The study of the motion of viscous fluids in tubes is an important part of hydrodynamics, necessary both in engineering (oil and water pipelines) and in biology (blood circulation). One of the key characteristics of such flow is the hydrodynamic resistance — the quantity that shows what pressure difference is required to provide a given flow rate. For steady laminar flow in a cylindrical tube, the pressure difference is related to the flow rate in the same way that voltage is related to current in electricity:
Thus, the “Ohm’s law for hydrodynamics” has the following form:
$$\Delta P = Q \cdot Z.$$
This analogy makes it possible to transfer well-developed methods for calculating complex electrical circuits to the analysis of hydrodynamic systems. While in electricity charge transfer is determined by the properties of the conductor material, in a fluid the motion is defined by internal friction — viscosity.
Divide the fluid into many thin layers of equal thickness $\Delta r$, which move relative to each other with different velocities. An internal friction force arises between adjacent layers:
$$F_{fr}=\eta\,\frac{\Delta v}{\Delta r}\,S,$$
where $\eta$ is the viscosity coefficient of the fluid, $\Delta v$ is the velocity difference between the layers, and $S$ is the contact area between the layers.
In many hydrodynamic problems, it is convenient to consider the ratio $G$ of the pressure difference to the tube length:
$$G=\frac{\Delta P}{L},$$
where $\Delta P$ is the pressure difference over a tube segment of length $L$. Then the volumetric flow rate $Q$ for laminar flow in a cylindrical tube depends on the radius $R$, the viscosity $\eta$ and $G$ as follows:
$$Q = C\, R^{a}\, G^{b}\, \eta^{c},$$
where $C$ is a dimensionless constant, and $a, b, c$ are integer powers.
Consider steady flow of an incompressible viscous fluid through a horizontal cylindrical tube of radius $R$ and length $L$. A pressure difference $\Delta P$ is the same across the entire cross-sectional area at the ends of the tube. The fluid velocity $v(r)$ depends on the distance $r$ from the tube axis: the velocity is maximal at the center, and at the wall $r=R$ it is equal to zero.
To calculate the volumetric flow rate $\Delta Q$ of fluid through a thin cylindrical layer between radii $r$ and $r+\Delta r$ (see Fig. 3), we can note that the area of the end face of this layer is equal to $2\pi r \,\Delta r$. Multiplying it by the velocity $v$, we obtain:
$$\Delta Q = 2\pi r \,\Delta r \cdot v.$$
This formula can be rewritten as
$$\Delta Q = \pi v\,\Delta(r^2),$$
where the value $v \Delta(r^2)$ is proportional to the area beneath the graph $v(r^2).$
The obtained expression is called Poiseuille's law.
A viscous liquid flows with a volumetric flow rate $Q$ through a tube with radius $R$ and length $L$. This tube splits into $N$ identical parallel narrow tubes, each of them with length $\beta L$ and radius $\alpha R$.
In the task B6, you obtained Poiseuille's law for laminar flow of a viscous incompressible fluid in a cylindrical tube. These ideas can be applied to a real biological system — human blood circulation.
Note: if you have not managed to calculate the dimensionless coefficient $C$, you can assume it to be equal to 1 (this is an incorrect value) in order to continue solving the problem.
Blood circulation can be conveniently viewed as a hydrodynamic system: the heart creates a pressure gradient, blood flows through a network of vessels, and the total flow rate is the same at all levels — from the aorta to the venous return. In this task, we examine a systemic circulation in a simplified model that is sufficient for quantitative estimates:
This model does not describe the pulsatility and elasticity of blood vessels, but it clearly shows where resistance is concentrated, how velocity changes at different levels, and how blood properties affect heart function.
Simplified levels of the vessels of the systemic circulation:
Level Radius $R$, mm Length $L$, cm Number of vessels $n$ Aorta $1.25 \cdot 10 ^1$ $4.00\cdot10^1$ $1.00$ Major arteries $2.00$ $2.00\cdot 10^1$ $1.00\cdot10^2$ Arterioles $3.00\cdot10^{-2}$ $6.00\cdot 10^{-1}$ $5.00\cdot10^5$ Capillaries $3.50\cdot10^{-3}$ $2.00\cdot10^{-1}$ $1.00\cdot 10^{10}$
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During physical activity, the volumetric flow rate increases to $Q=4Q_0$. The body achieves this by increasing the frequency and strength of heart contractions, as well as increasing the radius of the arterioles by $20\%$.
Gas transfer is one of the most important functions of blood. The circulatory system is an organ system that enables blood to move between the immediate environment of each cell and the tissues, where exchange with the external environment takes place.
Next, we will consider only the human cardiovascular system.
The partial pressures of oxygen and carbon dioxide change as these gases move through the circulatory system. The figure below shows some parts of the human circulatory system marked with numbers. The partial pressures of gases in these parts are given below with letters A-E.
A. $P(\text{O}_2) = 100$ mm Hg, $P(\text{CO}_2) = 40$ mm Hg;
B. $P(\text{O}_2)< 40$ mm Hg, $P(\text{CO}_2) > 46$ mm Hg;
C. $P(\text{O}_2) = 160$ mm Hg, $P(\text{CO}_2) = 0.3$ mm Hg;
D. $P(\text{O}_2) = 40$ mm Hg, $P(\text{CO}_2) = 46$ mm Hg;
E. $P(\text{O}_2) = 105$ mm Hg, $P(\text{CO}_2) = 40$ mm Hg.
It should be noted that gravity also affects blood flow. When we stand or sit, gravity acts on the blood in our legs, for example, making it difficult for it to flow upward. The table below lists some possible factors that contribute to the return of blood through the veins to the heart.
A. Contraction of the skeletal muscles surrounding the vein. E. Lower viscosity of venous blood compared to arterial blood. B. Low wall thickness of veins compared to arteries. F. Contraction of the walls of certain veins. C. The functioning of valves inside blood vessels. G. Low blood oxygen saturation. D. Negative pressure in the chest during inhalation. H. Negative pressure in the atria during ventricular systole.
Arterial pressure in the body is not constant, but depends on many factors. In particular, certain hormones affect arterial pressure.
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The table in the answer sheet contains a list of human hormones. For each hormone, indicate its chemical nature and the organ responsible for its production in the table: write down an "X" in the appropriate cells. In addition, write down an "X" in the last row corresponding to those hormones that cause an increase in arterial pressure (AP).
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The diffusion of oxygen from the air into the blood occurs in the alveoli. Fick's law states that when there is a concentration gradient $\Delta n$, the particle flux $j$ (the number of particles per unit time per unit area) is proportional to $\Delta n$:
$$j = \dfrac{\Delta N}{\Delta S\Delta t} = D \dfrac{\Delta n}{d}. $$
Here, $D$ is the diffusion coefficient, and $d$ is the thickness of the barrier.
Let us assume that one mole of oxygen provides energy $Q = 470$ kJ/mol, and the power generated by a person is $W = 1$ kW. The difference in oxygen concentration between the alveolar air and the capillaries corresponds to a pressure difference of $\Delta p = 8$ kPa, and the body temperature is $T = 37\,^\circ\mathrm{C}$.
For regular functioning of the body's cells, a constant $\text{pH}$ value of blood and intercellular fluid is required. The hydrogencarbonate buffer solution, consisting of carbonic acid $\text{H}_2\text{C}\text{O}_3$ and hydrogencarbonate $\text{H}\text{C}\text{O}_3^-$, plays the most important role in maintaining homeostasis. At a temperature of $37\,^\circ\mathrm{C}$, for carbonic acid $\text{p}K_{a_1} = 3.57$. Dissociation at the second stage of carbonic acid can be neglected.
To understand how a buffer solution works, let us consider two solutions:
(A) 1 liter of $0.15~\text{M}~\text{H}_2\text{C}\text{O}_3$;
(B) 1 liter of buffer solution obtained by mixing 500 ml of $0.30~\text{M}~\text{H}_2\text{C}\text{O}_3$ and 500 ml of $0.30~\text{M}~\text{Na}\text{H}\text{C}\text{O}_3$.
Let 0.2 liters of $0.1~\text{M}~\text{HCl}$ be added to each solution.
Carbonic acid is formed in the body as a result of the dissolution (aq) of carbon dioxide in water:
$$\text{CO}_{2\,(\text{aq})} + \text{H}_2\text {O} \rightleftharpoons \text{H}_2\text{C}\text{O}_3.$$
The equilibrium constant for carbon dioxide hydration is $K_h = 3.0\cdot10^{-3}$.
The concentration of carbon dioxide in the blood corresponds to its partial pressure $p(\text{CO}_2) = 5.3$ kPa. The relationship between the partial pressure of gas $p$ and its concentration $C$ in the solution in equilibrium is given by Henry's law:
$$C = kp,$$
where $k$ is Henry's constant. The Henry's constant for the dissolution of carbon dioxide at a temperature of $37\,^\circ\mathrm{C}$
is equal to $k = 2.3\cdot10^{-4}$ mol/(m${}^3\cdot$ Pa).
Another buffer system in blood is a mixture of protonated and deprotonated hemoglobin. The chemical equilibrium equations for oxyhemoglobin and deoxyhemoglobin are as follows:
$$\text{Hb}\cdot\text{O}_2\cdot\text{H}^{+} \rightleftharpoons\text{Hb}\cdot\text{O}_2 + \text{H}^+,\quad \text{p}K_a = 8.18; $$
$$\text{Hb}\cdot\text{H}^{+} \rightleftharpoons\text{Hb} + \text{H}^+,\quad \text{p}K_a = 6.62. $$
Since oxygen is poorly soluble in water, organisms need special adaptations to transfer it efficiently. Large multicellular organisms have evolved proteins that store and transfer oxygen. However, unlike some transition metals, the side chains of amino acids cannot reversibly bind oxygen. Therefore, iron is used to transfer gases in the human body. Due to the high reactivity of free iron ions, it is used in a bound form. Heme is a protein-bound compound into which iron is incorporated.
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The gross formula of heme is $\text{C}_a\text{H}_b\text{O}_c\text{N}_d\text{Fe}$, and the mass fractions of the elements in the compound are $w(\text{C}) = 66.24\%$, $w(\text{H}) = 5.23\%$, $w(\text{O}) = 10.38\%$, $w(\text{N}) = 9.09\%$. From this information, determine the numbers $a, b, c$ and $d$.
There are four levels of protein structural organization: primary, secondary, tertiary, and quaternary.
Below is a list of chemical bonds that appear in the structures mentioned above. Write down the letters representing the chemical bonds in the appropriate cells of the table below. Note that the same letters may be used in different cells.
A. Disulfid bridges (–S–S–).
B. Ionic bonds.
C. Hydrogen bonds within a molecule.
D. Hydrophobic interactions.
E. Peptide bonds.
F. Intermolecular hydrogen bonds.
Myoglobin is an oxygen-binding protein found in skeletal muscles and heart muscles.
Under normal conditions, in the absence of damage or inflammation of muscle tissue, myoglobin hardly enters the bloodstream.
In turn, hemoglobin transfers gases in the blood and consists of four subunits. Each subunit consists of heme and a polypeptide chain associated with it.
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An important feature of many proteins is the reversible binding of other molecules. A molecule that reversibly binds to a protein is called a ligand, and the binding site on the protein molecule is called the binding center. Myoglobin can both bind oxygen molecules and release them, which means that there is an interaction between the protein and the ligand.
Let us consider, in general terms, the equilibrium in solution between protein $\text{P}$, ligand $\text{L}$ and complex $\text{PL}$. Ligand binding is described by the reaction
$$\text{P} + \text{L} \rightleftharpoons \text{PL}$$
with an equilibrium constant $K_a$ ($a$ — association reaction). The fraction of occupied ligand binding sites is called the degree of saturation and is denoted by $\theta$:
$$\theta = \dfrac{[\text{PL}]}{[\text{PL}] + [\text{P}]}.$$
For gases, the formulas obtained remain valid, but instead of concentrations, it is necessary to work with partial pressures. For example, the oxygen pressure at which the degree of myoglobin saturation is $50\%$ is equal to $p_{50}({\text{O}_2}) = 2$ mm Hg.
Carbon monoxide $\text{CO}$ is extremely dangerous to humans. Carbon monoxide can also bind with myoglobin, and the corresponding constant is $p_{50}({\text{CO}}) = p_{50}({\text{O}_2}) / 200$.
The temperature dependence of the equilibrium constant is described by the Van 't Hoff equation:
$$\dfrac{\Delta \ln(K_a)}{\Delta T} = \dfrac{\Delta_r H^\circ}{RT^2},$$
where $\Delta_r H^\circ$ is the standard molar enthalpy of reaction (the index $r$ denotes the change in value during the association reaction). Assuming that $\Delta_r H^\circ$ is almost independent of temperature in the specified range, the equation can be transformed to the form
$$\ln K_a = -\dfrac{\Delta_r H^\circ}{RT} + \text{const}.$$
The table in the answer sheet shows the experimental values of the association constants at different temperatures in a specific buffer solution.
$T, ^\circ \mathrm{C}$ 10 20 30 35 40 $K_a$ 3.09 1.38 0.66 0.48 0.33
For a protein with $n$ binding sites, it can be approximated that $n$ ligands attach in one stage:
$$\text{P} + n\text{L} \rightleftharpoons \text{PL}_n.$$
A graph in the coordinates $\log_{10}\left(\dfrac{\theta}{1-\theta}\right)$ versus $\log_{10}([\text{L}])$ is called a Hill graph.
It turns out that there is cooperative interaction between subunits in hemoglobin. When $\text{O}_2$ binds to one of the units, the others slightly change their shape, thereby increasing their affinity for oxygen. Conversely, when all subunits are in a bound state and one subunit releases oxygen, the others also tend to release oxygen. The slope of the Hill graph $n_H$ represents a measure of cooperativity.
Hemoglobin can be found in two states: R and T. Oxygen has a greater affinity for hemoglobin in the R state, and in the absence of oxygen, the T state is more stable. It is due to the transitions between these states that hemoglobin binds sufficient oxygen in the lungs and releases it in the tissues.
Give the reason why myoglobin cannot be used as an effective oxygen carrier (efective binding and release of oxygen molecules) from the lungs to the tissues.
A. The myoglobin molecule has a hyperbolic oxygen saturation curve.
B. The concentration of myoglobin in the blood is significantly lower than the concentration of hemoglobin.
C. The myoglobin molecule is lighter, which makes it too mobile.
D. The myoglobin molecule is too small, which can cause it to enter other tissues.
In fact, hemoglobin transfers not only oxygen, but also protons $\text{H}^+$. To account for the effect of $\text{pH}$ on oxygen binding and release, we can consider the equilibrium equation in the form
$$\text{Hb}\cdot\text{H}^{+} + \text{O}_2 \rightleftharpoons \text{Hb}\cdot\text{O}_2 + \text{H}^+, $$
where $\text{Hb}\cdot\text{H}^+$ is the protonated form of hemoglobin.
The figure below shows the hemoglobin saturation curves at different $\text{pH}$ values. The upper and the lower curves correspond to $\text{pH}~7.2$ and $\text{pH}~7.6$.
The observed influence of $\text{pH}$ on oxygen binding and release is called the Bohr effect and determines the different degree of saturation of hemoglobin in different parts of the human body.
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