**Desensitization of a balance with Langmuir binding of weights**

Richard G. Lanzara,
Ph.D.

A balance is described
with Langmuir binding of weights to each side. This simple physical system,
when coupled with a previously derived equation for the equivalent displacement
of weight that perturbs the equilibrium of a balance, may model two-state,
receptor systems in pharmacology for receptor activation. Surprisingly, this
system also shows desensitization similar to many drug-receptor systems.
Although understanding the biophysics behind receptor signaling is an important
endeavor by itself, this analogy with a simple balance may bring new insights
into other areas of biological chemistry and physics as well.

Our senses function
through a distinct class of molecules known as G protein-coupled receptors
(GPCRs). These receptors sense our environment and control most of our
physiology. They are targets for about one-third of modern drug development.
The responses of many of these GPCRs to stimuli typically show hyperbolic,
dose-response curves that approach a maximum value; peak and then decline back
to baseline values. This phenomenon often appears as either normal or skewed
bell-shaped, dose-response curves. Pharmacologists have named this decline
towards baseline "desensitization" [1-15]. Although many of the most important
biological events show receptor desensitization, this phenomenon remains poorly
understood [1-12]. Desensitization also appears counterintuitive by suggesting
that more of an activating drug or molecule lessens the response. Therefore, it
remains difficult for pharmacologists to reconcile how a molecule can both
activate and deactivate those same receptors, often within a millisecond to
microsecond time frame [12]. If there were a simple physical system that shows
a similar response, it may offer important insights into this problem.

A simple balance may
offer such a physical system. The balance can be conceptualized to represent
receptor equilibrium and has the additional advantage of revealing
relationships between a physical and chemical, two-state equilibrium [14-16]. When
a receptor chemically "senses" a hormone or drug molecule, the
molecule binds preferentially to one state of the receptor (or one side of the
balance) and perturbs the initial receptor equilibrium (so that it is out of
balance) [14,15]. This perturbation produces an induced shift in the initial
receptor equilibrium toward the preferred binding state. This has the effect of
enriching the population of the preferred receptor state. For a cellular signal
to be transmitted, there must also exist a series of poised equilibria that
transmit these perturbations from the external signals into intracellular
messages.

The quantifiable nature
of this perturbation (shift) is analogous to a previously derived equation for
a weight, ∆w, that equivalently perturbs a balance in equilibrium,

(1)
where S1
and S2
are the supplemental weights added to each side of a simple balance and w1
and w2
are the weights on the balance in an initial equilibrium before the addition of
the supplemental weights [13]. The
nature of ∆w provides a mechanism to calculate perturbations in similar
two-state, equilibrium systems [13-15]. By introducing Langmuir binding equations
for S1
and S2
into Equation (1), we can transform a simple balance into a two-state, biophysical
receptor model that describes many aspects of drug-receptor interactions
[13-16]. Letting w1 = R1, w2 = R2 where R1 and R2
represent the maximum amount of weight allowable for each side. Substituting
the Langmuir binding expressions for the supplemental weights, S1=R1(S)/(S+K1),
and S2=R2(S)/(S+K2)
into equation (1), where K1 and K2 are the affinity constants
for the two sides of the balance, and letting ∆w = ∆R yields,

(2)

where ∆R represents the
change in the amount of weight equivalent to the perturbation of the initial
equilibrium by unequal (K1≠K2) molecular binding. This equation is
identical to equation (6) in reference [14] that was originally derived for
drug-receptor responses where "∆RH" represented the change in the
initial, high affinity receptor state. This
model differs from other two-state receptor models in that the other models do
not calculate the net change in the amount of the receptor that is shifted by
the preferred binding. This is primarily because the other models do not make
the net change in the shift an unknown variable that can be solved explicitly.
This model corrects this problem by solving explicitly for the net shift, ∆R
[13-15].

Previously, Equation (2)
was used to model dose-response relationships in pharmacology [14,15]. However, before this study, Equation (2) was
not subjected to an experimental test with a physical balance. For the sake of
inquiry, Equation (2) was tested with a simple two-pan balance. K1
and K2
were arbitrarily set equal to 10 g and 100 g respectively, which represent the
unequal affinities (1/K1 and 1/K2) of each pan (A and B) for (S). R1
and R2
were each set arbitrarily equal to 100 g, and the amount of (S) varied up to
500 g for this experiment. Note that for this demonstration (S) represents an
amount of weight (or a concentration) potentially available for binding up to
the saturating conditions of R1 and R2. It should be noted that
the Langmuir binding equations for S1 and S2 allow a fraction of the
available maximum weights R1 and R2 to weight pans A and B
respectively (see Table 1). The Langmuir binding equations are plotted for S1
and S2
in Figure 1. As seen in Figure 1 and Table 1, the theory fits the experimental
data for the balance very well.

**Table 1. Experimental and theoretical values for the balance.**

Pan
A Pan B measured shift shift theory
(∆R)

__(S)* weight (g)* weight (g)* (deg.) (% normalized) (% normalized)__

1
9 1 18
37 40

10 50 9 43
88 91

30 75 23 49
100 100

50 83 33 47
96 95

100 91 50 39
80 82

300 97 75 27
55 56

* Note that (S) represents a
"concentration of available weight" that "weights" each of
the pans A and B according to the Langmuir equations for S1
and S2.
The maximum weight available for each pan is given by R1 and R2
respectively.

**Figure 1**

FIG. 1. Plots of the Langmuir binding to pans A and B of the
balance with the theory (Theory ∆R) from Eq. (2) compared with the experimental
measurements (experiment shift). The insert is the same curve as the theory (∆R)
plotted with a reverse y-axis and an expanded scale.

The insert in the upper
right corner of Figure 1 is the theory plotted on a reverse axis with an
expanded scale. From the physical point of view, the insert demonstrates the
striking similarity to similar figures from the tracings of excitatory
postsynaptic currents produced in neurons by the neurotransmitter molecule
glutamate [12], and similar pharmacological dose-response curves [4,10,14]
(also see below). This demonstrates the similarity of response curves produced
with this balance model to the nonlinear response curves that are ubiquitous in
many complex cellular systems. Although this is an arbitrary physical system,
it could easily be made more realistic by decreasing the constants K1
and K2
to create a more realistic system for the binding of molecules to receptors on
a cell surface [14, 15]. From a purely physical perspective, these observations
suggest that suitable physical systems may be created to mimic the behaviors of
the more complex biological ones.

As a further example,
Figure 2 shows the balance as a slightly more complicated but more realistic
model for a hypothetical cellular receptor. The parameter, (S), represents the
concentration of molecules in solution and ∆R is the theoretical response
calculated by Equation (2). Also plotted in Figure 2 are the Langmuir binding
plots for S1 and S2 and the total binding on a logarithmic scale.
Desensitization occurs in the presence of continued binding and is observed as
a bell-shaped curve as seen for ∆R in Figure 2.

**Figure 2**

FIG. 2. Shows Langmuir binding plots of

S1
and S2
for K1
and K2
of 1x10

**and 1x10**^{-9}**respectively. The parameter, (S), represents the concentration of molecules in solution. The total binding is given as S1+S2, and ∆R is the theoretical response from Eq. (2). Note that at 50% of the response (∆R) the total binding is only 20% or less.**^{-7}
Another interesting
observation is that the response of the system occurs at only a fraction of the
total binding (compare 50% ∆R response with the curve for S1+S2 in FIG. 2).
This has been called the phenomenon of "spare receptors" in
pharmacology and has puzzled pharmacologists for many years. However, we can
now see what produces the apparent spare receptor reserve. Since only a
fraction of the receptor molecules (or weight in the case of the balance) is
transferred as an equivalent net shift or response, ∆R, the magnitude of this
response will always be some smaller fraction of the total receptor pool or
weight of the system. Interestingly, this also demonstrates that a balance can
show the curiosity, which is known as "spare receptors", when modeled
with Langmuir binding.

The peak response of
these curves can be found by taking the first derivative of Equation (2) with
respect to S and setting it equal to zero. The peak occurs at (K1K2/2)

**which can be useful to know for modeling and experimental purposes. Note that inhibitors can also be introduced into Equation (2) by including the inhibition factor (1+ I/ Ki) for an antagonist (I with its dissociation constant, Ki) multiplied times each of the dissociation constants, K1 and K2. This reproduces the effects of inhibitors in pharmacological systems and leads to the discovery of many useful effects [14]. Since biological receptors can be modeled by this approach [13-15], pharmacologists and physiologists may find Figure 3 interesting for its similarity to the electrophysiological measurements of excitable cells [2,4,10,11,12]. The family of curves displayed in Figure 3 demonstrate the effect of decreasing the available receptor pool (R1 and R2) by 20 to 80%.**^{1/2}**Figure 3**

FIG. 3. The plot for a family of curves that demonstrate the
response curve (∆R) from FIG. 2 with 80%, 50%, and 20% of the total receptor
pool (R1 and R2) on a reverse axis with an expanded scale. The
insert shows ∆R plotted at a larger scale.

The similarity between
the responses observed in complex pharmacological systems and the balance model
strongly suggests that cellular receptors mimic miniature chemical balances in
a coupled, two-state equilibrium that can be shifted by perturbations from
unequal molecular binding, or possibly other modulating factors. It also
suggests that many of the complex behaviors observed in these biological
responses and similar systems are a direct result of the physicochemical
responses to the perturbations affecting the underlying equilibrium of these
systems. Although not explored here, this may also suggest potential
applications in the areas of rapid chemical kinetics and theory of enzyme
reactions.

In conclusion, treating a
simple balance with the restriction of Langmuir binding creates systems that
behave surprisingly similar to many receptor systems that desensitize and
suggests that a simple physical model displays those characteristics previously
thought peculiar to many complex biological processes [1-15]. Desensitization
is found in many unusual and amazing places - our senses, drug receptors, the
neurochemical synapses within our brains, and a simple balance. This may be the
first time that the behavior of a balance was described as a desensitizing
system and experimentally tested. A balance that shows responses that
desensitize similar to biological receptors offers a physical link between GPCR
activation and desensitization, and may have implications for many other areas
of chemistry and physics as well.

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