Nutrient Uptake by Microorganisms according to Kinetic Parameters from Theory as Related to Cytoarchitecture

Abstract

The abilities of organisms to sequester substrate are described by the two kinetic constants specific affinity, a°, and maximal velocity V_max. Specific affinity is derived from the frequency of substrate-molecule collisions with permease sites on the cell surface at subsaturating concentrations of substrates. V_max is derived from the number of permeases and the effective residence time, τ, of the transported molecule on the permease. The results may be analyzed with affinity plots (v/S versus v, where v is the rate of substrate uptake), which extrapolate to the specific affinity and are usually concave up. A third derived parameter, the affinity constant K_A, is similar to K_M but is compared to the specific affinity rather than V_max and is defined as the concentration of substrate necessary to reduce the specific affinity by half. It can be determined in the absence of a maximal velocity measurement and is equal to the Michaelis constant for a system with hyperbolic kinetics. Both are taken as a measure of τ, with departure of K_M from K_A being affected by permease/enzyme ratios. Compilation of kinetic data indicates a 10⁸-fold range in specific affinities and a smaller (10³-fold) range in V_max values. Data suggest that both specific affinities and maximal velocities can be underestimated by protocols which interrupt nutrient flow prior to kinetic analysis. A previously reported inverse relationship between specific affinity and saturation constants was confirmed. Comparisons of affinities with ambient concentrations of substrates indicated that only the largest a°_S values are compatible with growth in natural systems.

“When theory and observations come together, science often takes a great step forward.”

Stephen Hawking, 1997

The ability of microorganisms to collect dissolved substrates, both relative to other organisms and absolute as based on some cell property such as dry weight, is a component of organism descriptions that reflects their role in the environment. Affinities of the organisms for nutrients express this ability and often set the concentrations of nutrients as well (13). The curvilinear nature of the kinetics for single-nutrient limited growth led Monod to propose a hyperbolic relationship as a phenomenological expression of growth that accommodates saturation. Inclusion of a yield “constant” (62) extended the relationship to the dependency of the rate of nutrient uptake on nutrient concentration. The idea of catalytic efficiencies decreasing with concentration was expressed by Langmuir (36), who based formulations on collision frequency. The Henri concept of enzyme saturation (26) and the Michaelis-Menten relationship (41) derived from the rate constants for substrate transformation, together with “the constant of affinity” (43), expressed by the concentration that drives the rate of catalysis at half the maximal rate, give formulations that are characteristic of enzyme operation (18). For nutrient absorption by microorganisms, various definitions of affinity have been proposed (7) with a range of units. Recent examples are those of a dissociation constant (70), the ratios of growth rate (58) or maximal uptake rate to the Michaelis constant, K_m (17), and concentration expressed as K_m alone (68). Affinity is often specified as a property of a population which reflects qualitative properties of a “substrate uptake enzyme” (49, 54) and is justified by an assumption that the enzyme or permease is a limiting enzyme, so that the quality of the permease reflects the kinetic properties of the cell. However, adherence to the hyperbolic kinetics used for single enzymes is not always appropriate for whole cells. Reasons include uptake rates that may not be asymptotic with concentration except over a range of concentration that is small, uptake rates that fill metabolic pools or deplete external substrate before rates can be measured (52), variation in the regions of significant flux control along the pathway depending on external concentrations, metabolic energy reserves, stoichiometry of the enzyme and permease components along the pathway, and complex coupling between substrate uptake and the rate of organism growth. Therefore, K_m has been regarded as an inaccurate constant for microbial processes (40).

The purpose of this communication is to clarify the connections between the uptake of nutrients by a cell and the physical and biochemical principles relevant to the process, to consider definitions of the constants that specify the rates, to apply these considerations to recently measured values, and to organize the resulting kinetic constants as an index of the ability of various organisms to collect nutrients. Because the literature often contains hidden units or ones that change with conditions (1), fusion of the microbial kinetic literature with that of enzyme and chemical kinetics requires assumptions that we seek to clarify by formulation and example. Related treatments include those of Pirt (48), with a focus on industrial processes, and Koch, who reviews historical aspects (33, 34) and gives an improved perspective of whole-cell biophysics (31, 33) as well as mathematical models describing the kinetics of growth that is slow (34). Dynamics in natural systems have recently been formulated by using steady-state assumptions (13) and transient-state modeling (62). An excellent review of transport mechanisms together with the related biophysics and bioenergetics as well as computational examples has been written by Cramer and Knaff (19).

THEORY

Basis in Collision Frequency

The relation between the rate of substrate uptake at a particular concentration and specific affinity as the intervening rate constant can be modeled from the rate of collision between two particles. One represents a substrate, the other represents a cell, and spherical shape (32) is assumed. The number of moles experiencing such collisions with a particle over time and expressed as a rate v_p is given by equation 1:

1

where k_p is a per-particle rate constant and S is the concentration of substrate. For nomenclature, see Table 1. This rate is set predominantly, according to collision frequency theory, by the radius of the larger particle or cell, r_x, together with the molecular diffusion constant, D, of the smaller particle or substrate. When the collisions are expressed as moles, the rate of collisions with a single cell is (4)

2

The number of moles of substrate colliding with an organism or cell on an organism mass basis (Fig. 1) is (number of collisions with a cell of radius r) ÷ (wet mass of cell with radius r). Then the rate constant describing the collisions from equation 2, k_s = 4πr_xD/1,000, assumes the units

where ρ is cell density. Converting this single-organism rate constant to units of cell mass appropriate to cell growth by combining equations 1 and 2 gives

3

Specific affinity has been described as a rate constant that defines the accumulation of substrate in terms of mass over a time frame of hours (6). Redefining the rate constant k_s by converting from moles to mass with the molecular weight of the substrate M and taking the cell density of bacteria as 1.08 g cm⁻³ gives 3 × 60² s h⁻¹/(1.08 × 1,000) = 10 liters s/g of cells h. The specific affinity simplifies to

4

where a^o_max is the specific affinity of a population of perfectly absorbing spheres of specified size and mass and the units become consistent with commonly made observations. It is a pseudo-first-order rate constant, which gives the uptake rate of 1 g of cells at a particular concentration of substrate, S. Since equation 1 specifies a per-particle rate, the specific affinity is a second-order rate constant. Then the specific uptake rate, v, of a particular population of cells of known volume to give biomass X is

5

For a cell radius of 0.4 μm and a substrate with a molecular weight of 100 with a diffusion constant of 10⁻⁵ cm² s⁻¹, the upper limit to the specific affinity according to equation 4 is

Microorganisms differ from perfectly absorbing spheres according to the resistance, R, that the cell envelope provides to penetration by nutrients. Other constraints can include permease saturation, product inhibition from the intermediates of pathway enzymes downstream and their connections with rates of macromolecule formation and arrangement, and limitations in energy reserves such as electrochemical gradients and potentials such as proton motive force and ATP concentration, which may be coupled to the flow (Fig. 1). When resistivity, the ratio of nutrient absorption to the theoretically attainable value, is written as an absorption coefficient, ζ = 1 − R, the rate of substrate absorption becomes

6

Resistivity then ranges from zero for an organism that is completely covered with sites to unity for an organism with none. The rate is conceptualized as a resistivity because the rate of nutrient import would be given by the frequency of collision with the cell surface if the cell surface provided no resistance. The value of a°_max from equation 5 is about 10³ larger than most specific affinities observed (see below, and note that the reported values are per milligram of cells) because of the barrier that the cell envelope provides to penetration by hydrophilic nutrients.

TABLE 1.

Nomenclature

Term	Definition	Dimensions
a°s	Specific affinity for substrate S; base or unsaturated value; aKAs value at S = KA	liters mg of cells−1 h−1
a°max	Specific affinity of a perfectly absorbing sphere	liters mg of cells−1 h−1
as	Specific affinity for substrate S as reduced by saturation	liters mg of cells−1 h−1
c	Constant	5DM rs2 (2 rx4)−1
D	Molecular diffusion constant	cm2 s−1
k	Rate constant	liters particle−1 t−1, M−1 t−1
kcat	Catalytic constant	Moles of substrate transformed (moles of protein−1 s−1)
KA	Affinity constant: concentration of S at aos/2	g liter−1
Km	Michaelis-Menten constant: concentration of S at Vmax/2	g liter−1
M	Molecular mass	Daltons
μ	Growth rate	h−1
N	Number of molecules of a particular permease	molecules cell−1
n	Number of molecules of a particular pathway enzyme	molecules cell−1
ρ	Organism density	g of cell material cm−3
R	Resistivity	None
ζ	Absorption coefficient	None
rx	Radius of a spherical cell	cm
rs	Effective radius of a permease site	cm
S	Concentration of substrate; Si, inside substrate concentration	g liter−1 or molar
τ	Residence time	Site g of cells h g of substrate−1, or cell site s molecule−1
v	Rate of substrate uptake by a cell (vx) or population of cells	g of substrate liter−1 h−1, g of substrate cell−1 h−1
Vmax	Maximal rate of substrate accumulated	g of substrate accumulated g of cells−1 h−1
X	Biomass	g of cells (wet wt) liter−1
Y	Cell yield	g of cells produced (g of substrate consumed)−1

FIG. 1.

Model of substrate uptake kinetics by a cell growing on a single substrate.

Effect of Permeases

Particular nutrient types are transported by associated membrane proteins for transport, and the active sites of these permeases cover only part of the limit-membrane surface. Assuming that the reduction in specific affinity is proportional to the ratio of the area of regions on the cell surface capable of accepting the substrate to the total surface area of the cell and that factors such as saturation become negligible as the substrate concentration, S, approaches zero, the base or unimpeded value for the specific affinity is defined as a^o_s and is given by a°_max ζ:

7

where the theoretical absorption coefficient, ζ, is determined by the number, N, of proteins or transporters (T, Fig. 1), the radius of the effective substrate collection area of the proteins, r_s, and the radius of the cell, r_x, as specified by Nπr_s²/4πr_x². Rearranging equation 7, the number of proteins is given by the specific affinity according to the measured uptake rate as compared to the calculated value for the whole cell surface

8

Using the value for the rate constant of a completely absorbing sphere from equation 4 as a^o_max gives a value for N, the effective number of permeases associated with a particular specific affinity:

9

Oligobacteria are capable of growth at the very low concentrations of dissolved organic material found in aquatic systems (8). Taking the radius of transport sites, r_s, as 10 Å, the example above, and the specific affinity required for growth of an oligobacterium in seawater as 4,000 liters g of cells⁻¹ h⁻¹ (see below),

and observed specific affinities might be attained with a fairly small portion of the cell surface covered with transport proteins. For example, a 0.8-μm-diameter oligobacterium surface area, 4 × 3.14 × (4 × 10³)² or 2 × 10⁸ Ų, would contain 410 patches of 900 Ų for 60-kDa permeases, which amounts to 0.2% of its surface area. Bacterial permeases in the 0.1 to 0.5% range of membrane proteins have been reported (3). With membrane proteins at 14% of the total (20), this amounts to 800 to 4,000 permease molecules per cell. Considering that oligotrophs may have a larger permease content, that bacteria can be sufficiently large to affect the form of rate equation 1, that effective areas of the active site are uncertain, that a scarcity of downstream enzymes may reduce the specific affinity (see below), and that effective concentrations may be reduced by the outer membrane of gram-negative bacteria, the calculations are not inconsistent.

Substrate collisions with permeases that are already occupied is a factor in reducing the base value of the specific affinity a^o_s for a substrate to progressively smaller values a_s as the concentration of substrate becomes large. Derivations of Michaelis and Menten specify the direct effects of saturation and can be modified to emphasize the dependence of nutrient uptake rate on the number of unoccupied transporters per cell. At steady state, assuming no change in the total number of transporters with S

10

This means that at any substrate concentration S, NS transporters will accrue from the total population of transporters, N. The rate constant for substrate release in the forward direction per cell, k₃, sets the flow through the total population of transporters at v = k₃[NS]. The forward flow is balanced by the rate of collision with free transporters (k₁ − k₂) [N] [S], where k₁ is the rate constant for substrate collisions with the effective area of free transporters and k₂ is the rate constant for substrate collisions with the same area but where the sites are occupied in addition to any back reaction occurring. Assuming that back reactions are negligible, the value of v at steady state is the same as the rate of substrate collisions with the free transporters: k₁ [N − NS] [S]. Writing equation 10 in the form

11

it may be seen that as k₂ approaches 0, a possible situation at low S when collisions with the effective area of the active site are not prevented by previously successful collisions, k₁ approaches k₃, [N − NS] approaches [N], and v = k₁ [N] [S]. Thus, the rate of substrate uptake increases with the number of transporters and the proportionality is direct when S is low. Units of the rate constant are liters site⁻¹ time⁻¹, identical to the units of specific affinity formulated above when multiplied by the factor (sites g of cells⁻¹). The value is reduced to a concentration-dependent value half as large when half the collisions are reflected due to site occupancy and k₂ = k₃. When k₂ is large, the number of free transporters [N − NS] approaches 0, [NS] approaches [N], and v = k₃ [N]. This latter term is the maximal velocity for the cell, or V_max when multiplied by cell population.

Saturation in Terms of Kinetic Constants

If it is assumed that the cell composition remains constant and that only the amount of transporter bound to substrate changes with a change in the concentration of substrate, substitution of K_m for the rate constants in equation 11 leads to the hyperbolic relationship

12

Factors unique to the transport kinetics of the cell as modeled are that transporter activity and density may change with S and that K_m might be thought of as an occupancy constant, i.e., the concentration of substrate large enough for half the permeases or pathway components to be sufficiently occupied with substrate processing to cause reflection of collisions that may otherwise have been successful. Multiplying through by V_max/K_m, and since the initial slope of the plot of v against S is the base value for the specific affinity a^o_s as defined for a hyperbola by a^o_s = V_max/K_m (11),

13

Equation 13 shows that the rate of transport may be completely defined by the two kinetic constants V_max and specific affinity a^o_s. Unlike V_max and K_m, the two parameters are mutually independent and are therefore primary kinetic constants. Equation 10 can be rearranged to give

14

so that a plot of v/S against v gives a straight line with intercepts V_max and a^o_s.

When substrate import is impeded by saturation, the rate constant of equation 1 changes from a^o_s as defined by equation 13 to the smaller value a_s as discussed above, i.e., from the initial slope of a plot of v against S (see Fig. 2A) to the slope of a line between 0 and the point (v, S) at any value of S. Assuming that the concentration-dependent partly saturated rate constant a_s remains v/S at any concentration or kinetic curve shape, substituting a_s = v/S gives

15

which is a general form of the rate equation. v/V_max quantifies the reduction in the specific affinity from a^o_s to a_s due to resistances to substrate flow from saturation as it may occur along the path of molecular flow from any particular external substrate concentration S. The specific affinity is related to substrate concentration by S = v/a_s at any particular rate v.

FIG. 2.

Graphical determination of kinetic constants for substrate uptake. (A) Plot of v against S for leucine uptake by E. coli. The inset shows the complete range of substrate concentrations analyzed. (B) Plot of v/S versus v (affinity plot). The equation of the curves is the sum of equation 13 and a linear function v = dS, where d is an arbitrary constant. The line was visually fit, by use of a plotting program, SigmaPlot, to the three plots simultaneously so that small and large values could be equally weighted. The base value of the specific affinity a^o_s, 0.008 liters mg of cells⁻¹ h⁻¹, was taken from the ordinate intercept of the affinity plot and used to draw the initial slope of the curve of v against S, which gives the specific affinity as well. V_max was indeterminate at >9 μg of substrate mg of cells⁻¹ h⁻¹. The half-maximal value of the specific affinity, a^KA_s, is 0.004 liters mg of cells⁻¹ h⁻¹. The affinity constant K_A is S at a^KA_s or v/a^KA_s and has a value of 2 μg of S mg of cells⁻¹ h⁻¹/0.004 liter mg of cells⁻¹ h⁻¹ or 500 μg of S liter⁻¹. In the plot of v against S, a^KA_s is the value of the abscissa at the intersection of the line with slope a^KA_s and the curve. The Michaelis constant, K_m, remains S at V_max/2 (>7,000 μg liter⁻¹).

Dependency of Saturating Concentrations on τ

V_max defines the reduction in rate of substrate import due to saturation because it specifies the time, τ, that the relevant protein is occupied by substrate according to τ = N V_max⁻¹ in the case of limitation by permease alone. If the activity of a particular enzyme is taken as reflective of maximal rate through the pathway, the maximal rate, V_max, is k_catN, where k_cat is the catalytic constant for that representative protein. The concentration-dependent specific affinity a_s becomes

16

with units

or, in terms of residence time of the substrate on the site,

17

Equation 9 can be rearranged to give a^o_s in terms of collision frequency as reflected by the ratio of the area of the permease sites to that of the cell:

18

If 5DMr_s²/2r_x⁴, which has the units liters cell (g of cells site h)⁻¹, is set equal to a constant c, the specific affinity, a^o_s = Nc. Since rate is related to substrate concentration by the saturation-dependent affinity a_s and v = a_sS, equation 17 becomes

19

or

20

INTERPRETATIONS OF KINETIC DATA

Kinetic Constants

Equation 20 shows the direct increase in organism affinity with the number of permeases or enzymes according to the increase in area available for substrate collection as specified by equation 7. Affinity approaches zero due to saturation when the substrate concentration is high. Increases in substrate concentration are most effective in reducing the affinity at intermediate concentrations, and the affinity is reduced by half at the affinity constant K_A. The concentration dependence of the reduction of affinity due to saturation depends on the residence time of the substrate molecules on the enzyme, and long residence times reduce concentrations that saturate. The effective value of τ is expected to reflect thermodynamic pressures along the connected members of the pathways, and a change in τ with substrate concentration will cause departure from hyperbolic kinetics. For example, if increased flow through a pathway fosters increased capacity of the pathway, τ will decrease and the plot of equation 14 will be a curve that is concave up. Since S = K_m for a pathway having hyperbolic kinetics, it can be seen that there is little effect of capacity, as given by N, on K_m. And since Scτ = 1 at K_A, it can be seen that K_A increases with decreasing residence time of substrate in the pathway.

Values for the constants may be solved for the example above. The value for c is

At K_A, the value for Scτ in equation 20 is unity, since V_max = Nc is reduced by half. Solving for τ at a substrate concentration of 10⁻⁶ g liter⁻¹,

and converting units

the substrate may take as long as 0.3 s to pass through the permease site while reducing the rate of absorption only by half. Computed values for K_A are not very different from those observed for the toluene/Cycloclasticus oligotrophus system (unpublished data).

Data Analysis

Plots of v against S (equation 13) give a direct indication of the kinetics of substrate uptake by a population of cells. Data from Wood (69) plotted in this way (Fig. 2A) favor a focus on measurements at high concentration, and the inset extends the range of concentration observed. They show the incomplete saturation of rates resulting in an indeterminate V_max as often observed. The original data are converted to the units shown to facilitate computation of the kinetic constants as outlined in the legend. The initial slope of the curve is the specific affinity a^o_s, and the affinity constant, K_A, is the concentration at which the specific affinity is reduced by half. Since the value of the affinity, a_s, is concentration dependent at all concentrations sufficient for saturation to occur, the substrate concentration associated with this parameter is required and is designated by a superscript (9). In this case, it is identified by the concentration that half-saturates the initial rate (K_A). Figure 2B shows the same data according to equation 14 and emphasizes rates at low concentrations. It can be called either an Eadie-Scatchard plot (57) or an affinity plot, since the ordinate extrapolates to the base value of the specific affinity a^o_s and the derivation is somewhat different.

When determinate, V_max is given by the abscissa intercept. Concentrations affecting saturation, using the maximal rate of substrate uptake at high concentration for comparison, may be computed from v/S when v = V_max/2 to give the Michaelis constant K_m. However, a value for V_max must be established. When a range of high concentrations is included, as in the present case, rates often continue to increase with concentration, causing difficulties in establishing what the value is or what influences control it. The spread between K_A and K_m, 0.5 to 7 mg/liter in this case, quantifies the departure from hyperbolic or Michaelian kinetics.

Data scatter is amplified in the affinity plot due to the division of small numbers of low precision, and a^o_s values can be difficult to establish for oligotrophs. For example, the specific affinity for C. oligotrophus, tabulated below, was sufficient for nearly total depletion of added substrate within 5 min when added to populations of only 1 mg of cells/liter, yet saturating concentrations were so low that substrate transformation rates were barely detectable even with the use of radioisotopes. When affinity plots are concave up, values for the specific affinity may be underestimated unless data are collected at very small concentrations of substrate.

Reported Kinetic Data

“It doesn’t do any good to do the theory if one doesn’t also confirm it with experimentation.”

Jaques Monod, 1950

Kinetic constants (Table 2) update earlier values tabulated for various organic substrates and organisms (7, 8, 56). Results are given in order of decreasing specific affinity. Therefore, organisms at any position in the table, other factors notwithstanding, are better able to compete for substrate than are organisms listed below them. Because of the difficulties in cultivating oligotrophs, data underrepresent this quantitatively dominant, but largely unstudied group of organisms.

TABLE 2.

Kinetic constants from the recent literature^a

Culture	Substrate	Specific affinity (aos) (liters mg of cells−1 h−1)c	Maximumal velocity (Vmax) (mg of S mg of cells−1 h−1)	Km (μg liter−1)	KA (μg liter−1)	Reference
Cycloclasticus oligotrophus	Toluene	47.4	1.2	10	1.3	Unpublished data
Escherichia coli	Glucose	7.3	4.8	50	≥50	38
Pseudomonas sp. strain P-15	Phenanthrene	1.2	0.06	>400	400	60
Seawater, Resurrection Bayb	Leucine	0.83	0.006	>2.6	2.6	Unpublished data
Pseudomonas sp. strain T2	Toluene	0.32	0.14	44	44	53
Rumen isolate SR	Arginine	0.30	0.41	13,900	1,044	63
Escherichia coli	Leucine	0.21	0.010	1,570	∼260	69
Sphingomonas sp. strain RB 2256	Glucose	0.2				56
Corynebacterium glutamicum	Glycine betaine	0.11	0.25	1,015		23
Sphingomonas sp. strain RB 2256	Alanine	0.06	0.029	440	440	56
Mixed methanotroph culture	Trichloroethylene	0.06	0.51	7,990		17
Desulfovibrio G-11 and syntropic benzoate isolate	Benzoate	0.025	0.045	1,732		66
Corynebacterium glutamicum	Tyrosine	0.02	0.011	543		67
Escherichia coli	Gluconate	0.017	0.08	4,900		64
Pseudomonas sp. strain B13 (chemostat)	3-Chlorobenzoate	0.016	0.032	6,768		61
Peter Lake	Leucine	0.016				47
Escherichia coli	Leucine (repressed)	0.008	0.009	7,000	500	69
Saccharomyces cerevisiae	Glucose	0.0016	0.54	5.6 × 106	1.1 × 105	64
Cycloclasticus oligotrophus	Acetate	0.0016	0.13	20,000	20,000	Unpublished data
Lactobacillus brevis 367	Trimethyl-β-galactoside	0.0010	0.14	1.1 × 105		28
Escherichia coli	p-Nitrophenyl phosphate	0.0072	0.65	44,630	149,000	40
Penicillium chrysogenum	Phenylacetic acid	3.3 × 10−4	>6.1	>5 × 106		27
Vibrio parahaemolyticus	Glucose	2.6 × 10−4	0.027	1.0 × 105		55
Halobacterium saccharovorum	Glucose	2.8 × 10−4	0.07	1.4 × 105	28,000	59
Seawater, Resurrection Bayc	Toluene	0.85 × 10−6				10
Escherichia coli K-12	Glycine	1 × 10−6	2 × 10−5	2.0 × 105		24

^aConversion factors used: 400 mg/liter (wet wt) OD⁻¹ (30), 3 mg of cells (wet wt) (mg dry wt)⁻¹, 1.9 mg of cells (dry wt) mg of protein⁻¹. 

^bBiomass is taken as 53% of the total population which used leucine, as determined by autoradiography, and corrected for the 0.8-μg-liter⁻¹ ambient leucine present (12). 

^cBiomass is taken as the total bacterial population, which was 7% toluene oxidizers. The specific affinity was corrected for saturation by using to the kinetic constants shown (constants were obtained with Cycloclasticus oligotrophus [data not shown]). 

Maximal velocities reflect the maximal uptake rate of a single substrate. Reported values were used to calculate specific affinities from V_max/K_m where kinetics were hyperbolic. Affinity plots will be concave down, linear, or concave up as K_m moves from small to large compared with K_A; when they differ, both values are shown. Combining four parameters to improve the fit to experimental data, the kinetic curve is defined by its initial slope a^o_s, its slope at concentration K^KA_s, its upper limit in rate, and the rate at concentration K_m.

Data Interpretations

The large range of specific affinities reported suggests a large difference in the ability to transport substrate from low concentrations. Appropriate units vary according to available information on the organisms and may be interconverted. The factor for converting liters mg of cells⁻¹ h⁻¹, the units used here, to liters milligram of C⁻¹ day⁻¹ is (5 mg of cells [wet] mg of cells [dry]⁻¹) (2 g of cells [dry] g of C⁻¹) (24 h day⁻¹) or approximately 240. For an aquatic organism growing at a specific rate of 0.05 h⁻¹, the required specific affinity at a substrate concentration of 10 μg liter⁻¹, and a yield of unity is a_s = μ/YS = 0.05/(1) (10⁻⁵) = 5 × 10³ or, with the dimensions used in Table 2, 5 liters mg of cells⁻¹ h⁻¹. While individual labile marine substrate concentrations may be lower (25), the 10 μg liter⁻¹ concentration of substrate chosen may be made up of several substrates used simultaneously. Experimental data indicate the advantage of simultaneous use of multiple substrates (37) to organisms competing for carbon and energy sources at low concentrations, and in the absence of competitive inhibition (56), the specific affinities for substrates used will be additive (9) depending on the substrates involved (38). The median specific affinity (Fig. 3A) is 0.07 liter g of cells⁻¹ h⁻¹, about 1% of that required for aquatic bacteria. Only the uppermost value of Fig. 3 is sufficient for growth on a single substrate in natural water systems at ambient measured concentrations of substrate which are near 1 μg/liter (14, 16).

FIG. 3.

Data trends from Table 2. (A) V_max plotted against specific affinity (inset), with the logarithmic transformation shown in the main figure. (B) Saturation constants plotted against specific affinity (inset), with the logarithmic transformation shown in the main figure. Solid symbols indicate K_m, and open symbols indicate K_A.

Specific affinities are easily depressed (52), and values for toluene utilization by C. oligotrophus could be increased by using very small populations over long periods or freshly collected cells from continuous culture at small populations (unpublished data). This is also seen by the 10⁶-fold difference among values reported for Escherichia coli, with the larger values being reported from continuous culture (38). Likewise, the value for acetate utilization was increased by a factor of 160 when the value was computed from the slope of the growth curve in extended batch culture rather than from measurements of the rate of uptake by harvested cells. Consequently, specific affinities measured by conventional procedures may be in error and may be responsible for measured values that are low compared with concentrations calculated to be necessary.

The very large value for toluene affinity is consistent with the idea that permeases are not required for hydrocarbon transport but, rather, that such lipid-soluble substrates diffuse directly to a dioxygenase for initial reaction (14). However, some workers suggest that such substrates may be transported actively with the aid of permeases (65). As expected from equation 20, there seems to be little dependence of maximal velocity on specific affinity (Fig. 3A). However, median values of 0.3 mg of substrate mg of cells (wet wt)⁻¹ h⁻¹ are only sufficient for a doubling time of 23 h, and the possibility of underestimation exists for V_max as well as a^o_s. As with specific affinity, V_max values appear to be easily underestimated due to cell trauma during harvest. For phosphate (52), maximal velocities were erratic and depressed. Values shown for both toluene and acetate uptake by C. oligotrophus could be greatly enhanced by using small populations of cells grown in continuous culture or by calculating V_max from rates of growth in batch culture (unpublished data), and the highest V_max value was again obtained in continuous culture.

Oligotrophs are often thought of as having small K_m values. As previously observed (8), there is a strong correlation between specific affinity and saturation constants (Fig. 3B). The large values for K_m (inset) are for Saccharomyces adapted to high concentrations of sugars and for systems such as acetate transport, thought to be governed by diffusion-limited steps. The highest specific affinities shown emphasize the ability of good oligotrophs to collect substrates. That K_m values for all the unmodified organisms exceed those for K_A results from the observation only of curvilinear kinetics that are concave up.

Interpretation of Kinetics from the Rate Constants

The value of k_cat may be computed from the estimate of 410 molecules of permease for the example illustrated in Fig. 1 and a V_max of 1.1 g of substrate g of cells⁻¹ h⁻¹ to give k_cat = V_max/N = 192 molecules site⁻¹ s⁻¹ and a residence time on the permease of τ = 1/192 = 5.3 ms. The catalytic constant is at the high end but is within the range of observed values (51, 70). The residence time, τ, in interactive portions of the pathway is then on the order of k_cat⁻¹ or 5.3 ms, probably a minimum since the value chosen for the specific affinity is near the upper known limits for oligotrophs. The forward rate constant, k₁, is k_cat/K_m (40) from k₃/[(k₂ + k₃)/k₁], assuming k₂ < k₁ and canceling. Using conditions specified in the previous section and a very good oligotroph with K_m = 10⁻⁸ M gives k₁ = 192 s⁻¹/10⁻⁸ M = 1.92 × 10¹⁰ M⁻¹ s⁻¹. By comparison, association constants of 10⁻⁷ to 10⁻⁸ M⁻¹ s⁻¹ have been measured for binding to periplasmic proteins (42), which in general have far larger saturation constants. The change of units for k₁ is due to a change from per particle or per cell to per mole of protein (permease). The rate equation in the present case is v_x = k₁ [N] [S], and the rate at [S] = 1 μg liter⁻¹ for a 100-Da substrate is

= 0.36 g of substrate g of cells⁻¹ h⁻¹, an adequate flux for an aquatic bacterium to divide several times per day (21).

For transported molecules, the difference between the whole-cell rate constant k_p in equation 1 and the permease-specific rate constant k₁ in equation 10 specifies the difference between the rate of nutrient transport by a completely absorbing cell and one with a finite number of permeases. If k₁ is a general property of permeases set by active-site area, then a^o_s depends most directly on the number of permeases (equation 20). The effect of the permease level on transport was formulated by Pirt (49), who assumed decreasing permease levels with increasing growth rate and a constant maximal growth rate. This resulted in an inverse relationship between the enzyme content of a cell and K_m, which translates to increased specific affinity even though the affinity was described as a property of the permease. Where the number of porins is small, the collision frequency at their location in the outer membrane helps set the specific affinity according to their number and pore size (40), and special diffusion properties may be involved as the molecules are swept in like a “string of pearls” (35). Some workers believe that substrate accumulation is enhanced over rates predicted by collision frequency theory due to improved probability of succeeding collisions with a cell following the first (4). Others disagree but note that there may be some enhancement due to two-dimensional diffusion (2), where substrate moves around, bound by surface effects, improving the probability of locating an active site. This issue remains unresolved.

Multiple Enzymes in Parallel

Reaction rates of a common substrate acted upon by two or more enzymes separate the kinetics of substrate uptake from those described for most enzymatic processes. The kinetics of microbial enzyme systems operating in parallel on a common substrate have been tabulated (49) and offered as an explanation of a type of nonhyperbolic kinetics that is often produced. The concept is that Michaelis constants for cells are a reflection of the substrate uptake enzymes in combination. These equations may be written in the form

21

where the subscripts 1 and 2 represent specific affinities and maximal velocities for each of two enzymes. Plots of equation 21 curve more gently than those of equation 13 (6) and give one reason for the concave-up affinity plots discussed below.

Multiple Enzymes in Sequence

For enzymatic pathways, the concept of a flux-controlling reaction (45) was modified by the hypothesis that flux control is shared by all the enzymes along the pathway, even though some may act more slowly than others (29). This may be rationalized by considering that intermediates preceding a slow step increase the thermodynamic driving force, thereby increasing the rate of the slow step. Equations for nutrient flux through two enzyme types in a cell that are separated by a metabolic pool, such as those steps responsible for k₃ and k_n in Fig. 1, have been explicitly solved to give net flux in terms of the kinetic constants of the separate enzymes (8). A key assumption was that intermediate or “pool” concentrations are directly linked to nutrient flux, as specified by the Onsager reciprocity or “minimum entropy production” relationship (46) that is consistent with increasing metabolic pool size with increasing flux (52). Findings from this model were than enzymatic steps so linked produce hyperbolic curves if the kinetics for the individual steps are also hyperbolic, but that the kinetics for the overall process retain kinetic properties from each step. Thus, the specific affinity as well as V_max and μ_max for an organism with a given number of permeases or initial enzymes with particular kinetic characteristics can be reduced through obligate coupling with a second enzyme that is present in small amounts or has a large Michaelis constant. Metabolic control analysis, reviewed by Quant (50), carries this global control a step further in having the control of flux through a pathway distributed among fluxes along all the connected pathways.

Kinetic data giving concave-down affinity plots result from equations (not shown) that are consistent with a sequential rate-controlling pathway derived by assuming that substrate is collected from a pool such as the periplasmic space that is filled at a diffusion-limited rate and removed from it by a sequence of saturatable steps and equating the two processes or by assuming that the residence time of the substrate in the pathway, τ, increases with the concentration of substrate. Such kinetics have been generated by controlling the diffusion of substrate to the periplasmic pool by regulating the number of porins (40).

Concave-up kinetic plots can be fit with equations consistent with sequential processes by assuming that V_max increases with substrate concentration, by assuming overflow metabolism with increasing amounts of leakage at high concentrations (32), or by assuming that τ decreases with substrate concentration as might be envisioned if the overall driving force or free-energy drop along the pathway increases as the energy available for transport increases with the increased substrate supply. Substituting V_a + K_va S for V_max into equation 13 gives this type of curve, where V_a is an apparent maximal velocity and k_va is a dimensionless pseudo-rate constant that gives the increase in V_max with substrate concentration. Alternatively, the affinity expressed by equation 20 may be modified to give an incremental increase in τ with the concentration of substrate which expresses the concept of increasing V_max without increasing the amount of limiting enzyme.

From equation 20, we may see that both the affinity constant and the Michaelis constant depend on the residence time of the substrate in the pathway. From Fig. 3B, we see that high-affinity organisms usually, but not always, have small affinity constants and Michaelis constants. This might be rationalized by considering that some organisms have balanced amounts of enzymes in each component along the substrate accumulation pathway, leading to short residence times. Others adapted to dilute environments may devote more material to the initial substrate collection part of the pathway, leading to restrictions further on and resulting in longer residence times for the pathway as a whole. Because less energy is required to construct and maintain unneeded metabolic enzymes, good oligotrophs are located in the uppermost part of the curve.

Thresholds

Energy is typically required for transport, as shown in Fig. 1. Unlike enzymatic catalysis, energy input is often in the form of a cotransported ion such as H⁺ and one that is potentially restrictive in dilute (energy-poor) systems. Endogenous energy requirements have been separated from those required for growth and measured by evaluation of yields at various rates of growth (5). Recently tabulated values of 0.016 to 0.021 g of cells (wet wt)/g of cells/h (49) are sufficient to consume most oligotrophs at growth rates specified by presumed substrate concentrations and specific affinities for substrates including those with large growth yields (62). However, the principle of decoupling material requirements for growth from those required for maintenance is parallel to decoupling energy requirements for maintaining an energized membrane from those required for transport, so the appropriate values for the calculation may be unknown. Organisms short on substrate may not have ions immediately available to help import substrate; i.e., they may have a deenergized membrane, causing a reduction in specific affinity and disproportionately low rates of substrate transport at low substrate concentrations. Such requirements should translate into either a threshold substrate concentration or a sigmoidal plot of v against S and affinity plots that are concave down at low rates. However despite theory favoring these sigmoidal curves, they are not seen in contemporary measurements, perhaps due to the experimental difficulties encountered in this area.

CONCLUSIONS AND FUTURE DIRECTIONS

Both theory and tabulations show that specific affinity is a good measure of the ability of the cell to collect substrate. It is thought to depend mainly on the amount of permease or initial enzyme in the metabolic sequence that is available for reaction with substrate. However, slow steps downstream may transmit restrictions. Maximal velocity is a capacity term quantifying flow at unrestrictive concentrations of substrate. It reflects both the amount of enzymes in the limiting sequence and the residence time of substrate in various components of the sequence. V_max, together with specific affinity, describes the dependency of the metabolic rate of a cell on the external concentration of substrate.

Small saturation constants are usually associated with larger affinities. This is taken as an indication of extensive material devoted to substrate collection through permeases or initial enzymes as compared with those further on. Organisms such as E. coli can have both high specific affinities and maximal velocities. However, this mechanism comes at a cost of much larger dry-weight-to-volume ratio than for some aquatic bacteria such as those isolated by extinction culture (15). Restrictive capacity downstream feeds back through product inhibition to slow the flow. The higher the ratio of permease to enzyme downstream, the smaller the saturation constants. The spread between values for the kinetic parameters K_A and K_m is taken as the result of the various forces controlling substrate flow through a long sequence of enzymatic steps and macromolecular rearrangements. K_A best indicates saturation of the uptake process at low concentrations, where the residence time in the pathway can be short even when restrictions occur further on, while K_m reflects concentrations sufficient to saturate the maximal rate of processing. Kinetic constants for organisms which are fragile are easily underestimated. However, the theory is sufficiently well developed to enable us to look for discrepancies which may help locate protocols in need of improvement. The theory can also be used to anticipate the effect of saturation on the rate constants that specify the affinity of organisms for substrate at any specific concentration of substrate. Values for kinetic constants reflect a wide range in the ability of organisms to collect substrate when the concentrations are low, and they indicate that few organisms examined are suitably equipped for life in aquatic environments. While discussions here are restricted to aquatic organisms, the theory applies to all cells sequestering substrate from aquatic environments.

A good understanding of aquatic organisms involves a number of requirements. Typical aquatic bacteria are difficult to grow in pure culture, and the number available from which conclusions may be drawn is extremely limited. Because their properties appear to be unusual compared with those commonly cultivated, a more robust selection would be useful. Second, little is known about the substrates that comprise the bulk of their sustenance. Moreover, with the possible exception of amino acids and hydrocarbons, the concentrations of these substrates are unknown. Since measured kinetics are strongly dependent on the conditions of measurement, it is important to eliminate this artifact from reported data. Therefore, kinetic data that agree with measured nutrient flux during undisturbed growth, both in the laboratory and in the environment, are needed. Comparisons require that measurements of nutrient flux are based on some property of biomass such as cell mass, because maximal velocities having units of reciprocal time alone and Michaelis constants that give concentrations associated with those rates are not easily interpreted. It appears that the ability of organisms to compete for substrates at low concentrations is reflected by the amounts and types of enzymes, usually permeases. At higher concentrations, the quantity of cytoplasmic enzymes used in nutrient processing becomes significant in order that maximal velocity might be increased, but at low concentrations, large amounts of these enzymes may become burdensome. That being the case, one of the important properties that can now be determined is the distribution of permease types and quantities, together with their activities and specificities. Such data link ambient substrate concentrations to organism physiology and specify members of the microbial community responsible for the regulation of specific solutes. This treatment argues that kinetic characteristics specify the ability of microorganisms to prevail through raw competition for nutrients. Additional undescribed characteristics may be equally important. These include structural adaptations that may retard attack by viruses and predators and the possibility that chemical communication among various species providing some control (22, 44) and stability (39) will exist as well. Only when the properties are characterized can we have a good understanding of how bacteria help regulate the chemistry and biology of the aquatic environments.