. 3—The concentrations of various intermediates during the
transport process [For simplicity, only the first intermediate EA1 is
shown for three different concentrations of A2, B1 and B2. The
important point to note in this graph is that the Michaelis-Menten
binding (black curve) is hyperbolic in nature, whereas the other
curves are not. High concentrations of ligands inhibit the transporter
very strongly (A1 = A2 = B1 = B2 = 0.9 gives <0.1 bound compared
to ~0.5 bound for Michaelis-Menten curve). k' = 0.1]
other ligands are present in different concentrations
(Fig. 3). The presence of other ligands in higher
concentrations shows an inhibitory effect on the
formation of EA1 intermediate. When compared with
the standard Michaelis-Menten curve, the other curves
do not show the hyperbolic nature. This is most likely
due to the additive nature of several parallel reactions
(transports in the present study), giving rise to same
product (transport of the molecule concerned being
the product). Individually, each reaction will give a
hyperbolic behaviour, when studied in isolation, but
the sum of several such processes is not necessarily
going to reflect in a hyperbolic graph.
The results of flux simulation are represented
graphically for symport and antiport (Fig. 4).
Symport
The simulation curves in Fig. 4a indicate the
behaviour of two species, depending on their
concentration gradient across a membrane in a
symport system. Here, the concentration gradient of A
and B across the membrane is 0.1 and -0.5. The
gradient of B is higher and hence it will move along
its concentration gradient and will drive the
movement of A against the concentration gradient.
Here, the blue and green curves form a pair (A1-A2)
and A moves against the concentration gradient. The
98
INDIAN J. BIOCHEM. BIOPHYS., VOL. 51, APRIL 2014
Fig. 4—Simulation curves for symport and antiport [Blue and green curves form a pair (A1/A2) and red and cyan forms the other pair
(B1/B2). Initial concentration gradients across the membrane of A and B are 0.1 and 0.5 respectively, where higher concentration is on
side 2 for both A and B. k' = 0.1. Initial concentrations of A1, A2, B1 and B2 are 0.4, 0.3, 0.1 and 0.6, respectively. (a) Symport - Curves
for A1/A2 are diverging and B1/B2 looks converging, i.e. the concentration gradient is increasing in case of A and decreasing for B. The
movement of ligands as inferred from the curve B2→ B1 and A2→A1. This indicates that for one set the concentration gradient is
decreasing and vice-versa and thus it is seen that the higher concentration gradient (B2-B1) is driving the flow of ligands (A1/A2) against
the concentration gradient. The steady state concentrations attained by A1, A2, B1 and B2 are 0.5, 0.2, 0.2 and 0.5, respectively; and (b)
Antiport - A1/A2 curves are converging first and then diverging and the other pair B1/B2 is converging throughout, i.e. the concentration
gradient is decreasing continuously in case of B, however, for A, it decreases first and then starts building up. The movement of ligands
as inferred from the curves is B2→ B1 and A1→A2. This indicates that the higher concentration gradient (B2-B1) is driving the movement
of A in the course which first decreases, becomes zero and then further rises since the concentration gradient of B is dominating. So,
initially movement of A is down the concentration gradient, but later is forced to move against the concentration gradient due to
concentration gradient of B. The steady state concentrations attained by A1, A2, B1 and B2 are 0.2, 0.5, 0.2 and 0.5, respectively]
free energy of the concentration gradient of B is partly
used up in building the free energy of the
concentration gradient of A, while the overall free
energy of the combined process decreases as a
spontaneous reaction.
Antiport
The simulation curves in Fig. 4b indicate the
behaviour of two species A and B, depending on their
concentration gradient across a membrane in an
antiport system. Here, the concentration gradient of A
and B across the membrane is 0.1 and -0.5. The
gradient of B is higher and hence it will move along
its concentration gradient and will drive the
movement of A in the opposite direction being
coupled in an antiport. As seen in the figure, A moves
along the concentration gradient initially, but
continues to move in the same direction even after
attaining same concentration on both sides of the
membrane and this movement against the
concentration gradient is driven by the movement of
B along its concentration gradient.
A surface plot is plotted by keeping the concentration
of A2 and B2 constant and varying the concentration of
A1 and B1 in the range of 1 to 10 (Fig. 5).
Allosteric effect
The binding of ligands to the transporter complex
resembles the allosteric binding process, first
proposed by Monod et al13. This involves
co-operativity, leading to strong binding of
subsequent ligand with a change in conformation of
different subunits. It is indeed important to include
interactions between different binding sites on the
transporter. The substrate binding to the transporter is
accordingly affected by this internal interaction and
this can be modelled according to the Monod's
allosteric model13. We have avoided this approach
basically to keep the transporter model simple and
avoiding additional computational complexities. It is
in fact quite straight forward to include any arbitrary
binding process in our present model.
Many transporters are also made up of multiple
subunits, which is an important requirement to attain
the symmetry of the transporter on either side of the
membrane. It is highly likely that a mechanism
similar to cooperative binding will be operative in
transporters in membranes. Co-transport of ligands
will then rapidly take place when the required
conformational conditions are achieved14. However,
KORLA & MITRA: KINETIC MODELLING OF COUPLED TRANSPORT
99
Fig. 5—The surface plot generated by plotting concentrations of A1 and B1 varying in the range of 1 to 10 [The concentrations of A2 and
B2 are however fixed at 3 and 6, respectively for both the plots. k' = 0.1. The contours are projected on xy plane. a: Surface plot for
symport flux; and b: Surface plot for antiport flux]
the experimental work in this direction is scant,
because of the difficulties in working with the
membrane-bound proteins. Experimental detection of
conformational changes in a protein due to binding of
multiple ligands is also very tricky.
Conclusion
We have simulated co-transport across a transporter.
This approach is based on Michaelis-Menten
approximation and can be used for kinetic simulation
of shuttles operating across the membranes, such as
malate-aspartate shuttle, citrate-pyruvate shuttle etc.
Acknowledgement
The authors thank Dr. M Durga Prasad for valuable
discussions. One of the authors (KK) acknowledges
the receipt of Junior Research Fellowship (JRF) from
University Grants Commission, India.
References
1 Srere P A & Mosbach K (1974) Annu Rev Microbiol 28,
61-84
2 Forrest L R, Krämer R & Ziegler C (2011) Biochim Biophys
Acta (BBA)-Bioenergetics 1807, 167-188
3 Albers R W (1967) Annu Rev Biochem 36, 727-756
4 Melkikh A V & Seleznev V D (2012) Progr Biophys Mol
Biol 109, 33-57
5 Sanders D, Hansen U P, Gradmann D & Slayman C L (1983)
J Membr Biol 77, 123-152
6 Krupka R M (1993) Biochim Biophys Acta (BBA)Bioenergetics 1183, 105-113
7 Lieb W R & Stein W D (1986) Transport and Diffusion
Across Cell Membranes (Stein W D, ed.), pp. 685, Academic
Press Inc., Orlando, FL.
8 Geck P & Heinz E (1976) Biochim Biophys Acta (BBA)Biomembranes 443, 49-63
9 Michaelis L & Menten M L (1913) Biochem Z 49, 333-369
10 Heinz E, Geck P & Wilbrandt W (1972) Biochim Biophys
Acta (BBA)-Biomembranes 255, 442-461
11 GNU Octave: http://www.gnu.org/software/octave/
12 Korla K & Mitra C K (2013) J Biomol Struct Dyn (ahead-ofprint), 1-17
13 Monod J, Wyman J & Changeux J P (1965) J Mol Biol 12,
88-118
14 Jardetzky O (1966) Nature 211, 969-970