Question

Carbon monoxide and hemoglobin Carbon monoxide is a deadly gas that binds hemoglobin roughly 240 times as tightly as oxygen does (this means that $\mathrm{CO}$ has $1/240$ th the dissociation constant of ${\mathrm{O}}_2$ or $240{\mathrm{K}}_{\mathrm{CO}}={\mathrm{K}}_{{\mathrm{O}}_2},$ where ${\mathrm{K}}_{{\mathrm{O}}_2}=26\mathrm{mmHg}$ (a) When both $\mathrm{CO}$ and ${\mathrm{O}}_2$ are present, use the Hill equation introduced in section 6.4 .3 to calculate the probability that hemoglobin will be saturated with oxygen. Similarly, compute the probability that hemoglobin will be saturated with $\mathrm{CO}$. Calculate the partial pressure of oxygen using the fact that atmospheric oxygen constitutes roughly $21\mathrm{\%}$ of air and assume a partial pressure of $\mathrm{CO}$ of $2{\mathrm{mmH}}_8$. Hemoglobin binding to carbon monoxide has a Hill coefficient of 1.4 and hemoglobin binding to oxygen has a Hill coefficient of 3.0 (b) Plot the probability of ${\mathrm{O}}_2$ binding to hemoglobin as a function of the partial pressure of $\mathrm{CO}$ assuming the oxygen partial pressure remains constant. (c) Show that $\mathrm{CO}$ and ${\mathrm{O}}_2$ will have an equal probability of binding when the condition $${\left(\frac{\left|O_2\right|}{K_d^{O_2}}\right)}^{n_{0_2}}{\left(\frac{K_d^{CO}}{|C{O}_1}\right)}^{n_{CD}}=1$$ is satisfied and work out the partial pressure of $\mathrm{CO}$ at which this occurs.

Short answer

Step 3 yields the probabilities of O2 and CO saturating hemoglobin. Step 4 shows that the partial pressure of CO at which CO and O2 have an equal probability of binding to hemoglobin can be calculated using the given condition. The plot from step 5 will illustrate how the probability of O2 binding varies with changes in CO partial pressure.

Step-by-step solution

Calculate the Dissociation Constants

The dissociation constant for CO is given by $K_{CO}=K_{O2}/240$, where $K_{O2}=26mmHg$. Hence, $K_{CO}=26/240=0.1083mmHg$.

Determine the Partial Pressures of Gases

The partial pressure of oxygen ($pO2$) is obtained from the given percentage composition of oxygen in air. Hence, $pO2=0.21\times 760mmHg=159.6mmHg$, where 760mmHg is the atmospheric pressure. It is given that the partial pressure of CO ($pCO$) is 2mmHg.

Apply the Hill Equation

The Hill equation is given as $p=\frac{(pO2{)}^n}{(pO2{)}^n+(K_{O2}{)}^n}$ for oxygen and $p^{\prime }=\frac{(pCO{)}^n}{(pCO{)}^n+(K_{CO}{)}^n}$ for carbon monoxide. Substituting the values into the Hill equation and using the Hill coefficients $n_{O2}=3.0$ and $n_{CO}=1.4$, the oxygen saturation of hemoglobin and carbon monoxide saturation of hemoglobin can be calculated.

Calculate the Equal Probability of Binding

The condition for equal probability of binding $CO$ and $O2$ to hemoglobin is given by ${\left(\frac{pO2}{K_{O2}}\right)}^{n_{O2}}{\left(\frac{K_{CO}}{pCO}\right)}^{n_{CO}}=1$. From this equation, the partial pressure of $\mathrm{CO}$ can be found when this condition is satisfied.

Plotting the Function

For part (b), the probability of $O2$ binding to hemoglobin as a function of the partial pressure of $CO$ can be plotted. This requires varying $CO$ pressure while keeping $O2$ pressure constant, and calculating the corresponding saturation probabilities using the Hill equation.

Key concepts

Hemoglobin

Hemoglobin is a critical protein in the blood that is responsible for transporting oxygen from the lungs to the rest of the body and returning carbon dioxide from the body to the lungs. This protein is found in red blood cells and gives blood its red color. Hemoglobin is composed of four subunits, each capable of binding one molecule of oxygen, which enhances its oxygen-carrying capacity.

Its function is crucial because it ensures that oxygen reaches tissues and cells, which need it to produce energy. This transport ability depends on the dynamic structure of hemoglobin, which changes shape after oxygen is bound. This is part of what allows for the cooperative binding of oxygen, meaning the binding of one oxygen molecule increases the likelihood of more binding.

One important aspect of hemoglobin's behavior is its affinity for different gases, such as oxygen and carbon monoxide. Affinity refers to how tightly hemoglobin holds onto these molecules. While oxygen binding is vital, when carbon monoxide (CO) binds, it prevents oxygen from attaching to hemoglobin. This happens because CO binds much more tightly than oxygen, making it particularly dangerous.

Dissociation Constants

Dissociation constants are a measure of the tendency of a complex to separate into its components. For hemoglobin, the dissociation constant (commonly denoted as $K_d$) indicates how likely it is for a gas molecule, such as oxygen or carbon monoxide, to detach from the hemoglobin.

A lower dissociation constant means a stronger affinity between hemoglobin and the gas, meaning the gas is less likely to be released. In the case of oxygen, the dissociation constant is relatively higher compared to carbon monoxide. This reflects why CO poses such a risk — its $K_d$ is much lower, making it about 240 times stronger in binding to hemoglobin than oxygen.

This difference in $K_d$ can be described through the formula: $K_{CO}=\frac{K_{O2}}{240}\text{.}$ This mathematical relationship shows the much greater binding potency of carbon monoxide, illustrating why even small concentrations of CO can significantly impair oxygen transport in the blood.

Hill Equation

The Hill equation is a mathematical model used to describe how different types of molecules bind to proteins like hemoglobin. It takes into account the cooperative nature of binding, meaning how the binding of one molecule can affect the binding of others.

Mathematically, the Hill equation is represented as: $p=\frac{(p_{extO2}{)}^n}{(p_{extO2}{)}^n+(K_{extO2}{)}^n}$ for oxygen and $p^{\prime }=\frac{(p_{extCO}{)}^n}{(p_{extCO}{)}^n+(K_{extCO}{)}^n}$ for carbon monoxide. Here, $p$ and $p^{\prime }$ represent the saturation probabilities for oxygen and CO, respectively, and $n$ is the Hill coefficient.

The Hill coefficient (\

Dissociation Constants

Dissociation constants ($K_d$) are a critical concept in understanding how tightly a ligand like oxygen or carbon monoxide is bound to hemoglobin. It indicates how readily a molecule will detach or "dissociate" from hemoglobin once it is bound.

A smaller $K_d$ means a stronger binding, implying that the molecule will remain associated with hemoglobin for a longer period. In our context, carbon monoxide (CO) has a much smaller $K_d$ compared to oxygen $(O_2)$, making CO a more potent competitor for binding sites on hemoglobin.

This is represented by the equation $K_{CO}=\frac{K_{O_2}}{240}$, where $K_{O_2}=26$ mmHg. Consequently, $K_{CO}$ is about 0.1083 mmHg, reflecting its stronger binding force. This difference in binding strengths plays a crucial role in situations where both gases are present, as even minimal amounts of CO can overshadow the effects of oxygen due to its tighter grip on hemoglobin.

Hill Equation

The Hill equation is essential in biochemistry for modeling how molecules like oxygen and carbon monoxide bind to receptors such as hemoglobin, especially when multiple molecules are involved. It illustrates the concept of cooperative binding, where the attachment of one ligand affects the binding of others.

For oxygen, the Hill equation is represented as $p=\frac{(p_{\text{O2}}{)}^n}{(p_{\text{O2}}{)}^n+(K_{\text{O2}}{)}^n}$, where $p$ is the probability of oxygen saturating hemoglobin, $n$ is the Hill coefficient (3.0 for oxygen in this scenario), and $K_{\text{O2}}$ is the dissociation constant for oxygen. Similarly, for carbon monoxide, the Hill equation is $p^{\prime }=\frac{(p_{\text{CO}}{)}^{n^{\prime }}}{(p_{\text{CO}}{)}^{n^{\prime }}+(K_{\text{CO}}{)}^{n^{\prime }}}$, with a Hill coefficient of 1.4.

These equations allow us to calculate how increases in gas partial pressure impact their binding probabilities to hemoglobin, thereby predicting potential competition outcomes in various environmental conditions.

Gas Partial Pressure

The concept of gas partial pressure is crucial to understanding how gases like carbon monoxide (CO) and oxygen (O2) interact with hemoglobin. Partial pressure refers to the pressure each gas in a mixture would exert if it occupied the entire volume alone. This pressure plays a significant role in determining how much of each gas binds to hemoglobin.

In our context, the partial pressure of oxygen ( po2) is 159.6 mmHg, determined by multiplying the percentage of oxygen in air (21%) by standard atmospheric pressure (760 mmHg). For carbon monoxide, the given partial pressure is 2 mmHg.

These values feed into the Hill equation to determine the likelihood of each gas binding to hemoglobin. When both gases compete, even the small partial pressure of CO becomes a threat due to its higher affinity for hemoglobin, highlighting the importance of understanding partial pressures in respiratory physiology.