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Saturday, December 14, 2019

Molecular Dynamic Simulations and the Quest for Calculating the G Protein-Coupled Receptor Response



As scientists, we occasionally need to step back and scrutinize important issues within developing scientific fields.  Molecular dynamics simulations of various G protein-coupled receptors (GPCRs) have attempted to find the key conformations of these receptors that describe the activation and modulation of their experimentally observed responses. Although this quest has been pursued for over half a century, most current articles on this subject stress that simulations have made great progress, but that much more needs to be done to accurately model drug-receptor interactions.

Although thousands of crystal structures have been deposited in databanks, such as the Protein Data Bank (PDB), with consistently better resolutions, the conditions that led to their successful crystallization are vastly different from their in vivo environments. This naturally raises questions about the relevance of these structures for our understanding of their natural functioning states. These artificially produced crystal structures are far different than the GPCR’s transmembrane embedded, pH-dependent and REDOX (Reduction (RED) and Oxidation (OX)) sensitive states in vivo.  After decades of work, we’re not much closer to discovering GPCR’s natural modes of activation that necessarily include these pH-dependent and REDOX sensitive states for these fascinating proteins.  

If one critically examines the dogma that has crept into the field of molecular simulations, then one must begin to question some of that dogma. For instance, the dogma surrounding the essential disulfide bond (linking together two cysteine amino acids by their thiol or sulfhydryl groups) found in many GPCRs, suggests that those studying the molecular dynamics of these proteins may have forgotten some of their biochemistry concerning the stability of disulfide bonds in vivo. Many of these disulfide bonds are easily broken and reformed under natural conditions. In addition, many GPCRs such as rhodopsin contain an odd number of extracellular cysteines so that there may always exist at least one free cysteine that remains reactive. A free cysteine also accounts for the pH-dependence and REDOX properties experimentally associated with many GPCRs.

There have been relatively few attempts to model these extracellular cysteines in their free acid (SH) and base (S-) states. The complexities of cysteine sulfhydryl chemistries in vivo add multiple layers of complexity and confusion when ascertaining the functions of GPCRs. Even the seemly simple treatment by Dithiothreitol (DTT) to liberate the two cysteine thiols, or sulfhydryls, from a disulfide bond, may also block their subsequent reactions with other drugs or ligands in binding or activation assays. In addition, once the sulfhydryl groups are free, their pKa, which determines when the free thiol, or SH, group becomes deprotonated, may vary over a very wide range depending on the polarization of neighboring groups and the surrounding membrane charges. These free thiols, or sulfhydryls, are also sensitive to oxidation under normal atmospheric conditions, which greatly complicates the experimental study of receptors under normal laboratory conditions.

Least we become too confident that our simulations are completely accurate and predictive, many molecular dynamic simulations are done with the hope of discovering unique conformational changes, or “mechanistic hypotheses”, for receptor activation, but this approach may be digging a much deeper hole than initially intended. The almost endless search for better, more meaningful simulations with more powerful computers, stretches the horizon of drug discovery into vastly more complex simulations with lipids, water, counter-ions and other proteins necessary for receptor activation. These endless cycles of refinement and simulation don’t provide us with a good model for the necessary molecular switch between an off and on state for receptor activation. We’re usually happy if we can see clear conformational differences between the agonist and antagonist binding with their targeted receptor molecules, but what about the partial agonists, allosteric modulators, inverse agonists, rapid desensitization and tachyphylaxis? These present enormous challenges to our present simulation methods. Many laboratory experiments are and were previously done with chiral mixtures of enantiomers, whereas, the molecular dynamic simulations usually use only one enantiomer that is considered the most active. What are the implications comparing these mixtures used for laboratory experiments versus molecular dynamic simulations using only one enantiomer?  

Similar criticisms can be made about QSAR (Quantitative Structure-Activity Relationship) studies of biological molecules, because they assume that the observed structure and properties that are modeled will help to predict the molecular behaviors. The idea that structure and its accompanying properties can predict function is appealing, but fraught with difficulties. We’re tempted by our ability to make molecular and protein structures with properties such as charges. We examine these charge patterns to see what we can learn about their reactive properties and then make inferences about how they react with another molecule, but in the environment of a cell, these charges may be fleeting or nonexistent if they are surrounded by a sea of catalytic molecules such as enzymes that can add or remove groups such as a phosphate or a plethora of many other functional groups.

We know the structures and properties of buildings and cars, but their functions depend upon many other variables such as their surrounding environments and the people who occupy those buildings and cars. Similarly, biological proteins and molecules are embedded within a cellular milieu with many other molecules that act as cofactors, energy providers, anchors, etc. These exist in a range of environments that have varying charges, fields, lipophilicities, gradients, etc. The whole is much greater than the sum of its parts when it comes to understanding the functioning of biological molecules.  

However, as hard-nosed scientists, we should ask the toughest questions, which suggests that we should ask what molecular mechanisms might function as a distinct on and off switch for the GPCRs? We know that something like a chemical, net shift must drive receptors from their inactive to active states. Current debates center around conformational selection versus conformational induced fit. The presence of constitutively active receptors due to receptor overexpression suggests that conformational selection may be the preferred mechanism for receptor activation. Alas, this doesn’t tell us what that active conformation is. Because the net binding energies of many drugs, or ligands, are similar in magnitude to the background thermal noise at normal temperatures, “mechanistic hypotheses” don’t describe a clean off and on switch. The primary criticism being that there are no distinct boundaries between the on and off states that determine the extent of movements of any residue, helix, beta sheet, or loop that are necessary to select the active conformational state. 

A better model for the molecular on and off switch would be something such as a distinct change from an acid to base state that would also be accompanied by an electrostatic change within the receptor. This would also be consistent with the experimental observations that many GPCRs show an increase in their activities at higher pH levels. There’s an ongoing need to critically analyze and incorporate much more of the available and reliable experimental data into our molecular dynamic simulations.

With the enormous scientific talent out there, we can and will explore more productive models for receptor activation. Having any model that provides some possible insights into the functions of GPCRs, may be pleasing, but we must not abandon many years of previous experimental observations that have been repeatedly checked and verified.

We may not be making more timely progress because the biochemists and enzymologists aren’t communicating enough with the pharmacologists and biophysicists and those who perform theoretical simulations.  By combining our collective expertise and maintaining a skeptical, but open mind, we will greatly enhance our understanding of how our sensory and drug-targeted receptors function.   

Richard G. Lanzara, MPH, Ph.D.

Saturday, September 21, 2019

Problems Inherent Within Pharmacological and Biological Assays

This essay addresses inherent problems within most pharmacological and biological assays that all scientists should know. A basic scientific principle is that scientific theories are overturned by experimental evidence that doesn’t support the current theory, but this is only true if the experiments are not only accurate and repeatable, but also accurately represent the natural reactions that happen within the whole animal or organism. This is especially important in the biological and pharmacological sciences, because there are numerous variables that influence the experimental outcomes that receive little or no mention in the mainstream scientific literature. In order to develop new drugs, we need to increase our awareness of the critical variables that are seldom mentioned in most experimental assays. Many of these factors may be why assays are often difficult or impossible to reproduce. When we look at the bigger picture, it is a wonder that we place so much trust in our assays for the important roles that they play in everything from drug development to medicine and research. To step back and see this bigger picture, I will briefly discuss three general factors that often alter the outcomes of these assays.


Three Factors Complicating Most Pharmacological and Biological Assays

1) Confounding Effects:

An overlooked subject that greatly influences pharmacological and biological assays is desensitization or substrate inhibition. This is observed when higher doses of drugs, ligands or substrate molecules show a decrease in the response of the receptor or activity of an enzyme. Most receptor and enzymatic reactions display some negative feedback such as desensitization or substrate inhibition. However, this often goes unnoticed because many assays measure the cumulative response, which is the area under the actual dose-response curve (or the area under the kinetic reaction curve) for different drug, ligand or substrate concentrations. Assays measuring a cumulative response mask the underlying desensitization or substrate inhibition within these systems.

It is often, wrongly, thought that by keeping the drug or substrate molecules at lower levels the effects of desensitization or substrate inhibition will be reduced or prevented.  This isn’t accurate, because the phenomena of desensitization and substrate inhibition is inherent within the chemical equilibria of these systems. Many scientists and medical doctors aren’t aware of the ubiquity of drug desensitization or enzyme substrate inhibition, because many experimental assays aren’t capable of clearly demonstrating these phenomena. These issues present a large problem in drug development and the proper pharmaceutical treatment of patients.

Interpretation of assay results is perhaps another one of the most confounding factors. Our interpretations are largely model dependent. We tend to see what our models allow us to see. Models, such as curve fitting, often create confounding factors that are very much dependent on the assumptions underlying these models. Any two-dimensional curve can be fitted by an arbitrary polynomial to arbitrarily high powers; however, this type of fitting does little to elucidate the underlying biophysical mechanisms of these systems. Statistically we can get a good fit with a bad model. Statistics alone cannot determine the quality of our research. A good model should be like a good pair of glasses that helps us see more clearly the underlying biophysical and physicochemical mechanisms of our assays. Bad models blur our understanding and create confusion.

Current trends using extremely simplified simulations to model and understand the reactions of enzymes or receptors may be hindering our understanding of the underlying basic principles. It seems unlikely that such relatively small differences of 2-5 kcal/mole in binding energies of drugs, ligands or substrates will produce signal-specific wiggles against the background thermal noise in these much larger molecules. Because of our limitations, our current models cannot accommodate many of the other important molecules included within these complex systems, such as the solvent, lipid, other proteins, counter-ions, redox, energy, and cofactor molecules that are necessary components for the activation of these systems in their natural state. Because these complex systems are simplified and run in a vacuum, we’re stuck trying to make sense of rather meaningless wiggles. These observations often serve to obfuscate rather than clarify our understanding of the underlying biophysical principles.  

2) Past Histories:

In general, assays can be divided into those using whole organisms (in vivo) or isolated preparations (in vitro), but there’s also the more general and less controlled variables of past histories and current environmental conditions surrounding these assays, which applies to both in vivo and in vitro assays. Past histories include considerations such as how the animals were handled, caged, fed, type of bedding, etc., or how the isolated preparation was prepared, such as type and amounts of buffers used, fluctuations in temperatures (and the duration and order of these fluctuations), osmotic pressures, pH, solvents; exposures to atmospheric (oxidation) conditions at what stages of the preparation and for what length of time, etc. How pure were our preparations, chemicals, buffers, solvent(s), cofactors, etc? Did the type of containment vessels (glass/plastic) alter the preparations?

Past exposures to various exogenous (xenobiotics, pollutants) and endogenous (steroids, fatty acids) chemicals may induce metabolic pathways such as the Cytochromes P450 enzyme systems (CPYs) that handles many exogenous and endogenous molecules. These induced pathways can profoundly alter our assays in ways that we don’t currently see or understand fully.  

These factors are usually not reported in the scientific literature to the detailed extent that is necessarily suggested here. In some cases, scientists have recognized these confounding facts and tried to account for them, but in general we should all be aware of these serious problems in interpreting any experimental results. Only by recognizing these problems and the limitations that they place on our current assays can we make future progress toward a better scientific understanding of the experimentally observed responses of receptor and enzyme systems.

3) Assay Conditions:

In general, the more procedures that required to isolate any biological sample, the more errors accumulate such that easy replication becomes very difficult or impossible.

Very little or no attention is given to the REDOX (Reduction-Oxidation reactions) environment of receptors or enzymes when measuring their activities in vitro. In vitro, there is little or no regard to the possible effects of light, high oxygen (including REDOX status), electromagnetic gradients, etc. The processes of isolating receptor or enzyme molecules often exposes them to oxidation conditions that generally go unaccounted for the possible effects on their redox sensitive groups.

The requirements of cofactors for enzyme reactions has been previously discovered, but there may be additional requirements such as the requirement for an energy source from molecules such as ATP or GTP or membrane energy gradients. Additional molecules may be necessary to sustain and regenerate these energy molecules or gradients across membranes. There may also be requirements for essential regional molecules such as gases (CO2, H2S, NO, etc.) that may have far greater tissue concentrations in vivo than in our in vitro assays.

Another problem that may seem simple, but is quite complicated, is simply determining the pH-dependence of a receptor response, or an enzymatic reaction. Often these reactions are done at various pHs to find the optimum pH for that specific reaction under the conditions of specific temperature, pressure, osmotic pressure, etc. To complicate matters, the binding drug, ligand, or substrate molecules may have their own titratable groups that are pH-dependent that differ from the receptor or enzyme molecules, which often have multiple titratable groups that may also act to influence each other. Other problems arise because the receptor or enzyme molecules are often membrane bound in vivo whereas the assays are performed in vitro. Other significant problems, which are very difficult to accommodate into in vitro assays, is that biological membranes often separate regions with different pHs, counter-ion concentrations, osmolarities, etc. Even the simpler problems such as determining the proper buffer(s) to use, the concentration, temperature correction(s), unwanted effects on other molecules, such as solvent(s), cofactors, pH-detector(s), etc. are daunting.

In general, it is also very difficult to perform assays under strictly in vivo conditions. The confounding problems with using the whole organism entails many additional factors that affect the experimentally measured responses, which include the pharmacokinetic factors such as the ADME (Absorption, Distribution, Metabolism and Excretion).  Each of the ADME factors have multiple complexities that can confound experimental observations. Just considering the distribution factor alone is often complicated by a drug, ligand, or substrate molecule having to cross one or more membrane barriers, and by the differing tortuosity of the route to tissue-embedded groups of receptor or enzyme molecules. These barriers can greatly delay or alter the drug or substrate molecules from reaching their target receptors or enzymes. The surrounding microenvironments often determine the further metabolism and replenishment rates of the drug, ligand, or substrate molecules, which are also dependent on the lipid compositions of membranes as important considerations. The correct biochemical and biophysical tensions across these membranes are also vitally important to ensure that these assays accurately reflect their natural biological activities. The correct ionic, osmotic, electrochemical, and pH gradients are the most obviously important ones, but there are many others.

Assays that test experimental drugs for potential further drug development are perhaps one of the most critical components of the drug discovery process; yet they remain poorly characterized for this as well as for other biological purposes. This is a very general description of several problems with experimental assays that I’ve noticed over the four plus decades of my career covering experimental, computational and theoretical approaches to many scientific problems. Some of these problems may seem simple but continue to remain largely marginalized and go unnoticed. They need to be recognized and openly discussed so that further progress can be made.  Other problems are much more complex than current experimental techniques can handle, but knowledge of these problems may spur improved assays or at least make us aware of the many problems inherent within our current pharmacological and biological assays.

Richard G. Lanzara, MPH, Ph.D.
President and Principal Scientific Officer
Bio Balance, Inc.

Friday, June 14, 2019

Quintessence - Life's Essential Balance Between Stability, Novelty and Fateful Encounters


We’re two scientists trying to understand Life and Evolution. On a fundamental level, we may all wonder if Life is somehow hard-wired into the structural design of the universe. This naturally leads to the question of what is it that is hard-wired into this vast universe that produces Life? Are there yet unknown chemistries that we haven’t explored? Are there unknown physical laws that govern the flow of energies into Life’s multiple forms? We attempt to answer these questions and more about Life’s most basic principles. This book is meant to engage and catalyze the reader’s mind on a journey of discovery, wonder and awe as we explore and marvel at the wonders of Life and Evolution. 


Thursday, October 5, 2017

How Understanding Weber's Law Led to an Understanding of the Receptor Response


An understanding of Weber's law in terms of the net shift of weights on a simple balance (see: Weber's Law Modeled by the Mathematical Description of a Beam Balance) led to a better measure for the receptor response.

By combining Langmuir binding to two receptor states representing high and low affinity states analogous to the two pans at the end of the arms of a simple balance, an expression can be derived that describes ligand binding and activation of receptors remarkably well (see: The Biophysical Basis for the Graphical Representations and Method for determining drug compositions to prevent desensitization of cellular receptors). It should be noted that these two states were previously given a biophysical representation (see: Activation of G Protein-Coupled Receptors Entails Cysteine Modulation of Agonist Binding and Molecular dynamics of a biophysical model for β 2-adrenergic and G protein-coupled receptor activation).

Whether or not these biophysical descriptions are accurate or not is somewhat immaterial to the basic understanding regarding the biophysical basis for these two possible states. I would argue that these descriptions satisfy a large number of essential criteria that are largely missed or glossed over in many attempts to model the possible transitions from inactive to active receptor states. Without going into too much detail, I will just mention that the free energy change between these states is quit low (estimated to be 2-3 kcal/mol), which suggests that an electrostatic change may be a better candidate for the initial change to produce receptor activation. This better accounts for the oxidation-reduction properties and pH-dependence of many of the G protein-coupled receptors. So if the exact details of the above models aren't perfect, I still think that they're on the right tract.

Over the years, this approach has been able to provide biophysical descriptions for ligand efficacy, affinity, spare receptors, tachyphylaxis and rapid desensitization; as well as the effects of receptor overexpression and agonist/antagonist combinations on the receptor response (see: Method for determining drug-molecular combinations that modulate and enhance the therapeutic safety and efficacy of biological or pharmaceutical drugs).

After almost thirty years of development, perhaps it is time to attempt the transition to a cleaner biophysical definition for the receptor response. I certainly realize that this is a quixotic quest, but I believe that it will ultimately be worth the effort!

Monday, September 18, 2017

Nonlinear Complex Phenomena in Biology and Physics


Richard G. Lanzara, Ph.D.

 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 w1 and w2 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 w1 and w2 are the initial weights in horizontal equilibrium. S1 and S2 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 S1 and S2 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, KR, sets the initial amounts of R1 and R2. The binding of S to R1 and R2 forms SR1and SR2, which will stress the initial equilibrium if K1 and K2 are unequal. Linking the physical parameters of the balance to the chemical parameters from the Diagram, w1 = R1 and w2 = R2 and substituting f(S) = SR1 = R1(S)/(S+K1), and g(S) = SR2 = R2(S)/(S+K2) into Eq. 3, where K1 and K2 are the dissociation constants of the molecule S for R1 and R2. 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 (K1 ≠ K2) (2). This provides a convenient method to calculate the initial stress applied to a two-state equilibrium in terms of competing dissociation constants, K1 and K2.
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 R1 and R2, 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, K1 and K2, are arbitrarily set to 10 and 100 (all units are in grams), which represent the unequal binding affinities (1/K1 and 1/K2) 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. R1 and R2 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 R1 and R2 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]/Ki) for a competitive antagonist, [I], binding to each state with the dissociation constant, Ki.  For competitive inhibition of Langmuir binding, this factor is multiplied times each of the dissociation constants, K1 and K2. The Langmuir functions then become, SR1 = R1(S)/(S+K1(1+[I]/Ki)), and SR2 = R2(S)/(S+K2(1+[I]/Ki)). 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 Ki, 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 Ki. These numbers represent the exponential values for the dissociation constants of the inhibitor for each receptor state, R1 and R2. The  (INV) plots, shown in gray, are from Eq. with K1 ≥ K2, which makes  negative.

If the K1 and K2 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.
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.

Friday, September 15, 2017

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


 Richard G. Lanzara, Ph.D.

 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 w1 and w2 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 w1 and w2 are the initial weights in horizontal equilibrium. S1 and S2 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 S1 and S2 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, KR, sets the initial amounts of R1 and R2. The binding of S to R1 and R2 forms SR1and SR2, which will stress the initial equilibrium if K1 and K2 are unequal. Linking the physical parameters of the balance to the chemical parameters from the Diagram, w1 = R1 and w2 = R2 and substituting f(S) = SR1 = R1(S)/(S+K1), and g(S) = SR2 = R2(S)/(S+K2) into Eq. 3, where K1 and K2 are the dissociation constants of the molecule S for R1 and R2. 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 (K1 ≠ K2) (2). This provides a convenient method to calculate the initial stress applied to a two-state equilibrium in terms of competing dissociation constants, K1 and K2.
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 R1 and R2, 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, K1 and K2, are arbitrarily set to 10 and 100 (all units are in grams), which represent the unequal binding affinities (1/K1 and 1/K2) 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. R1 and R2 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 R1 and R2 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]/Ki) for a competitive antagonist, [I], binding to each state with the dissociation constant, Ki.  For competitive inhibition of Langmuir binding, this factor is multiplied times each of the dissociation constants, K1 and K2. The Langmuir functions then become, SR1 = R1(S)/(S+K1(1+[I]/Ki)), and SR2 = R2(S)/(S+K2(1+[I]/Ki)). 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 Ki, 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 Ki. These numbers represent the exponential values for the dissociation constants of the inhibitor for each receptor state, R1 and R2. The  (INV) plots, shown in gray, are from Eq. with K1 ≥ K2, which makes  negative.

If the K1 and K2 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.