книги студ / color atlas of physiology 5th ed[1]. (a. despopoulos et al, thieme 2003)

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Since it is normally not possible to separately determine k and A of a biological membrane or cell layer, the product of the two (k · A) is often calculated as the ultrafiltration coefficient Kf

(m3 s–1 Pa–1) (cf. p. 152).

The transport of osmotically active particles causes water flow. Inversely, flowing water drags dissolved particles along with it. This type of solvent drag (! p. 24) is a form of convective transport.

Solvent drag does not occur if the cell wall is impermeable to the substance in question (σ = 1). Instead, the water will be retained on the side where the substance is located. In the case of the aforementioned epithelia, this means that the substances that cannot be reabsorbed from the tubule or intestinal lumen lead to osmotic diuresis (! p. 172) and diarrhea respectively. The latter is the mechanism of action of saline laxatives (! p. 262).

Oncotic Pressure / Colloid Osmotic Pressure

As all other particles dissolved in plasma, macromolecular proteins also exert an osmotic pressure referred to as oncotic pressure or colloid osmotic pressure. Considering its contribution of only 3.5 kPa (25 mm Hg) relative to the total osmotic pressure of the small molecular components of plasma, the oncotic pressure on a strictly semipermeable membrane could be defined as negligible. However, within the body, oncotic pressure is so important because the endothelium that lines the blood vessels allows small molecules to pass relatively easily (σ ! 0). According to equation 13.3, their osmotic pressure difference Δπ at the endothelium is virtually zero. Consequently, only the oncotic pressure difference of proteins is effective, as the endothelium is either partly or completely impermeable to them, depending on the capillary segment in question. Because the protein reflection coefficient σ #0 and the protein content of the plasma (ca. 75 g/L) are higher than that of the interstitium, these two factors counteract filtration, i.e., the blood pressure-driven outflow of plasma water from the endothelial lumen, making the endothelium an effective volume barrier between the plasma space and the interstitium.

If the blood pressure drives water out of the blood into the interstitium (filtration), the

plasma protein concentration and thus the oncotic pressure difference π will rise (! pp. 152, 208). This rise is much higher than equation 13.3 leads one to expect (! A). The difference is attributable to specific biophysical properties of plasma proteins. If there is a pres- sure-dependent efflux or influx of water out of or into the bloodstream, these relatively high changes in oncotic pressure difference automatically exert a counterpressure that limits the flow of water.

pH, pK, Buffers

The pH indicates the hydrogen ion [H+] concentration of a solution. According to Sörensen, the pH is the negative common logarithm of the molal H+ concentration in mol/kg H2O.

Examples:

1 mol/kg H2O = 100 mol/kg H2O = pH 0, 0.1 mol/kg H2O = 10–1 mol/kg H2O = pH 1, and so on up to 10-14 mol/kg H2O = pH 14.

Since glass electrodes are normally used to measure the pH, the H+ activity of the solution is actually being determined. Thus, the following rule applies:

pH = – log (fH · [H+]),

where fH is the activity coefficient of H+. Considering its ionic concentration (see above), the fH of plasma is !0.8.

The logarithmic nature of pH must be considered when observing pH changes. For example, a rise in pH from 7.4 (40 nmol/kg H2O) to pH 7.7 decreases the H+ activity by 20 nmol/ kg H2O, whereas an equivalent decrease (e.g., from pH 7.4 to pH 7.1) increases the H+ activity by 40 nmol/kg H2O.

The pK is fundamentally similar to the pH. It is the negative common logarithm of the dissociation constant of an acid (Ka) or of a base (Kb):

pKa = $log Ka

pKb = $log Kb.

For an acid and its corresponding base, pKa + pKb = 14, so that the value of pKa can be derived from that of pKb and vice versa.

The law of mass actions applies when a weak acid (AH) dissociates:

It states that the product of the molal concentration (indicated by square brackets) of the dissociation products divided by the concentration of the nondissociated substance remains constant:

Dimensions and Units

Converted into logarithmic form (and inserting H+ activity for [H+]), the equation is transformed into:

Based on the above definitions for pH and pKa, it can also be converted into

Because the concentration and not the activity of A– and AH is used here, pKa is concentration-dependent in nonideal solutions.

Equation 13.9 is the general form of the

Henderson–Hasselbalch equation (! p. 138ff.), which describes the relationship between the pH of a solution and the concentration ratio of a dissociated to an undissociated form of a solute. If [A–] = [AH], then the concentration ratio is 1/1 = 1, which corresponds to pH = pKa since the log of 1 = 0.

A weak acid (AH) and its dissociated salt (A–) form a buffer system for H+ and OH– ions:

Addition of H+ yields A– + H+ ! AH Addition of OH– yields AH + OH– ! A– + H2O. The buffering power of a buffer system is

greatest when [AH] = [A–], i.e., when the pH of the solution equals the pKa of the buffer.

Example: Both [A–] and [AH] = 10 mmol/L and pKa = 7.0. After addition of 2 mmol/L of H+ ions, the [A–]/ [AH] ratio changes from 10/10 to 8/12 since 2 mmol/

L of A– are consequently converted into 2 mmol/L of 379 AH. Since the log of 8/12 !-0.18, the pH decreases

by 0.18 units to pH 6.82. If the initial [A–] /[AH] ratio had been 3/17, the pH would have dropped from an initial pH 6.25 (7 plus the log of 3/17 = 6.25) to pH 5.7 (7 + log of 1/19 = 5.7), i.e., by 0.55 pH units after addition of the same quantity of H+ ions.

Concentration ratio of buffer pair [AH]:[A–]

10:0 9:1

8:2 7:3 6:4 5:5 4:6

3:7 2:8 1:9

pH

(pH=pK)

B. Buffering Curve. Graphic representation of the relationship between pH and the concentration ratio of buffer acid/buffer base [AH]/[A–] as a function of pH. The numerical values are roughly equivalent to those of the buffer pair acetic acid/ acetate (pKa = 4.7). The buffering power of a buffer system is greatest when [AH] = [A–], i.e., when the pH of the solution equals the pKa of the buffer (broken lines).

position of the 1 after the decimal point; therefore, 0.001 = 10-3. When writing numbers greater than 10, the exponent corresponds to the number of decimal positions to the left of the decimal point minus 1; therefore, 1124.5 = 1.245 · 103.

Exponents can also be used to represent units of measure, e.g., m3. As in the case of 103, the base element (meters) is multiplied by itself three times (m · m · m; ! p. 372). Negative exponents are also used to express units of measure. As with 1/10 = 10–1, 1/s can be written as s–1, mol/L as mol · L–1, etc.

There are specific rules for performing calculations with powers of ten. Addition and subtraction are possible only if the exponents are identical, e.g.,

(2.5 · 102) + (1.5 · 102) = 4 · 102.

Unequal exponents, e.g., (2 · 103) + (3 · 102), must first be equalized:

(2 · 103) + (0.3 · 103) = 2.3 · 103.

The exponents of the multiplicands are added together when multiplying powers of 10, and the denominator is subtracted from the numerator when dividing powers of ten.

Examples:

102 · 103 = 102+ 3 = 105 104 %102 = 104 - 2 = 102 102 %104 = 102 - 4 = 10-2

The usual mathematical rules apply to the multipliers of powers of ten, e.g.,

(3 · 102 · (2 · 103) = (2 · 3) · (102+ 3) = 6 · 105. Logarithms. There are two kinds of loga-

rithms: common and natural. Logarithmic calculations are performed using exponents alone. The common (decimal) logarithm (log or lg) is the power or exponent to which 10 must be raised to equal the number in question. The common logarithm of 100 (log 100) is 2, for example, because 102 = 100. Decimal logarithms are commonly used in physiology, e.g., to define pH values (see above) and to plot the pressure of sound on a decibel scale (!p. 363).

Natural logarithms (ln) have a natural base of 2.71828..., also called base e. The common logarithm (log x) equals the natural logarithm of x (ln x) divided by the natural logarithm of 10 (ln 10), where ln 10 = 2.302585. The following rules apply when converting between natural and common logarithms:

log x = (ln x)/2.3 ln x = 2.3 · log x.

When performing mathematical operations with logarithms, the type of operation is reduced by one rank—multiplication becomes addition, potentiation becomes multiplication, and so on.

Examples:

log (a · b) = log a + log b log (a/b) = log a - log b log an = n · log a

log n!"a = (log a)/n

Special cases: log 10 = ln e = 1 log 1 = ln 1 = 0 log 0 = ln 0 = & '

Graphic Representation of Data

C. Illustration of how to plot data on a coordinate system. The plot in this example shows the

relationship between body temperature (rectal, 381 at rest) and time of day.

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2y

x (linear)

ln x

D. Types of functions. D1: Linear function (violet), exponential function (red), logarithmic function (blue), and power function (green) showing linear plotting of data on both axes. The three curves can be made into a straight line (linearized) by logarithmically plotting the data on the y-axis (D2: exponential function) or on the x-axis (D4: logarithmic function) or both (D3: power function).

system, it can be impossible to plot small values individually without having the axes become extremely long. This problem can be solved by plotting the data as powers of 10 or logarithms. For example, 1, 10, 100, and 1000

382are written as 100, 101, 102, and 103 or as logarithms 0, 1, 2, and 3. This makes it possible to

obtain a relatively accurate graphic representation of very small numbers, and all the numbers fit on an axis of reasonable length (cf. sound curves on p. 363 B).

Correlations can be either linear or nonlinear. Linear correlations (! D1, violet line) obey the linear relationship

y = ax + b,

where a is the slope of the line and b is the point, or intercept (at x = 0), where it intersects the y-axis.

Many correlations are nonlinear. For simpler functions, graphic linearization can be achieved via a nonlinear (logarithmic) plot of the x and/or y values. This allows for the extrapolation of values beyond the range of measurement (see below) or for the generation of calibration curves from only two points. In addition, this method also permits the calculation of the “mean” correlation of scattered x-y pairs using regression lines.

An exponential function (! D1, red curve), such as

y = a · eb · x,

can be linearized by plotting ln y on the y-axis (!D2):

ln y = ln a + b · x,

where b is the slope and ln a is the intercept. A logarithmic function (! D1, blue curve),

such as

y = a + b · ln x,

can be linearized by plotting ln x on the x-axis (! D4), where b is the slope and a is the intercept.

A power function (! D1, green curve), such

as

y = a · xb,

can be graphically linearized by plotting ln y and ln x on the coordinate axes (! D3) because

ln y = ln a + b · ln x,

where b is the slope and ln a is the intercept.

Note: The condition x or y = 0 does not exist on logarithmic coordinates because ln 0 = '. Nevertheless, ln a is still called the intercept in the equation when the logarithmic abscissa (! D3,4) is intercepted by the ordinate at ln x = 0, i.e., x = 1.

Instead of plotting ln x and/or ln y on the x- and/or y-axis, they can be plotted on logarithmic paper on which the ordinate or abscissa (semi-log paper) or both coordinates (log-log paper) are plotted in logarithmic units. In such cases, a is no longer treated as the intersect because the position of a depends on site of intersection of the x-axis by the y-axis. All values (0 are possible.

Other nonlinear functions can also be graphically linearized using an appropriate plotting method. Take, for example, the Michaelis– Menten equation (! E1), which applies to

many enzyme reactions and carrier-mediated transport processes:

where J is the actual rate of transport (e.g., in

mol · m–2 · s–1), Jmax is the maximal transport rate, C (mol · m–3) is the actual concentration of

the substance to be transported, and KM is the concentration (half-saturation concentration)

at 1/2 Jmax.

One of the three commonly used linear rearrangements of the Michaelis–Menten equation, the Lineweaver–Burk plot, states:

Consequently, a plot of 1/J on the y-axis and 1/C on the x-axis results in a straight line (! E2). While a plot of J over C (! E1) does not

J = Jmax !C/(KM+C)

1/2Jmax

E. Two methods of representing the Mi- chaelis–Menten equation: The data can be plotted as a curve of J over C (E1), or as 1/J over 1/C in linearized form (E2). In the latter case, Jmax and KM are determined by extrapolating the data outside the range of measurement.

Graphic Representation of Data

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Despopoulos, Color Atlas of Physiology © 2003 Thieme

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Reference Values in Physiology

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Reference Values in Physiology