**A
Question of Balance? Nonlinear Complex Phenomena in Biology and Physics**

Richard G. Lanzara

Bio Balance, Inc.

30 West 86th Street

New York, New York 10024

**Abstract:** The
simple balance has many elemental and interesting relationships with complex
biological and physical problems. A fundamental equation of equilibrium,
derived from two equivalent ways to tilt a balance, models several nonlinear
phenomena in biology and physics with substitution of the appropriate functions
into the basic equation. This provides new insights into the future analysis of
these phenomena and may be a fruitful way to analyze many other areas as well.

*A balance that does not tremble
cannot weigh. *

*A man who does not tremble
cannot live.* - Erwin Chargaff

The simple balance has been studied since ancient times by both Archimedes (c287-212 B.C.) and Galileo (1564-1642). Since at least one of Archimedes’ notebooks is missing, the ancient Greeks may have understood several things that we never received down through the ages. Although it is one of the simplest and most examined of the physical systems, it may have deeper secrets to reveal by an unique analysis of its more fundamental properties. However, our own too rapid intuition may often lead us to err.

Take for example the following *gedanken* experiment with a hanging two-pan balance. With the balance
in an initial horizontal equilibrium and resting on a table, add just enough
weight to one pan of the balance so that the pan touches the surface of the
table. Next add two equal weights that are ten times larger than the weight
that caused the pan to touch the table. What happens to the balance when these
two larger weights are added equally to both sides of the balance?

This is a simple problem, but it illustrates our biases when we rely on intuition and forego scientific measurement and inquiry. When the larger weights are added equally to both pans of the balance, the pan that was touching the surface of the table will rise off of the table. The horizontal angle decreases and the balance becomes less tilted. Understanding why the horizontal angle decreases when equal weights are placed on each side of a balance that is tipped leads to some very interesting relationships.

In 1834, the physiologist E. H. Weber (1795-1878) studied the senses and the responses of humans to physical stimuli. He discovered that at least a 5% difference in weight was required for people to tell the difference between unequal weights. He hid the weights with a lightweight paper so the subjects could not see them. If the weight placed in the subject's hands was 100 grams for each hand, then he had to add 5 extra grams to one hand in order for people to sense that one hand held the larger weight. However, if the weight was 80 or 60 grams, he had to add 4 or 3 grams respectively for people to tell the difference. This law, which is also named the Weber-Fechner law, gained wide recognition when it was discovered that many of our sensory perceptions follow this law. However, the underlying basis for this law hasn't been clearly understood. Could it possibly be a basic physical law?

If we examine more closely the various ways that a
balance can be tilted, then the physical basis for the *gedanken* experiment and Weber's law may become evident. At the top of Fig. 1 is an equal arm
balance with equal sets of weights in horizontal equilibrium. Shown on the left
side of Fig. 1 is one way to tilt this system by placing unequal weights on the
right and left pans together with the original weights. This tips the balance
toward the side having the most weight that creates an angle a from the horizontal equilibrium. There is
an alternative but equivalent way to produce angle , which is by moving some of the original weight from one
side and placing it on the opposite side as shown in the right half of Fig. 1.

**Fig. 1.** Equivalent ways to tilt a balance to create
identical angles .

Therefore, we have for an equal arm balance the following equivalent ratios that both produce identical angles :

(1)

These ratios show why the pan of the balance was lifted off the table by
the addition of equal weights in our previous *gedanken* experiment. If w_{1} and w_{2} are
equally increased, then the ratios will be decreased along with the
corresponding angle .

Solving for the transfer of the fraction of weight, w, gives,

(2)

where w_{1} and w_{2} are the initial weights in horizontal
equilibrium. S_{1} and S_{2} are the additional weights added
to each side as shown in Fig. 1. Eq. 2 is a fundamental equation of physical
equilibrium that measures the net amount of stress applied to the initial
equilibrium.

In 1993, Eq. 2 was shown to obey Weber's law (1). Surprisingly, the manner by which biological receptors compress the sensory functions by a ratio-preserving process is strictly compatible with Eq. 2 (1). At that time, it was also suggested that a modified version of this equation could model the responses of biological receptors (1,2).

There is always
the impetus to take a simple system and elaborate on it. Therefore,
substituting mathematical functions, such as *f(S)* and *g(S)*, for the
parameters S_{1} and S_{2} in Eq. 2 gives,

(3)

This general expression compares the relative effects of the two
functions *f(S)* and *g(S)* on an equilibrium system, which allows us to consider
more complex variations of Eq. 2. Two of these variations are presented below.

More than half a century ago Langmuir (1881-1957)
proposed the chemical binding isotherm equation, such as SR** **= R(S)/(S+K), as a description for the absorption of
molecules onto surfaces. Since then it has been used universally in
pharmacology and chemistry to describe independent, single-site, binding of one
molecule to another. If the weights are applied to the pans of the balance
according to the Langmuir equation, then we can measure the stress produced by
unequal weighting to the two pans of a physical balance similar to the unequal
binding of a molecule to either side of a chemical equilibrium.

**Diagram** A two-state chemical equilibrium with binding of
molecule S:

The analogy between the physical and chemical balances requires a more detailed
consideration to relate each part of the two systems to one another. As shown
in the Diagram, the equilibrium constant, K_{R}, sets the initial
amounts of R_{1} and R_{2}. The binding of S to R_{1}
and R_{2} forms SR_{1} and
SR_{2}, which will stress the initial equilibrium if K_{1} and
K_{2} are unequal. Linking the physical parameters of the balance to
the chemical parameters from the Diagram, w_{1} = R_{1} and w_{2} = R_{2} and
substituting *f(S)* = SR_{1} = R_{1}(S)/(S+K_{1}),
and *g(S)* = SR_{2} = R_{2}(S)/(S+K_{2})
into Eq. 3, where K_{1} and K_{2} are the dissociation
constants of the molecule S for R_{1} and R_{2}. Then letting w = R yields,

(4)

where R represents the change in the amount of "weight"
equivalent to the perturbation produced by asymmetrical molecular binding (K_{1} ≠ K_{2}) (2). This provides
a convenient method to calculate the initial stress applied to a two-state
equilibrium in terms of competing dissociation constants, K_{1} and K_{2}.

When a ligand binds with a greater affinity to one side of a two-state chemical equilibrium this stresses the initial equilibrium toward the side with the higher affinity. However, this greatly depends upon how we define the chemical species that comprise the chemical equilibrium. This binding preference also produces the phenomena known as Le Chatelier’s principle. However, there is a critical difference between Le Chatelier's principle and R. Le Chatelier's principle states that the original equilibrium will shift to relieve the stress applied to the equilibrium; whereas, R determines the amount of state that must be transferred to produce an equivalent stress on the original equilibrium.

Eq. 4 was tested to see if it could generate responses
compatible with those for biological receptors. For this demonstration, (S)
represents an amount of weight available for Langmuir binding to R_{1}
and R_{2}, which is similar to the idea that the chemical concentration
represents an amount of a chemical species available to combine with another
chemical species. The dissociation constants, K_{1} and K_{2},
are arbitrarily set to 10 and 100 (all units are in grams), which represent the
unequal binding affinities (1/K_{1} and 1/K_{2}) of (S) for
pans A and B. The numbers aren't
important, they are easy to adjust for specific examples and are provided here
for demonstration purposes only. R_{1} and R_{2} were each set
equal to 100, and the amount of (S) was allowed to vary up to 500.

**Fig. 2.** Plots of R from Eq. 4 also showing plots of the weight on each pan
from Langmuir binding (Pan A and Pan B) and the total weight (Pan A+B). **(A)
**logarithmic x-axis **(B) **linear x-axis

With such a simple system one does not expect to see a detailed model emerge, but on the contrary, as one explores this system further many complex characteristics of receptor interactions become evident. In Figs. 2A and 2B, the curves for Pans A and B show hyperbolic binding as expected for Langmuir binding curves. Plots of the total weight (Pan A+B) are characteristic of two-site binding curves seen in several biological and pharmacological experiments. The plot for R on the logarithmic scale shows a bell-shaped curve that rises to a maximum and declines. On the linear scale (Fig. 2B), the plot of R shows a curve that rises to a maximum and then gradually declines. These are common patterns seen in many experiments that measure the responses of biological receptors (2).

Figs. 2A and 2B also display several other characteristics that are unique to response curves. First, the maximum of R is below the maximum values for any of the other curves. In Fig. 2A, the straight lines indicate the positions on the R curve where the 50% and 100% responses occur. The 50% response for R occurs at about 3 grams and the 100% response occurs at about 30 grams. Since these points occur where there is a relatively small fraction of the total binding, this suggests a physical rationale for the phenomena of spare receptors, which is a phenomenon in pharmacology that has puzzled pharmacologists for decades.

Second, the curve for R declines with the addition of extra weight. This shows that a physical balance desensitizes when the weights are applied according to the Langmuir binding equations. Desensitization, which is the fade of the response in the presence of continuous stimulation, is an essential physiological mechanism that regulates our responses to hormones and appears in a large number of important biological receptors (2). That this phenomenon occurs in our example with curves that are very similar to each other is not proof that they are similar phenomena. However, as one probes more deeply, the similarities continue to accrue.

There is evidence at the molecular level that the two
chemical states R_{1} and R_{2} result from the pH-dependence
of a common residue within receptors (3,4). We have previously constructed a two-state molecular model
that shows these two chemical states as an acid and base state. Fig. 3 shows
the molecualr electrostatic potentials of these states along with a potential
binding molecule. In this molecular model the acid and base states act as the
switch for receptor activation (Fig. 3) (4). Agonist ligands activate receptors
by showing a preference for the base state. This is the more electronegative
state shown in red in Fig. 3 that attracts the positively charged end of the
binding molecule shown in blue. This preferential attraction of the binding
molecule for the base state places an initial stress on the original receptor
equilibrium that is registered at the receptor either by a shift of the
equilibrium or by a change in the underlying dynamics of the receptor (4).

**Fig. 3.** The acid and base states of the molecular model for
the two-state chemical equilibrium for receptor activation (4). The molecular
electrostatic potentials are plotted as positive 25 (blue) and negative -25 kJ
mol^{-1} (red) meshes. A potential binding molecule (multicolored) is
also shown approaching the acid and base states.

Extending the balance analogy further, the phenomena of
inhibition and inverse agonism can be tested by adding the factor of (1+[I]/K_{i})
for a competitive antagonist, [I], binding to each state with the dissociation
constant, K_{i}. For
competitive inhibition of Langmuir binding, this factor is multiplied times
each of the dissociation constants, K_{1} and K_{2}. The
Langmuir functions then become, SR_{1} = R_{1}(S)/(S+K_{1}(1+[I]/K_{i})),
and SR_{2} =
R_{2}(S)/(S+K_{2}(1+[I]/K_{i})). An example of this is
shown in Fig. 4 for " + [I](-7,-9)" and " + 10[I](-7,-9)", where "" is shorthand for R. The two plots of with the inhibitor,
[I] and 10[I], display the parallel shift to the right typically seen in
competitively inhibited dose-response curves. However, there are often examples
where the inhibitor does not have exactly equal affinities for each state. If
we allow the inhibitor to have different dissociation constants in place of the
single K_{i}, we can create series of plots that show a wide range of
variations seen in many biological and pharmacological dose-response
experiments (Fig. 4).

**Fig. 4.** Plots of R, replaced by "", from Eq. 4
with the addition of the expressions for the competitive inhibitor, [I]. In the
legend, the numbers in the parentheses after [I] replace the single parameter K_{i}. These numbers represent the
exponential values for the dissociation constants of the inhibitor for each
receptor state, R_{1} and R_{2}. The (INV) plots,
shown in gray, are from Eq. 4 with K_{1} ≥ K_{2}, which makes negative.

If the K_{1} and K_{2}
values are reversed, then Eq. 4 is negative, which produces the inverse
agonist responses as shown in gray in Fig. 4. This suggests that the binding
ligand prefers the other state, which has been previously observed for inverse
agonists. Surprisingly, our simple balance model appears to describe the
relatively complex nonlinear properties of inverse agonism and modulation
observed in many biological receptors. Also, this model has previously
described receptor activation, fast receptor desensitization and a general method
for preventing desensitization (2). However, whether the balance represents a
realistic model for biological receptors is not the main issue.

The overall analogy suggests that we are measuring something fundamental to both a physical balance and the chemical equilibrium of biological receptors. Is this really so far fetched that the weighting of a balance corresponds to the perturbations of ligand binding to receptors? Perhaps it is how the underlying equilibrium in either system becomes stressed that is the core concept most important to measure.

What about testing other competing sets of functions? If
instead of using Langmuir functions, we examine the two Gaussian functions, *f(S)=* and *g(S)* =. Substituting into Eq. 3 gives,

(5)

For some arbitrary values, the resultant R has a positive and negative side as shown in Fig. 5. The values of the parameters are not particularly important for this example. The important point is that we can substitute a new function in place of the Langmuir functions and achieve another interesting result. Just as the balance can move either up or down, so the graph of R shows that weighing competing Gaussian probabilities produces a sine-like wave.

**Fig. 5. **Plots of the Gaussian functions and R from Eq. 5.

The x-axis is arbitrarily labeled "Relative Concentration", but this could have just as easily been labeled "Relative Probabilities" depending on the interpretation given to the functions of S. Could this be the source of the negative probability that Feynman found in quantum theory (5)? It is interesting to consider that two competing Gaussian probabilities yield a new character R that can be negative and describes the stress placed upon the underlying equilibrium of the probabilities between two states.

Obviously some of these examples are more developed than others, but the important point is that a fundamental equation of equilibrium derived from a simple balance may provide new insights into complex phenomena in biology and physics. The extension of this approach to other areas may prove fruitful as well.

References:

1. R.
G. Lanzara, *Math. Biosci.* **122**,
89 (1994) - Link.

2. R.
G. Lanzara, *Int. J. Pharmacol.* **1(2)**,
122 (2005) - Link.

3. L.
A. Rubenstein, R. G. Lanzara, *J. Mol. Struct. (Theochem.)* **430**, 57 (1998) - Link.

4. L. A. Rubenstein, R. J. Zauhar, R.
G. Lanzara, *J. Mol. Graphics Modell.* - Link.

5. R. Feynman, In *Quantum
Implications: Essays in Honor of David Bohm*,
B. Hiley, F. D. Peat, Eds. (Routledge and Kegan Paul, London, 1987) - Link.