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Improved estimates of phytoplankton carbon based on 14C incorporation into chlorophyll a
£ theor BioL (1984) 110, 425-434
Improved Estimates of Phytoplankton Carbon Based on ~4C Incorporation into Chlorophyll a EDWARD A. LAWS
Department of Oceanography and Hawaii Institute of Marine Biology, 1000 Pope Road, Honolulu, Hawaii 96822, U.S.A. (Received 18 November 1983, and in revised form 3 April 1984) A theoretical model of phytoplankton production, zooplankton grazing, and bacterial utilization of dissolved organic carbon (DOC) shows that both upper and lower bounds to the concentration of phytoplankton carbon can be obtained from primary production measurements and the rate of incorporatien of t4C into chlorophyll a carbon. An expression is derived to account for the error created by zooplankton grazing and bacterial production on both the upper and lower bound estimates of phytoplankton carbon. Application of this error correction to field data suggests that highly accurate estimates of phytoplankton carbon may be possible in cases where phytoplankton production is approximately balanced by zooplankton grazing and bacterial uptake of excreted DOC. Introduction
Several years ago Redalje & Laws ( 1981 ) introduced a method for estimating phytoplankton growth rates and carbon biomass based on the incorporation of t4C into chlorophyll a carbon. The technique makes use o f the fact that after incubations o f a few hours, the specific activities of the phytoplankton carbon and the chlorophyll a carbon are identical. Redalje (1983) for example has shown that in a culture o f Mantoniella sp. grown on a 12 h: 12 h light:dark cycle these specific activities are equal after an incubation of only two hours. One important advantage o f the Redalje & Laws (1981) technique for calculating growth rates is that the calculated growth rate is unaffected by zooplankton grazing and bacterial uptake of excreted dissolved organic carbon (DOC). This conclusion follows from the fact that the specific activity of the chlorophyll a carbon is not altered by zooplankton grazing or bacterial production. When growth rates are calculated from the rate o f incorporation of ~4C into total particulate material, zooplankton grazing and bacterial production will affect the calculated growth rate, because some of the organic ~4C activity appears in the zooplankton and bacteria. Jackson (1983) has provided a thoughtful discussion of this problem with respect to zooplankton grazing. 425
0022-5193/84/190425 + l 0 $03.00/0
© 1984 Academic Press | nc. (London) Ltd.
426
E.A. LAWS
The method proposed by Redalje & Laws (1981) for estimating the concentration of phytoplankton carbon consists of dividing the particulate t4C activity by the specific activity of the chlorophyll a carbon. Unfortunately the value of the phytoplankton carbon concentration calculated by this method is affected by zooplankton grazing and bacterial production. The concentration of phytoplankton carbon will be overestimated to the extent that ~4C activity appears in particles other than phytoplankton cells. This overestimation would become serious if phytoplankton production is roughly balanced by grazing and excretion, and if the duration of the incubation is more than one doubling time. The magnitude of the overestimation will of course depend on the efficiency with which labeled organic carbon is retained in particles other than phytoplankton cells (Jackson, 1983). The purpose of this communication is to suggest an alternative way to calculate phytoplankton carbon biomass. It will be shown that the effects of zooplankton grazing and bacterial production are such as to make the concentration calculated by the alternative method a lower bound on the phytoplankton carbon concentration, whereas the concentration calculated by the Redalje & Laws (1983) method is an upper bound. An approximate equation will be derived to account for the error created by zooplankton grazing and bacterial production.
Theory A list of symbols used in the following theoretical development is provided in the nomenclature. Let Cp(t) be the concentration of phytoplankton carbon at time t. Assume that d
dt Cp(t) = [/z(t) - g(t)]Cp(t)
(1)
where/z(t) is the net rate of carbon fixation per unit phytoplankton carbon and g(t) is the rate at which phytoplankton carbon is transferred to other particles per unit phytoplankton carbon. Although in the past one might have assumed that g(t) describes the effect of zooplankton grazing, recent studies have made it clear that significant amounts of fixed carbon may be transferred from phytoplankton to bacteria via the dissolved organic carbon (DOC) pool, either as the direct result of phytoplankton excretion, or as the result of the feeding activities of zooplankton (Sieburth et al., 1977; Lampert, 1978; Williams, 1981). Thus g(t) must be viewed as a general transfer function expressing the combined effects of zooplankton grazing and bacterial uptake of excreted DOC. /z(t) and g(t) are not assumed to
PHYTOPLANKTON
CARBON
ESTIMATES
427
have any particular functional dependence, and equation (1) is therefore completely general. The solution to equation (1) is
C.(t) = C Oexp [ M ( t ) - G(t)]
(2)
where
M(t) = G(t) =
fo ;o
/z(x) dx
(3)
g(x) dx
(4)
and cO is the phytoplankton carbon concentration at time t = 0. Now let A* be the ~4C activity in the phytoplankton at time t, let A* be the activity of t4C in the inorganic carbon, and let I be the concentration of inorganic carbon. Both A* and I are assumed to be constant. Then
dA*(t)
I~(t)Cp(t)A*o fI
g(t)A*(t)
(5)
where f is the isotope discrimination factor ( f ~ 1.05). The solution of equation (5) is ,
0
A*(t) = Ao Cp exp [-G(t)]{exp [ M ( t ) ] - 1}.
fI
(6)
From equations (2) and (6) one can see that the specific activity of the phytoplankton carbon, which I will call a*(t), is given by the equation
a*(t) = A*(t) =A*o Il - e-M(t)]. Cp(t) f I L
(7)
Redalje & Laws' (198 l) method for estimating phytoplankton growth rates amounts to solving equation (7) for M(t), and then setting the growth rate equal to M(t)/t. It is obvious from equation 3 that the Redalje & Laws (1981) growth rate, which I will call /~(t), is equal to
!
~(t) = t
Io
/z(x) dx.
(8)
Therefore the Redalje & Laws (1981) growth rate is in fact the time average of the phytoplankton growth rate over the course of the incubation. Now for the moment assume that ~4C remains in the particulate phase once it has been converted into phytoplankton carbon. In other words, assume that respiration and excretion losses associated with zooplanktonic
428
E.A. LAWS
and bacterial metabolism are negligible. Let A*(t) be the activity of particulate organic t4C at time t. Then dA*(t)= dt
A~(t)Cp(t) fI
(9)
The solution of equation (9) is
A*(t)
A0~t rI'
dx.
f I .Jo
(lO)
Now i f / z ( x ) and Cp(x) are uncorrelated, then
Io [/z (x) -/2(t)]ECp(x) - Cp(t)] dx = 0
(1 1)
where Cp(t) is the time average of the phytoplankton carbon concentration over the course of the incubation, and is equal to
Io
(12)
Io Ix(x)Cp(x) dx = t/2( t)Cp( - t)
(13)
--~p(t) _ 1
-t
Cp(x) dx.
If equation (11) is satisfied, then
and
A*(t) - A*t/2(t)Cp(t)
(14)
fi Now the average rate/5(t) of photosynthesis over the course of the incubation is
P(t) = A*(t)f[ A*ot
(15)
From equations (14) and (15) it follows that
Cp( t) = P( t)/ /2( t).
(16)
Thus the average phytoplankton carbon concentration over the course of the incubation can be estimated by dividing the photosynthetic rate /5(t) by the growth rate/2(t). The accuracy of this calculation of course depends on the extent to which equation (11) is satisfied. Is there any reason to expect that Cp(t) and tz(t) are uncorrelated? Certainly over time intervals of one day or less there is good reason to think that tz(t) and Cp(t) are uncorrelated. This conclusion follows from
PHYTOPLANKTON CARBON ESTIMATES
429
the fact that the fractional rate of change of phytoplankton carbon at time
t is primarily a function of environmental conditions at time t, but the phytoplankton carbon concentration at time t is clearly a function of antecedent environmental conditions. During the photoperiod one would expect Cp(t) to be positively correlated with time. However, be(t) would be expected to show a positive correlation with time during the morning and a negative correlation with time during the afternoon, in response to the natural variation in irradiance. Thus on average be(t) and Cp(t) would be very poorly correlated during the photoperiod. At night one would expect Cp(t) to be negatively correlated with time. However, be(t) would not be expected to show any systematic correlation with time in the dark. Thus the left-hand side of equation (11) should be very nearly zero over the course of both the day and night. It is instructive at this time to examine the effect of zooplankton grazing and bacterial uptake of labeled DOC on the calculated phytoplankton carbon biomass. At time t the particulate ~4C activity not confined to phytoplankton cells equals A*(t)-A*(t) if none of the ~4C consumed by zooplankton and bacteria is respired or excreted. Assume that a fraction r of the t4C consumed by the zooplankton and bacteria is in fact lost to respiration or excretion. The calculated value of 15(0 would therefore be reduced by a fractional amount r[A*(t)-A*(t)]/A*(t)= r[l-A*(t)/A*(t)]. Let /3=A*(t)/A*(t), and let fi(t) and Cp(t) be the calculated mean photosynthetic rate and phytoplankton carbon concentration, respectively, over the course of the incubation. Then / 5 ( 0 = P ( t ) [ l - r ( l - f l ) ] , and from equation (16) A
Cp(t) = Cp(t)[1
- r(1 -/3)].
(17)
Note that zooplankton grazing and bacterial production have no effect on the calculated value o f / 2 ( 0 , since zooplankton grazing and bacterial production have no effect on the chlorophyll a carbon specific activity, which is equal to ot*(t) (equation 7) after an incubation of a few hours. Redalje & Laws' (1981) estimate of phytoplankton carbon, which I will call t~p(t), is equal to the particulate 14C activity at time t divided by a* (t). From equation (7) it is obvious that Cp(t)= A*( t)/ a*( t), and therefore Cp(t) will overestimate Cp(t) to the extent that a*(t)-r[a*(t)-A*(t)] overestimates A*(t). It is straightforward to show that Cp(t) and Cp(t) are related by the equation
Note that the error caused by zooplankton grazing and bacterial production
430
E.A. LAWS
is linearly related to r in the case of both C?p(t) and C?p(t), but that the magnitude of the error is positively correlated with r in the former case (equation (17)) and negatively correlated with r in the latter case (equation (18)). The error in Cp(t) is zero when r = 0, because the calculation of Cp(t) assumes that all carbon fixed by phytoplankton is retained in the particulate phase. The error in ~?p(t) is zero when r = 1, because the calculation of Cp(t) assumes that all particulate ~4C activity is confined to the phytoplankton. Thus zooplankton grazing and bacterial production create no error in the calculation of Cp(t) as long as the zooplankton and bacteria respire or excrete all the 14C they consume. The magnitude of the error in both Cp(t) and Cp(t) is negatively correlated with/3, and approaches a maximum value as/3 ~ 0. However, the magnitude of the fractional error in Cp(t) when/3 = 0 is equal to r, whereas the fractional error in Cp(5) approaches infinity as/3 -~ 0. The latter result occurs because the condition /3 = 0 implies that all the phytoplankton carbon has been transferred to zooplankton and bacteria, and therefore the true phytoplankton concentration at time t is zero. However, the fact that the phytoplankton concentration at the end of the incubation is zero obviously does not imply that the average phytoplankton concentration over the course of the incubation [Cp(t)] has been zero. Hence the fractional error in Cp(t) remains finite even when/3 ~ 0. It is now useful to ask whether Cp(t) or Cp(t) is a more accurate estimator of phytoplankton carbon biomass. From equations (17) and (18) it is obvious that the magnitude of the fractional errors in Cp(t) and Cp(t) are equal if r(l -/3) = (1 - r ) ( ~ - - 1).
(19)
Equation (19) implies that
r=(l +/3)-'.
(20)
If r is less than or greater than (1 +/3)-', the magnitude of the fractional error in Cp(t) will be respectively less than or greater than the fractional error in Cp(t). It is obvious from equations (5) and (9) that A*(t) will equal A*(t) only if g(t) = 0. Otherwise A*(t) < A*(t). Hence the maximum value of/3 is 1, and the minimum value of (1 +/3) -1 is 0.5. Thus if r < 0 . 5 , the magnitude of the fractional error in Cp(t) will be less than the fractional error in Cp(t). However, most studies of the ecological growth efficiencies of bacteria and herbivorous zooplankton suggest that r > 0.5. Data summarized by C o m e r & Davies (1971) imply a zooplanktonic growth efficiency of about 23%. Williams' (1981) model of planktonic food webs assumes a growth efficiency of 30%. Jackson (1983), based on studies by Checkley
PHYTOPLANKTON
CARBON
ESTIMATES
431
(1980), assumed a growth efficiency of 37%. It therefore seems reasonable to assume that only about 30% of captured food is converted into new tissue by herbivorous zooplankton. However, a fraction of the excreted carbon will be released in particulate form, and will therefore contribute to the particulate 14C activity. Experiments conducted by Conover (1966) and Lampert (1978) suggest that 10-15% of captured carbon is released as DOC; the summary of Corner & Davies (1971) implies that 20-25% of captured carbon is released as DOC or particulate organic carbon (POC). I conclude that about 10% of captured carbon is released as POC. It therefore seems reasonable to assume that about 40% of the phytoplankton carbon captured by zooplankton is retained in the particulate phase, and hence for zooplankton grazing r---0-6. It is apparent from a great many studies that the growth efficiencies of bacteria depend very much on the carbon substrate. When bacteria are grown on small organic molecules such as simple sugars and amino acids, growth efficiencies are in the range 50-95% (Hobbie & Crawford, 1969; Williams, 1970; Payne, 1970; Crawford, Hobbie & Webb, 197d; Williams & Yentsch, 1976). However, when natural phytoplankton excretion products are provided as the source of carbon, growth efficiencies are in the range 35% (Herbland, 1975) to 45% (Sorokin, 197 l). It therefore seems reasonable to set r = 0.6 for both bacterial production and zooplankton grazing, and I will use this value of r in estimating the errors in Cp(t) and Cp(t). The value of/3 will of course depend on the nature of both /z(t) and g(t). Let us make the simplifying assumption that / z ( t ) = g ( t ) , i.e. that grazing and excretion losses exactly balance phytoplankton production. This assumption is clearly an approximation to the real world situation, but should not lead to seriously misleading conclusions in systems where M ( t ) = G(t), i.e. where the average rate of production is approximately equal to the average rate of consumption over the course of the incubation. If/~(t) = g(t), then Cp(t) is constant, and it is straightforward to show from equations (6) and (10) that
A*(t) 1 - e -MC') /3=A*(t)- M(t)
(21)
If the duration of the incubation is exactly one doubling time, then M(t) = In(2), and/3 = 0.72. Taking r = 0.6, I conclude from equations (17) and (18) that Cp(t) will underestimate Cp(t) by about 17%, and that Cp(t) will overestimate Cp(t) by 16%. If the duration of the incubation is half a doubling time, the corresponding errors in Cp(t) and t~p(t) are estimated to be 9% and 7%, respectively. These estimates of the errors created by zooplankton grazing and bacterial production suggest that estimates of
432
E.A. LAWS
phytoplankton carbon should be based on incubations lasting no more than approximately one doubling time. Under these conditions the errors in Cp(t) and Cp(t) should be no more than about 15-20%. If the error created by zooplankton grazing and bacterial production is no more than about 15-20%, it seems reasonable to make an approximate correction for zooplankton grazing and bacterial production using equations (17) and (18), with r assigned a value of 0.6 and /3 calculated from equation (21). The following example illustrates the application of the method. Results and Discussion
Table 1 tabulates estimated phytoplankton carbon concentrations [Cp(t)] and M ( t ) values based on 24 h incubations in the Southern California Bight as reported by Redalje (1983). I have calculated values of Cp(t) by simply dividing P(t) by /2(t), where /5(t) is the estimated daily photosynthetic rate a n d / 2 ( t ) is the average daily growth rate. Since t = 1 day in this case, /2 (t) is numerically equal to M(t). Note that since M ( t ) < In (2) in all cases, the estimated errors in Cp(t) and Cp(t) due to zooplankton grazing and bacterial production are expected to be less than 15-20% in all cases. The last two columns of Table 1 show the calculated values o f ~ p ( t ) and Cp(t) when a correction for zooplankton grazing is applied to Cp(t) and Cp(t), respectively. The remarkable aspect of these last two columns of Table l is that the calculated values of Cp(t) and Cp(t) differ by no more than one unit in the second significant figure. In fact there is no a priori reason why (~p(t) and Cp(t) should be identical, since the average value of the phytoplankton carbon concentration over the course of the incubation [(~p(t)] should not necessarily equal the phytoplankton carbon concentration at the end of the TABLE 1
Calculated values of Cp( t) and Cp( t) after incubations of 24 hours based on experiments reported by Redalje (1983) in the Southern California Bight. Values of Cp(t) and Cp(t) have been calculated using equations (17) and (18), with r = 0 . 6 and fl calculated from equation (21) Depth of incubation
M(t)
Cp(t) (~g/I)
Cp(t)
(~g/l)
Cp(t) (~g/l)
Cp(t)
(m)
(~g/l)
(~g/1)
6 II 21
0-506 0-463 0-312
58 50 25
74 62 29
67 57 27
67 56 27
PHYTOPLANKTON
CARBON
433
ESTIMATES
incubation [Cp(t)]. However, Redalje's (1983) chl a values were remarkably constant over the course of the entire 24 h incubations at each of the depths sampled. The mean (+1 standard deviation) chl a concentrations at 6 m, I1 m and 21 m were 1.34+0.13, 1.29±0.09 and 0.95 ± 0.10 txg/l, respectively. These figures are based on samples taken at the beginning of the incubation, and at 3, 6, 12 and 24 h after the start of the incubations. The constancy of the chl a concentrations implies that the phytoplankton biomass was in fact relatively constant, and hence the fact that Cp(t) and Cp(t) are nearly equal is not surprising. The apparent constancy of the phytoplankton biomass supports the assumption that I~(t)~-g(t), an assumption which was of course used to derive a simplified expression for the error created by zooplankton grazing and bacterial production. These results suggest that highly accurate estimates of [?hytoplankton carbon may be possible through calculation of Cp(t) and Cp(t), and a correction for zooplankton grazing and bacterial production effects using equations (17) and (18), respectively, with r = 0 . 6 and/3 determined from equation (21). Nomenclature
Symbol Definition A*(t) 14C activity in the phytoplankton at time t ~4C activity in the inorganic carbon A* A*(t) Particulate organic ~4C activity at time t in the theoretical absence of particulate ~4C losses due to respiration and excretion by zooplankton and bacteria a*p(t) Specific activity of the phytoplankton carbon at time t
A*( t)/ A*( t)
cat) I
co cat) p(t) Cp(t)
Phytoplankton carbon concentration at time t Phytoplankton carbon concentration at time t = 0 Mean phytoplankton carbon concentration over the course of the incubation = (1 / t) S'o Cp (x) dx Phytoplankton carbon concentration at time t estimated by Redalje & Laws (1981) technique = Cp(t)[1 +(1 - r)(1/fl - 1)] Calculated mean phytoplankton carbon concentration over the course of the incubation in the presence of particulate ~4C losses to respiration and excretion by zooplankton and bacteria=
P( t)/ l~( t)
Units Ci/l Ci/l Ci/l
Ci/g Dimensionless g/l g/1 g/l g/1
g/l
434 f g(t)
G(t) I
/5(0 /5(t)
~,(t) M(t) 12( t)
E. A. LAWS 14C i s o t o p e d i s c r i m i n a t i o n f a c t o r Dimensionless R a t e at w h i c h p h y t o p l a n k t o n c a r b o n is t r a n s f e r r e d d -j to o t h e r p a r t i c l e s p e r unit p h y t o p l a n k t o n c a r b o n at t i m e t t So g ( x ) d x Dimensionless Concentration of inorganic carbon g/l A v e r a g e rate o f p h o t o s y n t h e s i s o v e r the c o u r s e o f g/l/d t h e i n c u b a t i o n = A * ( t ) f I / ( A * t) C a l c u l a t e d a v e r a g e rate o f p h o t o s y n t h e s i s o v e r t h e g/l/h c o u r s e o f t h e i n c u b a t i o n in the p r e s e n c e o f p a r t i c u late 14C losses to r e s p i r a t i o n a n d e x c r e t i o n b y z o o p l a n k t o n a n d b a c t e r i a = / 5 ( t ) [ 1 - r( 1 - fl)] F r a c t i o n o f 14C c o n s u m e d b y z o o p l a n k t o n a n d D i m e n s i o n l e s s b a c t e r i a t h a t is lost to r e s p i r a t i o n a n d e x c r e t i o n N e t rate o f c a r b o n fixation p e r unit p h y t o p l a n k t o n dc a r b o n at t i m e t S'o/z(x) d x Dimensionless A v e r a g e p h y t o p l a n k t o n g r o w t h rate o v e r t h e c o u r s e d- t o f the i n c u b a t i o n = M ( t ) / t
Hawaii Institute of Marine Biology Contribution No. 686.
REFERENCES
CHECKLEY, D. M. (1980). Limnol. Oceanogr. 25, 430. CONOVER, R. J. (1966). In: Some Contemporary Studies in Marine Science, pp. 187-194 (Barnes, H., ed). London, Allen & Unwin. CORNER, E. D. S. & DAVIES, A. G. (1971). Adv. mar. Biol. 9, 101. CRAWFORD, C. C., HOBBLE,J. E. & WEBB, K. L. (1974). Ecology 55, 551. HERBLAND, A. (1975). J. exp. mar. Biol. Ecol. 19, 19. HOBBLE, J. E. & CRAWFORD, C. C. (1969). Limnol. Oceanogr. 14, 528. JACKSON, G. A. (1983). J. Plankton Res. 5, 83. LAMPERT, W. (1978). Limnol. Oceanogr. 23, 831. PAYNE, W. J. (1970). A. Rev. Microbiol. 24, 17. REDALIE, D. G. (1983). Mar. Ecol. Prog. Ser. 11, 217. REDALJE, D. G. & LAWS, E. A. (1981). Mar. Biol. 62, 73. SIEBURTH, J. MCN., JOHNSON, K. M., BURNEY, C. M. & LAVOIE,D. M.,(1977). Helgolander wiss. Meeresunters. 30, 565. SOROKIN, Y. I. (1971). Int. Revue. ges. Hydrobiol. 56, 1. WILLIAMS, P. J. le B. (1970). J. mar. Biol. Ass. U.K. 50, 859. WILLIAMS, P. J. le B. (1981). Kieler Merresforsch., Sonderh. 5, 1. WILLIAMS, P. J. le B. & YENTSCH, C. S. (1976). Mar. Biol. 35:31.
Improved Estimates of Phytoplankton Carbon Based on ~4C Incorporation into Chlorophyll a EDWARD A. LAWS
Department of Oceanography and Hawaii Institute of Marine Biology, 1000 Pope Road, Honolulu, Hawaii 96822, U.S.A. (Received 18 November 1983, and in revised form 3 April 1984) A theoretical model of phytoplankton production, zooplankton grazing, and bacterial utilization of dissolved organic carbon (DOC) shows that both upper and lower bounds to the concentration of phytoplankton carbon can be obtained from primary production measurements and the rate of incorporatien of t4C into chlorophyll a carbon. An expression is derived to account for the error created by zooplankton grazing and bacterial production on both the upper and lower bound estimates of phytoplankton carbon. Application of this error correction to field data suggests that highly accurate estimates of phytoplankton carbon may be possible in cases where phytoplankton production is approximately balanced by zooplankton grazing and bacterial uptake of excreted DOC. Introduction
Several years ago Redalje & Laws ( 1981 ) introduced a method for estimating phytoplankton growth rates and carbon biomass based on the incorporation of t4C into chlorophyll a carbon. The technique makes use o f the fact that after incubations o f a few hours, the specific activities of the phytoplankton carbon and the chlorophyll a carbon are identical. Redalje (1983) for example has shown that in a culture o f Mantoniella sp. grown on a 12 h: 12 h light:dark cycle these specific activities are equal after an incubation of only two hours. One important advantage o f the Redalje & Laws (1981) technique for calculating growth rates is that the calculated growth rate is unaffected by zooplankton grazing and bacterial uptake of excreted dissolved organic carbon (DOC). This conclusion follows from the fact that the specific activity of the chlorophyll a carbon is not altered by zooplankton grazing or bacterial production. When growth rates are calculated from the rate o f incorporation of ~4C into total particulate material, zooplankton grazing and bacterial production will affect the calculated growth rate, because some of the organic ~4C activity appears in the zooplankton and bacteria. Jackson (1983) has provided a thoughtful discussion of this problem with respect to zooplankton grazing. 425
0022-5193/84/190425 + l 0 $03.00/0
© 1984 Academic Press | nc. (London) Ltd.
426
E.A. LAWS
The method proposed by Redalje & Laws (1981) for estimating the concentration of phytoplankton carbon consists of dividing the particulate t4C activity by the specific activity of the chlorophyll a carbon. Unfortunately the value of the phytoplankton carbon concentration calculated by this method is affected by zooplankton grazing and bacterial production. The concentration of phytoplankton carbon will be overestimated to the extent that ~4C activity appears in particles other than phytoplankton cells. This overestimation would become serious if phytoplankton production is roughly balanced by grazing and excretion, and if the duration of the incubation is more than one doubling time. The magnitude of the overestimation will of course depend on the efficiency with which labeled organic carbon is retained in particles other than phytoplankton cells (Jackson, 1983). The purpose of this communication is to suggest an alternative way to calculate phytoplankton carbon biomass. It will be shown that the effects of zooplankton grazing and bacterial production are such as to make the concentration calculated by the alternative method a lower bound on the phytoplankton carbon concentration, whereas the concentration calculated by the Redalje & Laws (1983) method is an upper bound. An approximate equation will be derived to account for the error created by zooplankton grazing and bacterial production.
Theory A list of symbols used in the following theoretical development is provided in the nomenclature. Let Cp(t) be the concentration of phytoplankton carbon at time t. Assume that d
dt Cp(t) = [/z(t) - g(t)]Cp(t)
(1)
where/z(t) is the net rate of carbon fixation per unit phytoplankton carbon and g(t) is the rate at which phytoplankton carbon is transferred to other particles per unit phytoplankton carbon. Although in the past one might have assumed that g(t) describes the effect of zooplankton grazing, recent studies have made it clear that significant amounts of fixed carbon may be transferred from phytoplankton to bacteria via the dissolved organic carbon (DOC) pool, either as the direct result of phytoplankton excretion, or as the result of the feeding activities of zooplankton (Sieburth et al., 1977; Lampert, 1978; Williams, 1981). Thus g(t) must be viewed as a general transfer function expressing the combined effects of zooplankton grazing and bacterial uptake of excreted DOC. /z(t) and g(t) are not assumed to
PHYTOPLANKTON
CARBON
ESTIMATES
427
have any particular functional dependence, and equation (1) is therefore completely general. The solution to equation (1) is
C.(t) = C Oexp [ M ( t ) - G(t)]
(2)
where
M(t) = G(t) =
fo ;o
/z(x) dx
(3)
g(x) dx
(4)
and cO is the phytoplankton carbon concentration at time t = 0. Now let A* be the ~4C activity in the phytoplankton at time t, let A* be the activity of t4C in the inorganic carbon, and let I be the concentration of inorganic carbon. Both A* and I are assumed to be constant. Then
dA*(t)
I~(t)Cp(t)A*o fI
g(t)A*(t)
(5)
where f is the isotope discrimination factor ( f ~ 1.05). The solution of equation (5) is ,
0
A*(t) = Ao Cp exp [-G(t)]{exp [ M ( t ) ] - 1}.
fI
(6)
From equations (2) and (6) one can see that the specific activity of the phytoplankton carbon, which I will call a*(t), is given by the equation
a*(t) = A*(t) =A*o Il - e-M(t)]. Cp(t) f I L
(7)
Redalje & Laws' (198 l) method for estimating phytoplankton growth rates amounts to solving equation (7) for M(t), and then setting the growth rate equal to M(t)/t. It is obvious from equation 3 that the Redalje & Laws (1981) growth rate, which I will call /~(t), is equal to
!
~(t) = t
Io
/z(x) dx.
(8)
Therefore the Redalje & Laws (1981) growth rate is in fact the time average of the phytoplankton growth rate over the course of the incubation. Now for the moment assume that ~4C remains in the particulate phase once it has been converted into phytoplankton carbon. In other words, assume that respiration and excretion losses associated with zooplanktonic
428
E.A. LAWS
and bacterial metabolism are negligible. Let A*(t) be the activity of particulate organic t4C at time t. Then dA*(t)= dt
A~(t)Cp(t) fI
(9)
The solution of equation (9) is
A*(t)
A0~t rI'
dx.
f I .Jo
(lO)
Now i f / z ( x ) and Cp(x) are uncorrelated, then
Io [/z (x) -/2(t)]ECp(x) - Cp(t)] dx = 0
(1 1)
where Cp(t) is the time average of the phytoplankton carbon concentration over the course of the incubation, and is equal to
Io
(12)
Io Ix(x)Cp(x) dx = t/2( t)Cp( - t)
(13)
--~p(t) _ 1
-t
Cp(x) dx.
If equation (11) is satisfied, then
and
A*(t) - A*t/2(t)Cp(t)
(14)
fi Now the average rate/5(t) of photosynthesis over the course of the incubation is
P(t) = A*(t)f[ A*ot
(15)
From equations (14) and (15) it follows that
Cp( t) = P( t)/ /2( t).
(16)
Thus the average phytoplankton carbon concentration over the course of the incubation can be estimated by dividing the photosynthetic rate /5(t) by the growth rate/2(t). The accuracy of this calculation of course depends on the extent to which equation (11) is satisfied. Is there any reason to expect that Cp(t) and tz(t) are uncorrelated? Certainly over time intervals of one day or less there is good reason to think that tz(t) and Cp(t) are uncorrelated. This conclusion follows from
PHYTOPLANKTON CARBON ESTIMATES
429
the fact that the fractional rate of change of phytoplankton carbon at time
t is primarily a function of environmental conditions at time t, but the phytoplankton carbon concentration at time t is clearly a function of antecedent environmental conditions. During the photoperiod one would expect Cp(t) to be positively correlated with time. However, be(t) would be expected to show a positive correlation with time during the morning and a negative correlation with time during the afternoon, in response to the natural variation in irradiance. Thus on average be(t) and Cp(t) would be very poorly correlated during the photoperiod. At night one would expect Cp(t) to be negatively correlated with time. However, be(t) would not be expected to show any systematic correlation with time in the dark. Thus the left-hand side of equation (11) should be very nearly zero over the course of both the day and night. It is instructive at this time to examine the effect of zooplankton grazing and bacterial uptake of labeled DOC on the calculated phytoplankton carbon biomass. At time t the particulate ~4C activity not confined to phytoplankton cells equals A*(t)-A*(t) if none of the ~4C consumed by zooplankton and bacteria is respired or excreted. Assume that a fraction r of the t4C consumed by the zooplankton and bacteria is in fact lost to respiration or excretion. The calculated value of 15(0 would therefore be reduced by a fractional amount r[A*(t)-A*(t)]/A*(t)= r[l-A*(t)/A*(t)]. Let /3=A*(t)/A*(t), and let fi(t) and Cp(t) be the calculated mean photosynthetic rate and phytoplankton carbon concentration, respectively, over the course of the incubation. Then / 5 ( 0 = P ( t ) [ l - r ( l - f l ) ] , and from equation (16) A
Cp(t) = Cp(t)[1
- r(1 -/3)].
(17)
Note that zooplankton grazing and bacterial production have no effect on the calculated value o f / 2 ( 0 , since zooplankton grazing and bacterial production have no effect on the chlorophyll a carbon specific activity, which is equal to ot*(t) (equation 7) after an incubation of a few hours. Redalje & Laws' (1981) estimate of phytoplankton carbon, which I will call t~p(t), is equal to the particulate 14C activity at time t divided by a* (t). From equation (7) it is obvious that Cp(t)= A*( t)/ a*( t), and therefore Cp(t) will overestimate Cp(t) to the extent that a*(t)-r[a*(t)-A*(t)] overestimates A*(t). It is straightforward to show that Cp(t) and Cp(t) are related by the equation
Note that the error caused by zooplankton grazing and bacterial production
430
E.A. LAWS
is linearly related to r in the case of both C?p(t) and C?p(t), but that the magnitude of the error is positively correlated with r in the former case (equation (17)) and negatively correlated with r in the latter case (equation (18)). The error in Cp(t) is zero when r = 0, because the calculation of Cp(t) assumes that all carbon fixed by phytoplankton is retained in the particulate phase. The error in ~?p(t) is zero when r = 1, because the calculation of Cp(t) assumes that all particulate ~4C activity is confined to the phytoplankton. Thus zooplankton grazing and bacterial production create no error in the calculation of Cp(t) as long as the zooplankton and bacteria respire or excrete all the 14C they consume. The magnitude of the error in both Cp(t) and Cp(t) is negatively correlated with/3, and approaches a maximum value as/3 ~ 0. However, the magnitude of the fractional error in Cp(t) when/3 = 0 is equal to r, whereas the fractional error in Cp(5) approaches infinity as/3 -~ 0. The latter result occurs because the condition /3 = 0 implies that all the phytoplankton carbon has been transferred to zooplankton and bacteria, and therefore the true phytoplankton concentration at time t is zero. However, the fact that the phytoplankton concentration at the end of the incubation is zero obviously does not imply that the average phytoplankton concentration over the course of the incubation [Cp(t)] has been zero. Hence the fractional error in Cp(t) remains finite even when/3 ~ 0. It is now useful to ask whether Cp(t) or Cp(t) is a more accurate estimator of phytoplankton carbon biomass. From equations (17) and (18) it is obvious that the magnitude of the fractional errors in Cp(t) and Cp(t) are equal if r(l -/3) = (1 - r ) ( ~ - - 1).
(19)
Equation (19) implies that
r=(l +/3)-'.
(20)
If r is less than or greater than (1 +/3)-', the magnitude of the fractional error in Cp(t) will be respectively less than or greater than the fractional error in Cp(t). It is obvious from equations (5) and (9) that A*(t) will equal A*(t) only if g(t) = 0. Otherwise A*(t) < A*(t). Hence the maximum value of/3 is 1, and the minimum value of (1 +/3) -1 is 0.5. Thus if r < 0 . 5 , the magnitude of the fractional error in Cp(t) will be less than the fractional error in Cp(t). However, most studies of the ecological growth efficiencies of bacteria and herbivorous zooplankton suggest that r > 0.5. Data summarized by C o m e r & Davies (1971) imply a zooplanktonic growth efficiency of about 23%. Williams' (1981) model of planktonic food webs assumes a growth efficiency of 30%. Jackson (1983), based on studies by Checkley
PHYTOPLANKTON
CARBON
ESTIMATES
431
(1980), assumed a growth efficiency of 37%. It therefore seems reasonable to assume that only about 30% of captured food is converted into new tissue by herbivorous zooplankton. However, a fraction of the excreted carbon will be released in particulate form, and will therefore contribute to the particulate 14C activity. Experiments conducted by Conover (1966) and Lampert (1978) suggest that 10-15% of captured carbon is released as DOC; the summary of Corner & Davies (1971) implies that 20-25% of captured carbon is released as DOC or particulate organic carbon (POC). I conclude that about 10% of captured carbon is released as POC. It therefore seems reasonable to assume that about 40% of the phytoplankton carbon captured by zooplankton is retained in the particulate phase, and hence for zooplankton grazing r---0-6. It is apparent from a great many studies that the growth efficiencies of bacteria depend very much on the carbon substrate. When bacteria are grown on small organic molecules such as simple sugars and amino acids, growth efficiencies are in the range 50-95% (Hobbie & Crawford, 1969; Williams, 1970; Payne, 1970; Crawford, Hobbie & Webb, 197d; Williams & Yentsch, 1976). However, when natural phytoplankton excretion products are provided as the source of carbon, growth efficiencies are in the range 35% (Herbland, 1975) to 45% (Sorokin, 197 l). It therefore seems reasonable to set r = 0.6 for both bacterial production and zooplankton grazing, and I will use this value of r in estimating the errors in Cp(t) and Cp(t). The value of/3 will of course depend on the nature of both /z(t) and g(t). Let us make the simplifying assumption that / z ( t ) = g ( t ) , i.e. that grazing and excretion losses exactly balance phytoplankton production. This assumption is clearly an approximation to the real world situation, but should not lead to seriously misleading conclusions in systems where M ( t ) = G(t), i.e. where the average rate of production is approximately equal to the average rate of consumption over the course of the incubation. If/~(t) = g(t), then Cp(t) is constant, and it is straightforward to show from equations (6) and (10) that
A*(t) 1 - e -MC') /3=A*(t)- M(t)
(21)
If the duration of the incubation is exactly one doubling time, then M(t) = In(2), and/3 = 0.72. Taking r = 0.6, I conclude from equations (17) and (18) that Cp(t) will underestimate Cp(t) by about 17%, and that Cp(t) will overestimate Cp(t) by 16%. If the duration of the incubation is half a doubling time, the corresponding errors in Cp(t) and t~p(t) are estimated to be 9% and 7%, respectively. These estimates of the errors created by zooplankton grazing and bacterial production suggest that estimates of
432
E.A. LAWS
phytoplankton carbon should be based on incubations lasting no more than approximately one doubling time. Under these conditions the errors in Cp(t) and Cp(t) should be no more than about 15-20%. If the error created by zooplankton grazing and bacterial production is no more than about 15-20%, it seems reasonable to make an approximate correction for zooplankton grazing and bacterial production using equations (17) and (18), with r assigned a value of 0.6 and /3 calculated from equation (21). The following example illustrates the application of the method. Results and Discussion
Table 1 tabulates estimated phytoplankton carbon concentrations [Cp(t)] and M ( t ) values based on 24 h incubations in the Southern California Bight as reported by Redalje (1983). I have calculated values of Cp(t) by simply dividing P(t) by /2(t), where /5(t) is the estimated daily photosynthetic rate a n d / 2 ( t ) is the average daily growth rate. Since t = 1 day in this case, /2 (t) is numerically equal to M(t). Note that since M ( t ) < In (2) in all cases, the estimated errors in Cp(t) and Cp(t) due to zooplankton grazing and bacterial production are expected to be less than 15-20% in all cases. The last two columns of Table 1 show the calculated values o f ~ p ( t ) and Cp(t) when a correction for zooplankton grazing is applied to Cp(t) and Cp(t), respectively. The remarkable aspect of these last two columns of Table l is that the calculated values of Cp(t) and Cp(t) differ by no more than one unit in the second significant figure. In fact there is no a priori reason why (~p(t) and Cp(t) should be identical, since the average value of the phytoplankton carbon concentration over the course of the incubation [(~p(t)] should not necessarily equal the phytoplankton carbon concentration at the end of the TABLE 1
Calculated values of Cp( t) and Cp( t) after incubations of 24 hours based on experiments reported by Redalje (1983) in the Southern California Bight. Values of Cp(t) and Cp(t) have been calculated using equations (17) and (18), with r = 0 . 6 and fl calculated from equation (21) Depth of incubation
M(t)
Cp(t) (~g/I)
Cp(t)
(~g/l)
Cp(t) (~g/l)
Cp(t)
(m)
(~g/l)
(~g/1)
6 II 21
0-506 0-463 0-312
58 50 25
74 62 29
67 57 27
67 56 27
PHYTOPLANKTON
CARBON
433
ESTIMATES
incubation [Cp(t)]. However, Redalje's (1983) chl a values were remarkably constant over the course of the entire 24 h incubations at each of the depths sampled. The mean (+1 standard deviation) chl a concentrations at 6 m, I1 m and 21 m were 1.34+0.13, 1.29±0.09 and 0.95 ± 0.10 txg/l, respectively. These figures are based on samples taken at the beginning of the incubation, and at 3, 6, 12 and 24 h after the start of the incubations. The constancy of the chl a concentrations implies that the phytoplankton biomass was in fact relatively constant, and hence the fact that Cp(t) and Cp(t) are nearly equal is not surprising. The apparent constancy of the phytoplankton biomass supports the assumption that I~(t)~-g(t), an assumption which was of course used to derive a simplified expression for the error created by zooplankton grazing and bacterial production. These results suggest that highly accurate estimates of [?hytoplankton carbon may be possible through calculation of Cp(t) and Cp(t), and a correction for zooplankton grazing and bacterial production effects using equations (17) and (18), respectively, with r = 0 . 6 and/3 determined from equation (21). Nomenclature
Symbol Definition A*(t) 14C activity in the phytoplankton at time t ~4C activity in the inorganic carbon A* A*(t) Particulate organic ~4C activity at time t in the theoretical absence of particulate ~4C losses due to respiration and excretion by zooplankton and bacteria a*p(t) Specific activity of the phytoplankton carbon at time t
A*( t)/ A*( t)
cat) I
co cat) p(t) Cp(t)
Phytoplankton carbon concentration at time t Phytoplankton carbon concentration at time t = 0 Mean phytoplankton carbon concentration over the course of the incubation = (1 / t) S'o Cp (x) dx Phytoplankton carbon concentration at time t estimated by Redalje & Laws (1981) technique = Cp(t)[1 +(1 - r)(1/fl - 1)] Calculated mean phytoplankton carbon concentration over the course of the incubation in the presence of particulate ~4C losses to respiration and excretion by zooplankton and bacteria=
P( t)/ l~( t)
Units Ci/l Ci/l Ci/l
Ci/g Dimensionless g/l g/1 g/l g/1
g/l
434 f g(t)
G(t) I
/5(0 /5(t)
~,(t) M(t) 12( t)
E. A. LAWS 14C i s o t o p e d i s c r i m i n a t i o n f a c t o r Dimensionless R a t e at w h i c h p h y t o p l a n k t o n c a r b o n is t r a n s f e r r e d d -j to o t h e r p a r t i c l e s p e r unit p h y t o p l a n k t o n c a r b o n at t i m e t t So g ( x ) d x Dimensionless Concentration of inorganic carbon g/l A v e r a g e rate o f p h o t o s y n t h e s i s o v e r the c o u r s e o f g/l/d t h e i n c u b a t i o n = A * ( t ) f I / ( A * t) C a l c u l a t e d a v e r a g e rate o f p h o t o s y n t h e s i s o v e r t h e g/l/h c o u r s e o f t h e i n c u b a t i o n in the p r e s e n c e o f p a r t i c u late 14C losses to r e s p i r a t i o n a n d e x c r e t i o n b y z o o p l a n k t o n a n d b a c t e r i a = / 5 ( t ) [ 1 - r( 1 - fl)] F r a c t i o n o f 14C c o n s u m e d b y z o o p l a n k t o n a n d D i m e n s i o n l e s s b a c t e r i a t h a t is lost to r e s p i r a t i o n a n d e x c r e t i o n N e t rate o f c a r b o n fixation p e r unit p h y t o p l a n k t o n dc a r b o n at t i m e t S'o/z(x) d x Dimensionless A v e r a g e p h y t o p l a n k t o n g r o w t h rate o v e r t h e c o u r s e d- t o f the i n c u b a t i o n = M ( t ) / t
Hawaii Institute of Marine Biology Contribution No. 686.
REFERENCES
CHECKLEY, D. M. (1980). Limnol. Oceanogr. 25, 430. CONOVER, R. J. (1966). In: Some Contemporary Studies in Marine Science, pp. 187-194 (Barnes, H., ed). London, Allen & Unwin. CORNER, E. D. S. & DAVIES, A. G. (1971). Adv. mar. Biol. 9, 101. CRAWFORD, C. C., HOBBLE,J. E. & WEBB, K. L. (1974). Ecology 55, 551. HERBLAND, A. (1975). J. exp. mar. Biol. Ecol. 19, 19. HOBBLE, J. E. & CRAWFORD, C. C. (1969). Limnol. Oceanogr. 14, 528. JACKSON, G. A. (1983). J. Plankton Res. 5, 83. LAMPERT, W. (1978). Limnol. Oceanogr. 23, 831. PAYNE, W. J. (1970). A. Rev. Microbiol. 24, 17. REDALIE, D. G. (1983). Mar. Ecol. Prog. Ser. 11, 217. REDALJE, D. G. & LAWS, E. A. (1981). Mar. Biol. 62, 73. SIEBURTH, J. MCN., JOHNSON, K. M., BURNEY, C. M. & LAVOIE,D. M.,(1977). Helgolander wiss. Meeresunters. 30, 565. SOROKIN, Y. I. (1971). Int. Revue. ges. Hydrobiol. 56, 1. WILLIAMS, P. J. le B. (1970). J. mar. Biol. Ass. U.K. 50, 859. WILLIAMS, P. J. le B. (1981). Kieler Merresforsch., Sonderh. 5, 1. WILLIAMS, P. J. le B. & YENTSCH, C. S. (1976). Mar. Biol. 35:31.