Osmotic pressure and water movement

Osmosis is the transport of solvent driven by a difference in solute concentration across a membrane that is impermeable to solute

Because of diffusion, a net movement of molecules occurs down concentration gradients, and substances tend to move in a way that abolishes concentration differences in different regions of a solution. Alternatively, it could be said that diffusion results in mixing . We now examine the consequences when a solute is prevented from diffusing down its concentration gradient. Fig. 3.1 A shows two aqueous compartments, 1 and 2, separated by a semipermeable membrane. Initially compartment 1 contains pure water and compartment 2 contains a solute dissolved in water. The membrane is permeable to water but impermeable to the solute. Because the solute concentration is higher in compartment 2 than in compartment 1, normally a net flux of solute molecules from 2 into 1 would occur. Because the membrane is impermeable to the solute, however, such a flux cannot take place; solute molecules cannot move from 2 into 1 to abolish the concentration difference between the two compartments.

The situation can be viewed from another perspective. When a solute is dissolved in water to form an aqueous solution, as the concentration of solute in the solution is increased, the concentration of water in the same solution must correspondingly decrease. In the two compartments shown in Fig. 3.1 A, whereas the solute concentration is higher in 2 than in 1, the water concentration is higher in 1 than in 2. Therefore, although the membrane does not permit the solute to move across from 2 into 1, it does allow water to move from 1 into 2. The difference in the water concentration allows us to predict, correctly, that a net flux of water will occur from 1 into 2. ^a

a This reasoning based on water concentration differences is heuristic but always gives the correct prediction.

The movement of water through a semipermeable membrane, from a region of higher water concentration to a region of lower water concentration, is called osmosis .

Water transport during osmosis leads to volume changes

In light of the foregoing, we can think about simple diffusion again. Because an increase in solute concentration dictates a corresponding decrease in water concentration, when a solute concentration gradient exists, there must be a water concentration gradient running in the opposite direction. We can therefore view simple diffusion as a net flux of solute molecules down their concentration gradient occurring simultaneously with a net flux of water molecules down the water concentration gradient.

In osmosis, the membrane does not permit solute movement, so the only flux is that of water. Because the net flux of water in one direction is not balanced by a flux of solute in the opposite direction, the net movement of water would be expected to contribute to a change in volume: as water moves from compartment 1 to compartment 2, the volume of solution in 2 should increase, as shown in Fig. 3.1 B.

Osmotic pressure drives the net transport of water during osmosis

In the two-compartment situation depicted in Fig. 3.1 , as osmosis proceeds, the solution volume in 2 will increase and a “head” of solution will build up ( Fig. 3.1 B). The head of solution will exert hydrostatic pressure , which will tend to push the water across the membrane back to compartment 1, thus reducing the net flux of water from 1 into 2. As the solution volume in 2 increases, a point will be reached when the hydrostatic pressure from the solution head is sufficient to counteract exactly the net flow of water from 1 into 2.

This view suggests that we can also think of osmotic flow of water as being driven by some “pressure” that forces water to flow from 1 into 2. This pressure, arising from unequal solute concentrations across a semipermeable membrane, is termed osmotic pressure . From the earlier discussion, we expect that the larger the solute concentration difference between two solutions separated by a semipermeable membrane, the greater the osmotic pressure difference driving water transport. Therefore the osmotic pressure of a solution should be proportional to the concentration of solute. Indeed, the osmotic pressure can be described fairly accurately by van’t Hoff’s law ^a

a Derived by Jacobus Henricus van’t Hoff, who received the Nobel Prize in Chemistry in 1901.

:

where π is the osmotic pressure, C _solute is the solute concentration, T is the absolute temperature ( T = Celsius temperature +273.15, in absolute temperature units, i.e., Kelvins), and R is the universal gas constant (0.08205 liter·atmosphere·mole ⁻¹ ·Kelvin ⁻¹ ). The various units of measurement used in the study of osmosis are described in Box 3.1 .

Importantly, the osmotic pressure of a solution is a colligative property—that is, a property that depends only on the total concentration of dissolved particles and not on the detailed chemical nature of the particles. Therefore, in Equation 3.1 , C _solute is the total concentration of all solute particles. For example, a nondissociable molecular solute such as glucose or mannitol at a concentration of 1 M gives C _solute = 1 M. In contrast, a 1 M solution of an ionic solute such as NaCl, which can fully dissociate into equal numbers of Na ⁺ and Cl ⁻ ions, actually has a solute particle concentration of C _solute = 2 M. Similarly, for a 1 M solution of MgCl ₂ , which can dissociate into constituent Mg ²⁺ and Cl ⁻ ions, C _solute = 3 M.

In Fig. 3.1 the osmotic pressure of the solution in compartment 1 is 0 ( C _solute = 0), whereas the osmotic pressure of the solution in compartment 2 is equal to RTC _solute . In practice, the magnitude of the osmotic pressure is also operationally equal to the amount of hydrostatic pressure that must be applied to the compartment with the higher solute concentration to stop the net flux of water into that compartment. The osmotic pressure of a physiological solution is typically very high ( Box 3.2 ).

When impermeant solute is present on both sides of a semipermeable membrane, water flux across the membrane will depend on the osmotic pressure difference between the two compartments. In turn, the osmotic pressure difference depends on the difference of solute concentrations across the membrane:

where Δπ is the osmotic pressure difference across the membrane, Δ C _solute is the difference in solute concentrations across the membrane, and R and T are as defined for Equation 3.1 .

In real life, membranes are never completely impermeable to solute. What happens when the membrane is only partially impermeable to solute? We can deduce the answer by considering the two extreme cases we already know: (1) If the membrane allows the solute molecules to pass through as freely as water molecules can, the flux of water is counterbalanced by the flux of solute in the opposite direction and the situation is no different from free diffusion—there would be no osmotic pressure difference; and (2) if the membrane is completely impermeable to solute but completely permeable to water, the maximum osmotic pressure that can be achieved is given by Equation 3.2 . The behavior of real-life membranes must lie somewhere between these two extremes. Because the relative permeability of the membrane to solute and water determines the actual behavior, we can define a parameter, called the reflection coefficient , represented by the symbol σ:

where P _solute and P _water are the membrane permeabilities to solute and water, respectively. When the membrane is absolutely impermeable to solute, P _solute = 0, and σ = 1—the membrane “reflects” all solute molecules and does not allow them to pass through, and full osmotic pressure should develop. When the solute molecules permeate the membrane as readily as do water molecules, P _solute / P _water = 1, and σ = 0. In this case the membrane allows free passage of both water and solute, and the osmotic pressure should be 0. By incorporating the reflection coefficient into the osmotic pressure equation, we can describe membrane behavior more accurately:

where Δπ is the effective osmotic pressure difference across a membrane.

The fact that, even at the same concentration, solutes with different reflection coefficients can generate different osmotic pressures gives rise to a distinction between osmolarity and tonicity , which is explained in Box 3.3 .

The direction of fluid flow through the capillary wall is determined by the balance of hydrostatic and osmotic pressures, as described by the starling equation

In the previous section we noted that hydrostatic pressure can counteract water movement driven by osmotic pressure. This suggests that hydrostatic and osmotic pressures can both drive water movement through a membrane. We can also conclude that the direction of net fluid flow must be determined by the balance of osmotic and hydrostatic pressures across a membrane barrier. This principle governs the movement of fluid into and out of capillaries and is thus fundamental to circulatory physiology.

In analyzing fluid movement across capillary walls, we only need to identify the forces that drive fluid out of the capillary into the fluid space of the surrounding tissue (the interstitium) and the forces that drive fluid from the interstitium into the capillary. Fig. 3.2 shows the four pressures that are in play; the direction of water movement driven by each pressure component is indicated by an arrow associated with that pressure. These pressures are commonly referred to as the Starling forces . ^a

a Named after Ernest Starling, a 19th-century physiologist who first investigated fluid movement driven by osmotic and hydrostatic forces and who is also known for the Frank-Starling law of heart function.

The capillary hydrostatic pressure (blood pressure) is P _c ; the hydrostatic pressure in the interstitium is P _i . The principal contributors to the osmotic pressure difference across the capillary wall are proteins dissolved in the plasma, which are too large to pass through the capillary walls ( Box 3.4 ). Osmotic pressure arising from dissolved proteins is commonly referred to as colloid osmotic pressure (COP) , or oncotic pressure . Plasma COP in the capillary is symbolized as π _c ; the COP of the interstitial fluid is symbolized as π _i . The capillary hydrostatic pressure pushes water out of the capillary into the interstitium, whereas any interstitial hydrostatic pressure pushes interstitial water into the capillary. The osmotic components operate in the opposite way: the capillary COP pulls interstitial water into the capillary, whereas interstitial COP pulls water out of the capillary into the interstitium. Movement of fluid out of the capillary is called filtration, and movement of fluid into the capillary is called absorption. As blood passes through a capillary, the balance of these four pressures determines whether filtration or absorption occurs:

$\begin{array}{c}\text{Net fluid movement}\text{∝}\left(\text{Pressures that drive fluid out}\right)\\ \text{-}\left(\text{Pressures that drive fluid in}\right)\text{.}\end{array}$

BOX 3.4

Dissolved Proteins Give Rise to the Colloid Osmotic (Oncotic) Pressure in the Plasma and the Interstitium

By far the most abundant solutes in blood plasma are low-molecular-weight ions and molecules (e.g., Na ⁺ , Cl ⁻ ). The total concentration of such small solutes is close to 300 mM. If a membrane barrier were completely impermeable to these solutes, an osmotic pressure close to 6000 mmHg (i.e., ∼7.6 atm; see Box 3.1 ) would develop. In reality, the wall of a typical capillary is quite permeable to small solutes (reflection coefficient, σ ≈ 0). Thus with respect to small solutes, the interstitial fluid has approximately the same composition as plasma, with the consequence that small solutes exert very little net osmotic effect across the capillary wall. It is reasonable that the walls of most capillaries are quite permeable to low-molecular-weight solutes, because the circulating blood brings small nutrient molecules (e.g., glucose, amino acids) that must leave the capillaries to nourish the cells in tissue. These cells, in turn, generate metabolic waste (e.g., carbon dioxide) that must enter the capillaries and be borne away by the blood.

Two types of proteins are found in significant amounts in the plasma. Albumin, with a molecular weight of approximately 69,000, is present at approximately 4.5 g/dL (grams/deciliter, same as grams/100 milliliters), or approximately 0.65 mM. Globulins, with a molecular weight of approximately 150,000, are present at approximately 2.5 g/dL, or approximately 0.17 mM. These proteins, being macromolecules, do not readily pass through the capillary wall (σ ≥ 0.9); therefore the plasma is protein-rich, whereas the interstitial fluid is low in proteins. Because the total protein concentration in plasma is close to 0.82 mM, we expect the osmotic pressure contributed by the proteins to be:

$\begin{array}{c}\text{π}=\text{σ}RTC\text{=}0.9\text{×}0.08205\left[\frac{\text{L}\cdot \text{atm}}{\text{mol}\cdot \text{K}}\right]\text{×}310.15\left[\text{K}\right]\\ \text{×}0.82\text{×}{10}^{\text{-}3}\left[\frac{\text{mol}}{\text{L}}\right]\text{=}0.0216\text{atm}\end{array}$

or 15.8 mmHg, which is quite a bit less than the π _c = 25 mmHg that is typically measured. This suggests that each protein molecule exists in solution not as a single particle, but rather as an ionic macromolecule with some associated small ions. Because the proteins cannot leave the capillary, the population of small “counterions” that are associated with them must also remain within the capillary (see Box 4.4 ), thereby increasing the total amount of effectively impermeant solute. In other words, by ignoring the fact that proteins can dissociate into more than one ionic particle, we underestimate the osmotic contribution of proteins.

The most important point is that proteins, because they do not readily pass through the capillary wall, are the predominant contributor to osmotic forces across the capillary wall.

We know that

$\text{(Pressures that drive fluid out})\text{=}{P}_{\text{c}}\text{+}{\text{π}}_{\text{i}}$

and

$\text{(Pressures that drive fluid in})\text{=}{P}_{\text{i}}\text{+}{\text{π}}_{\text{c}},$

so net fluid movement can be represented more formally and quantitatively as

$J_{\text{v}}\text{=}{L}_{\text{p}}S\left[\left(P_{\text{c}}\text{+}{\text{π}}_{\text{i}}\right)\text{-}\left(P_{\text{i}}\text{+}{\text{π}}_{\text{c}}\right)\right],$

where J _v is the volume of fluid filtered through the capillary wall and is known as volume flow ; L _p is the hydraulic conductivity , which characterizes the ease with which a membrane barrier allows water to pass through; and S is the area of membrane through which water flows. We can see from Equation 3.5 that when the pressures driving fluid out ( P _c + π _i ) exceed the pressures driving fluid in ( P _i + π _c ), J _v is a positive number and there is net fluid flow out of the capillary into the interstitium—i.e., filtration. Conversely, J _v being negative corresponds to absorption of fluid into the capillary. The terms enclosed in square brackets in Equation 3.5 can be rearranged:

$J_{\text{v}}\text{=}{L}_{\text{p}}S\left[\left(P_{\text{c}}\text{-}{P}_{\text{i}}\right)\text{-}\left({\text{π}}_{\text{c}}\text{-}{\text{π}}_{\text{i}}\right)\right]$

Equation 3.6 is often referred to as the Starling equation .

It is instructive to check the units of the various quantities in Equation 3.6 . P _c , P _i , π _c , and π _i are typically measured in units of millimeters of mercury (mmHg). L _p represents the volume of water that can pass through a certain area of membrane, when driven by a certain pressure, within a certain interval of time, so its units can be written as cm ³ ·cm ⁻² ·mmHg ⁻¹ ·sec ⁻¹ , which simplifies to cm·mmHg ⁻¹ · sec ⁻¹ . To be consistent with L _p , S, being an area, is measured in cm ² . Thus J _v must have units of cm ³ ·sec ⁻¹ —consistent with volume flow.

The quantities L _p and S are sometimes combined to give the filtration constant : K _f = L _p × S; it is easy to verify that K _f has units of cm ³ ·mmHg ⁻¹ ·sec ⁻¹ . The Starling equation incorporating K _f is

$J_{\text{v}}\text{=}{K}_{\text{f}}\left[\left(P_{\text{c}}\text{-}{P}_{\text{i}}\right)\text{-}\left({\pi }_{\text{c}}\text{-}{\pi }_{\text{i}}\right)\right]$

Analysis of the situation in Fig. 3.3 A illustrates the principles involved in applying the Starling equation. The arteriolar and venular hydrostatic pressures are 32 and 15 mmHg, respectively. The interstitial hydrostatic pressure is close to 0. The capillary and interstitial oncotic pressures are π _c = 25 mmHg and π _i = 2 mmHg, respectively ( Box 3.4 ).

We wish to determine whether fluid moves into or out of the capillary at two key points along the capillary, the arteriolar and venular ends of the capillary. Two observations are useful: (1) at the point where the capillary is connected to the arteriole, the capillary hydrostatic pressure should be essentially identical to the blood pressure in the arteriole, that is, P _c = 32 mmHg, and (2) at the point where the capillary is connected to the venule, the capillary hydrostatic pressure should be essentially the same as the blood pressure in the venule, that is, P _c = 15 mmHg. All other pressure components ( P _i , π _c , and π _i ) do not vary with location. We can determine the net effect of the various pressure components through a straightforward calculation. At the arteriolar end,

$\left(P_{\text{c}}\text{-}{P}_{\text{i}}\right)\text{-}\left({\pi }_{\text{c}}\text{-}{\pi }_{\text{i}}\right)\text{=}\left(32\text{-}0\right)\text{-}\left(25\text{-}2\right)\text{=+}9\text{mmHg}$

This means that a net positive pressure is driving fluid out of the capillary near the arteriolar end. At the venular end,

$\left(P_{\text{c}}\text{-}{P}_{\text{i}}\right)\text{-}\left({\pi }_{\text{c}}\text{-}{\pi }_{\text{i}}\right)\text{=}\left(15\text{-}0\right)\text{-}\left(25\text{-}2\right)\text{=-}8\text{mmHg}$

This means that a net negative pressure is driving fluid out, which is the same as saying that the net pressure effect is to drive fluid back into the capillary near the venular end. In this capillary, filtration occurs toward the arteriolar end and absorption takes place toward the venular end. It is worth emphasizing that because P _i , π _c , and π _i tend to be relatively constant, whereas normally only the capillary hydrostatic pressure, P _c , varies and is the primary determinant of whether filtration or absorption occurs.

If the capillary hydrostatic pressure decreases linearly from the arteriolar to the venular end, we can estimate the magnitude of the Starling forces along the entire length of the capillary. The result is shown in Fig. 3.3 B. At the arteriolar end, where the capillary hydrostatic pressure is high, the forces driving fluid out override the forces driving fluid in, and the result is filtration of fluid out of the capillary. Advancing along the capillary, capillary hydrostatic pressure wanes and fluid filtration correspondingly decreases. Eventually the capillary hydrostatic pressure declines to the point where the forces driving fluid in overtake the forces driving fluid out, and the result is absorption of fluid back into the capillary. In the example shown in Fig. 3.3 , filtration occurs over a little more than the first half of the length of the capillary and absorption takes place over the remainder. The end result is that over the length of the capillary, there is a slight excess of filtration over absorption, with a net flow of a small amount of fluid into the interstitium. The fluid that “leaks” into the interstitium is gathered by the lymphatic system and eventually returned into circulation. During inactivity, lymph production in humans amounts to approximately 4 liters over a 24-hour period. Because blood circulates at the rate of approximately 5 liters per minute, over a 24-hour period more than 7000 liters of blood will pass through the capillaries. This shows that capillary filtration and fluid reabsorption typically are finely balanced. Under certain circumstances the relative balance of filtration and absorption is disrupted, leading to edema , or excess accumulation of fluid in tissue ( Box 3.5 ).

BOX 3.5

Edema: Excess Accumulation of Fluid

The Starling equation ( Equation 3.6 ) shows that whether fluid leaves or enters the capillary is determined by the balance of capillary and interstitial hydrostatic and osmotic pressures. Altered pressure balance leads to altered fluid flow. When venous blood pressure is raised as a result of venous blood clots or congestive heart failure, the result is elevated hydrostatic pressure ( P _c ) in the capillary, which leads to more fluid filtration. Liver disease and severe starvation both can greatly reduce albumin production by the liver. Because albumin is the predominant contributor to plasma colloid osmotic pressure (π _c ), loss of albumin drastically lowers the retention of fluid in the capillaries. Finally, endogenous biochemical agents secreted during the allergic response (e.g., histamine released from mast cells), as well as some factors in insect venom (e.g., melittin from bees), markedly increase the permeability of capillary walls to solutes and water. Normally impermeant plasma proteins become more permeant (smaller σ) and thus can leave the capillary and enter the interstitium, which lowers π _c and raises π _i . Water can also move across the capillary wall more easily (increased L _p ). All these lead to increased movement of fluid from the capillaries into the interstitium and, consequently, edema.

Accumulation of fluid in the brain (cerebral edema) is dangerous. The brain is encased by the cranium; therefore enlargement of the brain resulting from edema can give rise to excessive intracranial pressure, which can lead to abnormal neurological symptoms. In contrast to capillaries in the peripheral circulation, cerebral capillaries have exceedingly low permeability to most solutes, including even small molecules such as glucose (MW 180). This blood-brain barrier makes it possible to reduce brain edema by introducing a small solute such as mannitol (MW 182) into the circulation. Because mannitol cannot cross the blood-brain barrier, it increases the osmotic pressure of blood plasma relative to the cerebrospinal fluid. Water is thus drawn out of the brain into the circulatory system, with a corresponding reduction in brain volume. Neurosurgeons routinely use this technique to treat brain edema or to reduce brain volume during neurosurgery.

In different tissues in the body, fluid filtration can vary widely because of differences in the structure of the capillary wall and variations in the local Starling forces ( Box 3.6 ). The highest filtration rates are in the glomerular capillaries of the kidney ( Box 3.7 ).

Box 3.6

Filtration and Absorption of Fluid in the Microvasculature

Research conducted over the past several decades has revealed the varied nature of fluid exchange in the microvasculature. In many tissues, the measured interstitial hydrostatic pressure, P _i , is slightly negative (–1 to –3 mmHg); this is at least partly attributable to steady uptake of interstitial fluid by lymphatic capillaries, which are in close proximity to the blood capillaries. In the same tissues, the plasma protein concentration in the interstitial fluid is higher than previously thought, giving rise to significant interstitial colloid osmotic pressure (π _i ≈ 10 mmHg). The combined effect of P _i and π _i is to increase filtration. Indeed, in many tissues, including skin, skeletal muscle, mesentery, and lung, filtration occurs throughout the entire length of the capillary and no net absorption occurs (this is easy to verify through the same Starling calculations performed in the main text). The filtered fluid is collected by the lymphatic system and eventually returned to the circulation. In contrast, sustained fluid absorption does occur in some tissues. The gut absorbs water through the intestinal epithelium; the absorbed water enters the interstitium and is continually absorbed by the mucosal capillaries. In the kidney, fluid passing out of the proximal tubules is continually absorbed by the peritubular capillaries. In both cases, fluid entering the interstitium increases P _i and dilutes interstitial proteins and thus reduces π _i ; this shift in Starling forces favors fluid absorption into the capillaries.

Vasomotion—rhythmic oscillations (5–20 min ⁻¹ ) of vascular tone caused by local contraction and relaxation of vascular smooth muscle—has been observed in the vascular beds of nearly all tissues. In particular, contraction of arterioles can slow or stop blood flow through downstream capillaries, with a corresponding drop in the capillary hydrostatic pressure, P _c . Arteriolar relaxation causes resumption of flow and a rise in P _c . Such rhythmic changes in P _c cause concomitant changes in filtration/absorption.

BOX 3.7

Filtration in the Glomeruli of the Kidney

Capillaries in the glomeruli of the kidney have the highest hydraulic conductivity of any vessels in the body. Blood passing through the glomerulus is filtered through the capillaries; the resulting “ultrafiltrate” contains no blood cells and is largely free of proteins. Each glomerulus measures approximately 90 μm in diameter but encompasses capillaries with a combined length of approximately 20,000 μm. These glomerular capillaries present a total of approximately 350,000 μm ² , or 0.0035 cm ² , of surface through which filtration can occur. A pair of healthy adult kidneys contains approximately 2.2 million glomeruli, which afford a total filtration area of approximately 7700 cm ² . The hydraulic conductivity of glomerular capillaries has been estimated at 3.3 × 10 ⁻⁵ cm·sec ⁻¹ ·(mmHg) ⁻¹ , and the average driving force for fluid filtration is approximately 7.5 mmHg. Combining these quantitative estimates yields a volume flow of 1.9 cm ³ (or mL) of ultrafiltrate per second. Thus over the course of 24 hours, approximately 165 liters of ultrafiltrate are formed. Because urine production is only approximately 1.5 liters per day, we can immediately deduce that greater than 99% of the volume of the initially formed ultrafiltrate is ultimately reabsorbed back into the circulation.