How To Find The Hill Coefficient

"Hill coefficient" sounds like a term that pertains to the steepness of a grade. In fact, it's a term in biochemistry that relates to the behavior of the binding of molecules, usually in living systems. It is a unitless number (that is, it has no units of measure like meters per second or degrees per gram) that correlates with the ​cooperativity​ of the binding between the molecules under examination. Its value is empirically determined, meaning that it is estimated or derived from a graph of related data rather than being used itself to help generate such data.

Put differently, the Hill coefficient is a measure of the extent to which the binding behavior between two molecules deviates from the ​hyperbolic​ relationship expected in such situations, where the velocity of the binding and subsequent reaction between a pair of molecules (often an enzyme and its substrate) initially rises very quickly with increasing substrate concentration before the velocity-vs.-concentration curve flattens out and approaches a theoretical maximum without quite getting there. The graph of such a relationship rather resembles the upper-left quadrant of a circle. The graphs of velocity-vs.-concentration curves for reactions with high Hill coefficients are instead ​sigmoidal​, or s-shaped.

There is a lot to unpack here regarding the basis for the Hill coefficient and related terms and how to go about determining its value in a given situation.

Enzymes are proteins that increase the rates of particular biochemical reactions by enormous amounts, allowing them to proceed anywhere from thousands of times more quickly to thousands of trillions times faster. These proteins do this by lowering the activation energy ​E​_a of exothermic reactions. An exothermic reaction is one in which heat energy is released and that therefore tends to proceed without any outside help. Although the products have a lower energy than the reactants in these reactions, however, the energetic path to get there is typically not a steady downward slope. Instead, there is an "energy hump" to get over, represented by ​E​_a.

Imagine yourself driving from the interior of the U.S., about 1,000 feet above sea level, to Los Angeles, which is on the Pacific Ocean and clearly at sea level. You cannot simply coast from Nebraska to California, because in between lie the Rocky Mountains, the highways crossing which climb to well over 5,000 feet above sea level – and in some spots, the highways climb up to 11,000 feet above sea level. In this framework, think of an enzyme as something capable of greatly lowering the height of those mountain peaks in Colorado and making the whole journey less arduous.

Every enzyme is specific for a particular reactant, called a ​substrate​ in this context. In this way, an enzyme is like a key and the substrate it is specific for is like the lock that the key is uniquely designed to open. The relationship between substrates (S), enzymes (E) and products (P) can be represented schematically by:

\(\text{E} + \text{S} ⇌ \text{ES} → \text{E} + \text{P}\)

The bidirectional arrow on the left indicates that when an enzyme binds to its "assigned" substrate, it can either become unbound or the reaction can proceed and result in product(s) plus the enzyme in its original form (enzymes are only temporarily modified while catalyzing reactions). The unidirectional arrow on the right, on the other hand, indicates that products of these reactions never bind to the enzyme that helped create them once the ES complex separates into its component parts.

Enzyme kinetics describe how quickly these reactions proceed to completion (that is, how quickly product is generated (as a function of the concentration of enzyme and substrate present, written [E] and [S]. Biochemists have come up with a variety of graphs of this data to make it as visually meaningful as possible.

Most enzyme-substrate pairs obey a simple equation called the Michaelis-Menten formula. In the above relationship, three different reactions are occurring: The combining of E and S into an ES complex, the dissociation of ES into its constituents E and S, and the conversion of ES into E and P. Each of these three reactions has its own rate constant, which are ​k​₁, ​k​_-1 and ​k​₂, in that order.

The rate of appearance of product is proportional to the rate constant for that reaction, ​k​₂, and to the concentration of enzyme-substrate complex present at any time, [ES]. Mathematically, this is written:

\(\frac{dP}{dt} = k_2[\text{ES}]\)

The right-hand side of this can be expressed in terms of [E] and [S]. The derivation is not important for present purposes, but this allows for the computation of the rate equation:

\(\frac{dP}{dt} = \frac{k_2[\text{E}]_0[\text{S}]}{K_m+[\text{S}]}\)

Similarly the rate of the reaction ​V​ is given by:

\(V= \frac{V_{max}[\text{S}]}{K_m+[\text{S}]}\)

The Michaelis constant ​K​_m represents the substrate concentration at which the rate proceeds at its theoretical maximum value.

The Lineweaver-Burk equation and corresponding plot is an alternative way of expressing the same information and is convenient because its graph is a straight line rather than an exponential or logarithmic curve. It is the reciprocal of the Michaelis-Menten equation:

\(\frac{1}{V} = \frac{K_m+[\text{S}]}{ V_{max}[\text{S}]} = \frac{K_m}{V_{max}[\text{S}]} + \frac{1}{V_{max} }\)

Some reactions notably do not obey the Michaelis-Menten equation. This is because their binding is influenced by factors that equation does not take into account.

Hemoglobin is the protein in red blood cells that binds to oxygen (O₂) in the lungs and transports it to tissues that require it for respiration. An outstanding property of hemoglobin A (HbA) is that it participates in cooperative binding with O₂. This essentially means that at very high O₂ concentrations, such as those encountered in the lungs, HbA has a much higher affinity for oxygen than a standard transport protein obeying the usual hyperbolic protein-compound relationship (myoglobin is an example of such a protein). At very low O₂ concentrations, however, HbA has a much lower affinity for O₂ than a standard transport protein. This means that HbA eagerly gobbles up O₂ where it is plentiful and just as eagerly relinquishes it where it is scarce – exactly what is needed in an oxygen-transport protein. This results in the sigmoidal binding-vs.-pressure curve seen with HbA and O₂, an evolutionary benefit without which life would certainly be proceeding at a substantially less enthusiastic pace.

In 1910, Archibald Hill explored the kinematics of O₂-hemoglobin binding. He proposed that Hb has a specific number of binding sites, ​​n​​:

\(P + n\text{L } ⇌ P\text{L}_n\)

Here, ​P​ represents the pressure of O₂ and L is short for ligand, which means anything that takes part in binding, but in this case it refers to Hb. Note that this is similar to part of the substrate-enzyme-product equation above.

The dissociation constant ​K​_d for a reaction is written:

\(\frac{[P][\text{L}]^n}{[P\text{L}_n]}\)

Whereas the fraction of occupied binding sites ​ϴ​, which ranges from 0 to 1.0, is given by:

\(ϴ = \frac{[\text{L}]^n}{K_d +[\text{L}]^n}\)

Putting all of this together gives one of many forms of the Hill equation:

\(\log\bigg(\frac{ϴ}{1- ϴ}\bigg) = n \log p\text{O}_2 – \log P_{50}\)

Where ​P​₅₀ is the pressure at which half of the O₂ binding sites on Hb are occupied.

The form of the Hill equation provided above is of the general form

\(y = mx + b\)

also known as the slope-intercept formula. In this equation, ​m​ is the slope of the line and ​b​ is the value of ​y​ at which the graph, a straight line, crosses the ​y​-axis. Thus the slope of the Hill equation is simply ​n​. This is called the Hill coefficient or ​n​​**_H**​. For myoglobin, its value is 1 because myoglobin does not bind cooperatively to O₂. For HbA, however, it is 2.8. The higher the ​n​​**_H**​, the more sigmoidal the kinetics of the reaction under study.

The Hill coefficient is easier to determine from inspection than by doing the requisite calculations, and an approximation is usually sufficient.