User:Abw20/sandbox

Topic: Hill equation (biochemistry)

Biochemical binding curves showing the characteristically sigmoidal curves generated by using the Hill equation to model cooperative binding. Each curve depicts a different Hill coefficient: light blue corresponds to n=0.5, red to n=1, gray to n=2, yellow to n=3, and dark blue to n=4. The vertical axis displays the Fraction (Theta) of occupied ligand-binding sites on a protein receptor, equal to ratio of the concentration of Ligand-bound Protein ([PLn_Total]) to the total concentration of Protein receptor ([P_Total]). The horizontal axis is the ratio of the ligand concentration producing half occupation (K_A) to the total Ligand concentration ([L]).

[Note: The intro is from the original article, although I have rearranged it some. The beginning of the Hill Equation section is from the original article too, but it lacks citations, which I plan to find. All of the derivation and limitations section are my additions. Some of my written material is bold and/or italicized. All section titles are original (the article does not have many sections).]

In biochemistry and pharmacology, the binding of a ligand to a macromolecule is often enhanced if there are already other ligands present on the same macromolecule (this is known as cooperative binding). The Hill equation, which was originally formulated by Archibald Hill in 1910 to describe the sigmoidal O₂ binding curve of haemoglobin, is used to describe the fraction of a macromolecule saturated by ligand as a function of the ligand concentration. The equation is useful for determining the degree of cooperativity of the ligand(s) binding to the enzyme or receptor. The Hill coefficient provides a way to quantify the degree of interaction between ligand binding sites.

The Hill equation is formally equivalent to the Langmuir isotherm, and as a result, it is sometimes referred to as the Hill-Langmuir equation.

Hill Equation (Mathematical Model)

[This section is already in the article, but it lacks citations. I will provide those]

A Hill coefficient of 1 indicates completely independent binding, regardless of how many additional ligands are already bound. Numbers greater than one indicate positive cooperativity, while numbers less than one indicate negative cooperativity. For example, the Hill coefficient of oxygen binding to haemoglobin (an example of positive cooperativity) falls within the range of 1.7-3.2.

The Hill equation is commonly expressed in the following ways:

$\theta ={[{\ce {L}}]^{n} \over K_{d}+[{\ce {L}}]^{n}}={[{\ce {L}}]^{n} \over (K_{A})^{n}+[{\ce {L}}]^{n}}={1 \over \left({K_{A} \over [{\ce {L}}]}\right)^{n}+1}$,

The equation's terms are defined as follows:

• $\theta$ - Fraction of the ligand-binding sites on the receptor protein which are occupied by the ligand.
• ${\ce {[L]}}$ - Free (unbound) ligand concentration
• $K_{d}$ - Apparent dissociation constant derived from the law of mass action (the equilibrium constant for dissociation), which is equal to the ratio of the dissociation rate of the ligand-receptor complex to its association rate (${\textstyle K_{\rm {d}}={k_{\rm {d}} \over k_{\rm {a}}}}$).
• $K_{A}$ - The ligand concentration producing half occupation (ligand concentration occupying half of the binding sites). Because ${\textstyle K_{A}}$ is defined so that ${\textstyle (K_{A})^{n}=K_{\rm {d}}={k_{\rm {d}} \over k_{\rm {a}}}}$ his is also the microscopic dissociation constant. In recent literature, this constant is sometimes referred to as ${\textstyle K_{D}}$.
• $n$ - The Hill coefficient, describing cooperativity (or possibly other biochemical properties, depending on the context in which the Hill equation is being used)

When appropriate, the value of the Hill coefficient describes the cooperativity of ligand binding in the following way:

• $n>1$ - Positively cooperative binding: Once one ligand molecule is bound to the enzyme, its affinity for other ligand molecules increases.
• $n<1$ - Negatively cooperative binding: Once one ligand molecule is bound to the enzyme, its affinity for other ligand molecules decreases.
• $n=1$ - Noncooperative binding: The affinity of the enzyme for a ligand molecule is not dependent on whether or not other ligand molecules are already bound. When n=1, we obtain a model that can be modeled by Michaelis–Menten kinetics, in which ${\textstyle K_{D}=K_{A}=K_{M}}$, the Michaelis-Menten constant. [[[In this case, the Hill equation (as a relationship between the concentration of a compound adsorbing to binding sites and the fractional occupancy of the binding sites) is equivalent to the Langmuir equation.]]]

Hill Plot

Taking the reciprocal of both sides of the Hill equation, rearranging, and inverting again yields: ${\theta \over 1-\theta }={[{\ce {L}}]^{n} \over K_{d}}={[{\ce {L}}]^{n} \over (K_{A})^{n}}$.

Taking the logarithm of both sides of the equation leads to an alternative formulation of the Hill equation:

$\log \left({\theta \over 1-\theta }\right)=n\log {[{\ce {L}}]}-\log {K_{d}}=n\log {[{\ce {L}}]}-n\log {K_{A}}$.

This last form of the Hill equation is advantageous because a plot of ${\textstyle \log \left({\theta \over 1-\theta }\right)}$ versus $\log {[{\ce {L}}]}$ yields a linear plot, which is called a Hill plot. Because the slope of a Hill plot is equal to the Hill Coefficient for the biochemical interaction, the slope is denoted by $n_{H}$. A slope greater than one thus indicates positively cooperative binding between the receptor and the ligand, while a slope less than one indicates negatives cooperative binding.

Derivation from Mass Action Kinetics (this is all my research and writing)

Consider a protein ($P$), such as Haemoglobin or a protein receptor, with ${\mathit {n}}$ binding sites for ligands ($L$). The binding of the ligands to the protein can be represented by the chemical equilibrium expression:

${\ce {{P}+{\mathit {n}}{L}<=>[k_{a}][k_{d}]{P}{L}_{\mathit {n}}}}$,

where $k_{a}$ (forward rate, or the rate of association of the protein-ligand complex) and $k_{d}$ (reverse rate, or the complex's rate of dissociation) are the reaction rate constants for the association of the ligands to the protein and their dissociation from the protein, respectively. From the law of mass action, the apparent dissociation constant $K_{d}$, an equilibrium constant, is given by:

$K_{\rm {d}}={k_{\rm {d}} \over k_{\rm {a}}}={{[{\rm {P}}][{\rm {L}}]^{\mathit {n}}} \over [{\rm {PL_{\mathit {n}}}}]}$.

At the same time, $\theta$, the fraction of the ligand binding sites on the receptor protein which are occupied by the ligand, is given by:

$\theta ={\mathrm {Occupied\ Binding\ Sites} \over \mathrm {Total\ Binding\ Sites} }={[{\rm {PL_{\mathit {n}}}}] \over {[{\rm {P}}]\ +\ [{\rm {PL_{\mathit {n}}}}]}}$.

By using the expression obtained earlier for the dissociation constant, we can replace ${\textstyle [{\rm {PL_{\mathit {n}}}}]}$ with ${\textstyle {[{\rm {P}}][{\rm {L}}]^{\mathit {n}} \over K_{\rm {d}}}}$ to yield a simplified expression for ${\textstyle \theta }$:

$\theta ={({[{\rm {P}}][{\rm {L}}]^{\mathit {n}} \over K_{\rm {d}}}) \over {[{\rm {P}}]\ +\ ({[{\rm {P}}][{\rm {L}}]^{\mathit {n}} \over K_{\rm {d}}})}}={{[{\rm {P}}][{\rm {L}}]^{\mathit {n}}} \over {K_{\rm {d}}[{\rm {P}}]\ +\ {[{\rm {P}}][{\rm {L}}]^{\mathit {n}}}}}={{[{\rm {L}}]^{\mathit {n}}} \over {K_{\rm {d}}\ +\ {[{\rm {L}}]^{\mathit {n}}}}}$,

which is a common formulation of the Hill equation.

Assuming that the protein receptor was initially completely free (unbound) at a concentration ${\textstyle [{\rm {P_{0}}}]}$, then at any time, ${\textstyle {[{\rm {P}}]+[{\rm {PL_{\mathit {n}}}}]}=[{\rm {P_{0}}}]}$ and ${\textstyle \theta ={[{\rm {PL_{\mathit {n}}}}] \over {[{\rm {P_{0}}}]\ }}}$. Consequently, the Hill Equation is also commonly written as an expression for the concentration ${\textstyle [{\rm {PL_{\mathit {n}}}}]}$of bound protein:

$[{\rm {PL_{\mathit {n}}}}]=[{\rm {P_{0}}}]\cdot {{[{\rm {L}}]^{\mathit {n}}} \over {K_{\rm {d}}\ +\ {[{\rm {L}}]^{\mathit {n}}}}}$ .

Note that all of these formulations assume that the protein has ${\mathit {n}}$ sites to which ligands can bind. In practice, however, the Hill Coefficient ${\mathit {n}}$ rarely provides an accurate approximation of the number of ligand binding sites on a protein. Consequently, it has been observed that the Hill Coefficient should instead be interpreted as an "interaction coefficient" describing the cooperativity among ligand binding sites.

Applications to Mathematical Modeling (mine)

Regulation of Gene Transcription

The Hill equation can be applied in modeling the rate at which a gene product is produced when its parent gene is being regulated by transcription factors (activators and/or repressors). If the production of protein from gene ${\textstyle X}$ is up-regulated (activated) by a transcription factor ${\textstyle Y}$, then the rate of production of protein ${\textstyle X}$ can be modeled as a differential equation in terms of the concentration of activated ${\textstyle Y}$ protein:

${d[X_{produced}] \over dt}=k\ \cdot {{[{\rm {Y_{active}}}]^{\mathit {n}}} \over {(K_{A})^{n}\ +\ {[{\rm {Y_{active}}}]^{\mathit {n}}}}}$,

where $k$ is the maximal transcription rate of gene ${\textstyle X}$.

Likewise, if the production of protein from gene ${\textstyle Y}$ is down-regulated (repressed) by a transcription factor ${\textstyle Z}$, then the rate of production of protein ${\textstyle Y}$ can be modeled as a differential equation in terms of the concentration of activated ${\textstyle Z}$ protein:

${d[Y_{produced}] \over dt}={{k} \over {1\ +\ ({[{\rm {Y_{active}}}] \over K_{A}})^{n}}}$.

Underlying (Physiological?) Mechanisms

[I'm going to leave this section out in favor of providing a SEE ALSO link at the end to COOPERATIVE BINDING]

Limitations (mine)

Because of its assumption that ligand molecules bind to a receptor simultaneously, the Hill equation has been criticized as a physically unrealistic model. Moreover, the Hill coefficient should not be considered a reliable approximation of the number of cooperative ligand binding sites on a receptor except when the binding of the first and subsequent ligands results in extreme positive cooperativity.

Unlike more complex models, the relatively simple Hill equation provides little insight into underlying physiological mechanisms of protein-ligand interactions. This simplicity, however, is what makes the Hill equation a useful empirical model, since its use requires little a priori knowledge about the properties of either the protein or ligand being studied. Nevertheless, other, more complex models of cooperative binding have been proposed. For more information and examples of such models, see Cooperative binding.