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Plots of turnover times versus added substrate concentrations provide only upper bounds to in situ substrate concentrations
J. theor. Biol. (1983) 101, 147-150
Plots of Turnover Times Versus Added Substrate Concentrations Provide Only Upper Bounds to in situ Substrate Concentrationst EDWARD
A. LAWS
University of Hawaii, Oceanography Department, Hawaii 96822, U.S.A. (Received 23 November
Honolulu,
1982)
In recent years it has become a common practice in the study of microbial activity in aquatic systems to estimate in situ substrate concentrations by plotting substrate turnover times (the substrate concentration divided by microbial uptake rate) against the concentrations of added radioactively labeled substrate. The rationale has been that the uptake rate will remain essentially constant if added concentrations of high specific activity substrates are small compared to in situ concentrations. Unfortunately, a correct mathematical analysis shows that the in situ concentrations estimated by this method are only upper bounds to the true in situ substrate concentrations, regardless of the size of the substrate spikes. The estimated in situ concentration will equal the true in situ concentration only if uptake rate is completely independent of substrate concentration, an unlikely situation in natural systems. Introduction In a publication several years ago, Dietz, Albright & Tuomineu (1977) described a “simple, sensitive, and versatile. . . approach to assay of heterotrophic microbial activities of natural waters”. Dietz et al. (1977) indicate that their technique, “allows one to determine in situ substrate concentrations (S,), turnover times (Z”,,), and velocities of utilization (V,) of any substrate that can be suitably radioactively labeled”. Dietz et al. (1977) note that their model, “is similar to the one described by Wright & Hobbie (1966), except that they added substrate in concentrations in excess of naturally occurring substrate, whereas we used substrate additions in concentrations less than or equal to that of the naturally occurring substrate.” The principal apparent theoretical advantage of the Dietz et al. (1977) approach over that of Wright & Hobbie (1966) is that, “the value of S,, can be computed and need not be experimentally determined” i Hawaii
Institute
of Marine
Biology
Contribution
No. 646
147
0022-5193/83/050147+04$03.00/0
@ 1983 Academic
Press Inc. (London)
Ltd.
148
E.
A.
LAWS
(Dietz et al. 1977). Wright & Burnison (1979) have criticized the Dietz et al. (1977) technique on the grounds that analytical limitations presently make it difficult to determine precisely the changes in turnover times caused by trace additions of substrate. As a result substrates are often added at concentrations which very likely cause significant changes in uptake rates, a condition which violates the assumptions of the Dietz ef al. (1977) model. The purpose of this communication is to point out that in cases where turnover times can be measured with good precision, the Dietz et al. (1977) technique will still not provide more than an upper bound to the value of S,, even when the substrate is added in only trace amounts. Theory
Assume that a radioactively labeled substrate having an activity Ad is added at concentration A to a sample of water in which the in siru substrate concentration is S,. If V is the uptake rate of the substrate and if there is negligible isotopic discrimination, then the uptake rate Va of the added substrate will be
v/F&.
(1)
”
If VA is constant over time, the net uptake of radioactivity t will be AdVAt ud=-.-.-.-=---A
AdVt S, +A’
Ud after time (2)
As defined by Dietz et al. (1977), the turnover time T of the total substrate is given by &+A T=V=z
Adt
(3)
Based on equation (3) Dietz et al. (1977) incorrectly conclude that a linear plot of T on the ordinate versus A on the abscissa, “intersects. . . the abscissa on the negative side at the in situ substrate concentration, S,“. This conclusion is clearly incorrect if V = 0 at A = 0, i.e. if the uptake rate at the in situ substrate concentration S, is zero. In that case T + COas A + 0. This condition is not unrealistic. Caper-on & Meyer (1972) for example have reported that the uptake of inorganic nitrogen by certain marine phytoplankton approaches zero at inorganic nitrogen concentrations of 50-70 nM, and Brown & Button (1979) have shown that growth of a PO:limited culture of Selenastrum capricornutum ceases at a PO:- concentra-
ERRORS
IN
ESTIMATING
SUBSTRATE
LEVELS
149
tion of about 10 nM. If V # 0 at A = 0, it is reasonable to assume that V is continuous and has a continuous first derivative at A = 0, in which case the derivative of equation (3) with respect to A at A = 0 is well defined and is given by the equation =-- 1 &(dV/dA)o vo Vi
(4)
where the subscripts on dT/dA, dV/dA and V indicate that the terms are to be evaluated at A = 0. Equation (4) is of course the slope of the tangent to the curve of T versus A at A = 0. The intersection of the tangent line with the abscissa occurs at a point C-Xgiven by the expression
Since it is reasonable to assume that (dV/dA)o is positive, equation (5) indicates that the magnitude of (Y will exceed S,. Dietz et al. (1977) concluded from equation (3) that, “if velocity of utilization [V] is constant, since background amounts of substrate were added, then the turnover time should increase in proportion to the total amount of substrate, (A +S,)“. Although it is undoubtedly true that the change in V can be kept arbitrarily small by keeping A sufficiently small, approximate constancy of V by no means implies that (dV/dA)o is small relative to Vo/S,. It is obvious from equation (5) that if (dV/dA), = Vo/S,, then the magnitude of (Y will seriously overestimate S,. For example, if the relationship between V and substrate concentration follows Michaelis-Menten kinetics, it is straightforward to show from equation (5) that CY= -(S, +K), where K is the Michaelis-Menten half-saturation constant. Furthermore, if V = 0 at some nonzero substrate concentration less than S,, it is quite possible that (dV/dA j. will exceed Vo/S,. In that case Q will be positive rather than negative, and will provide no information regarding the magnitude of S,. It is noteworthy that a theory mathematically equivalent to that developed by Dietz et al. (1977) had earlier been developed by Forsdyke (1968). Using the terminology of Forsdyke (1968), we have
where x are the observed counts appearing in a macromolecule, n is the maximum count that would be incorporated with no dilution, and y, p and u are dilution factors due to added cold substrate, endogeneous substrate
150
E.
A.
LAWS
and de nova synthesis respectively. Note that x and n in equation (6) are analgous to T-l and V respectively in equation (3). If n were indeed independent of y, a plot of x-i versus y + 1 would be linear and could be used to estimate p + CL However, if n depends on y, one encounters the same problem in interpreting the intercept (i.e. value of y + 1 at which x -* equals zero) as was the case in the Dietz et al. (1977) formulation. This conclusion is true even when added substrate concentrations are small (i.e. y + 1~ p + u), because one must extrapolate the graph to a point at which y + 1 has the same magnitude as p + u in order to apply the method. Summary aud Conclusions
The theories of Dietz et al. (1977) and Forsdyke (1968) are logically appealing at first glance, but a careful mathematical analysis reveals that the validity of both theories rests on the assumption that V is completely independent of A. Unfortunately this condition is probably violated in many applications in which the theories have been used. As noted by Wright & Burnison (1979), “it is not proper to assume that uptake velocity Lan be constant, unless the microbial transport systems are saturated with substrate, and unlikely situation when working at natural substrate levels”. It is true that V is approximately constant if A is sufficiently small, but the fact that V is approximately constant by no means implies that (d V/dA Jo is small relative to V,-JS,. We conclude that the theories of Dietz et al. (1972) and Forsdyke (1968) are based on faulty mathematics, and can provide no better than upper bounds to in situ substrate concentrations, even when substrates are added in trace amounts. REFERENCES BROWN, E. J. & BU~ON, D. K. (1979) J. Phycol. 15 305. CAPERON, J. & MEYER, J (1972). Deep-Sea Res. 19,601. DIETZ, A. S., ALBRIGHT, L. J. & TUOMINEU, T. (1977). Appl. environ. Microbial. 33, 817. FORSDYKE, D. R. (1968). Biochem. J. 107,197. WRIGHT, R. T. & BURNISON, B. K. (1979). In: Nutioe Aquatic Bacteria: Enumeration, Activity and Ecology. Costerton, J. W. & Colwell, R. R. eds. Washington D.C.: ASTM. Publication 695 p. 140. WRIGHT, R. T. & HOBBIE J. E. 1966 Ecology 47,447
Plots of Turnover Times Versus Added Substrate Concentrations Provide Only Upper Bounds to in situ Substrate Concentrationst EDWARD
A. LAWS
University of Hawaii, Oceanography Department, Hawaii 96822, U.S.A. (Received 23 November
Honolulu,
1982)
In recent years it has become a common practice in the study of microbial activity in aquatic systems to estimate in situ substrate concentrations by plotting substrate turnover times (the substrate concentration divided by microbial uptake rate) against the concentrations of added radioactively labeled substrate. The rationale has been that the uptake rate will remain essentially constant if added concentrations of high specific activity substrates are small compared to in situ concentrations. Unfortunately, a correct mathematical analysis shows that the in situ concentrations estimated by this method are only upper bounds to the true in situ substrate concentrations, regardless of the size of the substrate spikes. The estimated in situ concentration will equal the true in situ concentration only if uptake rate is completely independent of substrate concentration, an unlikely situation in natural systems. Introduction In a publication several years ago, Dietz, Albright & Tuomineu (1977) described a “simple, sensitive, and versatile. . . approach to assay of heterotrophic microbial activities of natural waters”. Dietz et al. (1977) indicate that their technique, “allows one to determine in situ substrate concentrations (S,), turnover times (Z”,,), and velocities of utilization (V,) of any substrate that can be suitably radioactively labeled”. Dietz et al. (1977) note that their model, “is similar to the one described by Wright & Hobbie (1966), except that they added substrate in concentrations in excess of naturally occurring substrate, whereas we used substrate additions in concentrations less than or equal to that of the naturally occurring substrate.” The principal apparent theoretical advantage of the Dietz et al. (1977) approach over that of Wright & Hobbie (1966) is that, “the value of S,, can be computed and need not be experimentally determined” i Hawaii
Institute
of Marine
Biology
Contribution
No. 646
147
0022-5193/83/050147+04$03.00/0
@ 1983 Academic
Press Inc. (London)
Ltd.
148
E.
A.
LAWS
(Dietz et al. 1977). Wright & Burnison (1979) have criticized the Dietz et al. (1977) technique on the grounds that analytical limitations presently make it difficult to determine precisely the changes in turnover times caused by trace additions of substrate. As a result substrates are often added at concentrations which very likely cause significant changes in uptake rates, a condition which violates the assumptions of the Dietz ef al. (1977) model. The purpose of this communication is to point out that in cases where turnover times can be measured with good precision, the Dietz et al. (1977) technique will still not provide more than an upper bound to the value of S,, even when the substrate is added in only trace amounts. Theory
Assume that a radioactively labeled substrate having an activity Ad is added at concentration A to a sample of water in which the in siru substrate concentration is S,. If V is the uptake rate of the substrate and if there is negligible isotopic discrimination, then the uptake rate Va of the added substrate will be
v/F&.
(1)
”
If VA is constant over time, the net uptake of radioactivity t will be AdVAt ud=-.-.-.-=---A
AdVt S, +A’
Ud after time (2)
As defined by Dietz et al. (1977), the turnover time T of the total substrate is given by &+A T=V=z
Adt
(3)
Based on equation (3) Dietz et al. (1977) incorrectly conclude that a linear plot of T on the ordinate versus A on the abscissa, “intersects. . . the abscissa on the negative side at the in situ substrate concentration, S,“. This conclusion is clearly incorrect if V = 0 at A = 0, i.e. if the uptake rate at the in situ substrate concentration S, is zero. In that case T + COas A + 0. This condition is not unrealistic. Caper-on & Meyer (1972) for example have reported that the uptake of inorganic nitrogen by certain marine phytoplankton approaches zero at inorganic nitrogen concentrations of 50-70 nM, and Brown & Button (1979) have shown that growth of a PO:limited culture of Selenastrum capricornutum ceases at a PO:- concentra-
ERRORS
IN
ESTIMATING
SUBSTRATE
LEVELS
149
tion of about 10 nM. If V # 0 at A = 0, it is reasonable to assume that V is continuous and has a continuous first derivative at A = 0, in which case the derivative of equation (3) with respect to A at A = 0 is well defined and is given by the equation =-- 1 &(dV/dA)o vo Vi
(4)
where the subscripts on dT/dA, dV/dA and V indicate that the terms are to be evaluated at A = 0. Equation (4) is of course the slope of the tangent to the curve of T versus A at A = 0. The intersection of the tangent line with the abscissa occurs at a point C-Xgiven by the expression
Since it is reasonable to assume that (dV/dA)o is positive, equation (5) indicates that the magnitude of (Y will exceed S,. Dietz et al. (1977) concluded from equation (3) that, “if velocity of utilization [V] is constant, since background amounts of substrate were added, then the turnover time should increase in proportion to the total amount of substrate, (A +S,)“. Although it is undoubtedly true that the change in V can be kept arbitrarily small by keeping A sufficiently small, approximate constancy of V by no means implies that (dV/dA)o is small relative to Vo/S,. It is obvious from equation (5) that if (dV/dA), = Vo/S,, then the magnitude of (Y will seriously overestimate S,. For example, if the relationship between V and substrate concentration follows Michaelis-Menten kinetics, it is straightforward to show from equation (5) that CY= -(S, +K), where K is the Michaelis-Menten half-saturation constant. Furthermore, if V = 0 at some nonzero substrate concentration less than S,, it is quite possible that (dV/dA j. will exceed Vo/S,. In that case Q will be positive rather than negative, and will provide no information regarding the magnitude of S,. It is noteworthy that a theory mathematically equivalent to that developed by Dietz et al. (1977) had earlier been developed by Forsdyke (1968). Using the terminology of Forsdyke (1968), we have
where x are the observed counts appearing in a macromolecule, n is the maximum count that would be incorporated with no dilution, and y, p and u are dilution factors due to added cold substrate, endogeneous substrate
150
E.
A.
LAWS
and de nova synthesis respectively. Note that x and n in equation (6) are analgous to T-l and V respectively in equation (3). If n were indeed independent of y, a plot of x-i versus y + 1 would be linear and could be used to estimate p + CL However, if n depends on y, one encounters the same problem in interpreting the intercept (i.e. value of y + 1 at which x -* equals zero) as was the case in the Dietz et al. (1977) formulation. This conclusion is true even when added substrate concentrations are small (i.e. y + 1~ p + u), because one must extrapolate the graph to a point at which y + 1 has the same magnitude as p + u in order to apply the method. Summary aud Conclusions
The theories of Dietz et al. (1977) and Forsdyke (1968) are logically appealing at first glance, but a careful mathematical analysis reveals that the validity of both theories rests on the assumption that V is completely independent of A. Unfortunately this condition is probably violated in many applications in which the theories have been used. As noted by Wright & Burnison (1979), “it is not proper to assume that uptake velocity Lan be constant, unless the microbial transport systems are saturated with substrate, and unlikely situation when working at natural substrate levels”. It is true that V is approximately constant if A is sufficiently small, but the fact that V is approximately constant by no means implies that (d V/dA Jo is small relative to V,-JS,. We conclude that the theories of Dietz et al. (1972) and Forsdyke (1968) are based on faulty mathematics, and can provide no better than upper bounds to in situ substrate concentrations, even when substrates are added in trace amounts. REFERENCES BROWN, E. J. & BU~ON, D. K. (1979) J. Phycol. 15 305. CAPERON, J. & MEYER, J (1972). Deep-Sea Res. 19,601. DIETZ, A. S., ALBRIGHT, L. J. & TUOMINEU, T. (1977). Appl. environ. Microbial. 33, 817. FORSDYKE, D. R. (1968). Biochem. J. 107,197. WRIGHT, R. T. & BURNISON, B. K. (1979). In: Nutioe Aquatic Bacteria: Enumeration, Activity and Ecology. Costerton, J. W. & Colwell, R. R. eds. Washington D.C.: ASTM. Publication 695 p. 140. WRIGHT, R. T. & HOBBIE J. E. 1966 Ecology 47,447