Journal of Plankton Research Vol.18 no.10 pp.1941-1959, 1996
Algal 14C and total carbon metabolisms. 1. Models to account for
the physiological processes of respiration and recycling
Peter J.le B.Williams and Dominique Lefevre
School of Ocean Sciences, University of Wales, Bangor, Menai Bridge, Gwynedd
LL595EY,UK
Abstract A consistent set of equations has been written to describe the net rate of algal 14COj uptake
(and where appropriate respiration and photosynthesis) which take into account separately complications due to respiration of the labelled photosynthetic products and the recycling of respiratory CO2.
Written specifically into the equations is the concept of'new' and 'old' carbon, the coefficient q is used
in the respiration model to allow for the differential respiration of organic material from the 'new' and
'old' carbon pools. Analytical integrals have been found for respiration and recycling models, and the
behaviour of the models studied over periods of 12 h (i.e. up to 70% of the intrinsic generation time).
The rate constant for respiration has a greater effect on the behaviour of the recycling than the respiration model. Over short time courses (up to 30% of the intrinsic generation time), the effects of respiration and recycling on net MCO2 uptake are quite distinct, especially at high P/R ratios, and not
complicated by assumptions over the value of q. Although the value of q will have a time-dependent
secondary effect on the modelled total carbon-specific respiration rate, this was found not to give rise
to major problems of interpretation. Beyond 50% of the intrinsic generation time, the separate treatment of respiration and recycling in the models becomes less satisfactory. It was concluded that the
present equations, which are not constrained by mass balance considerations, would not be appropriate for a model that combines the two processes. The pattern of recycling at low P/R values is identified as one of the major uncertainties in producing models of MC uptake. The effect of the release of
dissolved organic material can be anticipated in a general way. The models have been used to define
an experimental strategy to establish the separate effects of respiration and recycling on the time
course of net 14C uptake. The initial rates give the clearest resolution of the two processes and it would
appear that with photosynthetic rates in the region of 1 day-', incubation periods up to 3-6 h would be
suitable to determine the importance of recycling in controlling net MC uptake. With the present
models, only in the absence of recycling could the effect of respiration be studied and the value of q
established.
Introduction
The data base of 14C measurements is massive and important in quantifying
marine biogeochemical processes. Enormous cost and effort have been incurred
in its acquisition. Despite this investment, it is evident that we still lack a sound
basis for the interpretation of 14C uptake rates due to a lack of understanding of
the relationship between net 14C uptake and defined ecological processes such as
gross and net production. The flows of the tracer during 14C measurements of algal
carbon fixation are complex and incompletely understood, and remain a major
factor limiting the accuracy with which the primary method in biological oceanography can be interpreted. Williams (1993) summarized the flows of carbon in an
idealized algal cell and highlighted a number of general problem areas associated
with the interpretation of net 14CO2 metabolism. The extent to which newly fixed
carbon is used for respiration in preference to older material, and the extent of
recycling of respiratory 12CO2 (and eventually 14CO2) and its effect on the specific
activity of the substrate of RUBISCO, were concluded to be the main areas of
uncertainty.
© Oxford University Press
1941
PJJe B.WUKams and D.Leftvre
Loss of fixed carbon due to respiration of photosynthetic products
Major physiological uncertainties centre round the respiration and recycling of
the photosynthetic products. Although Bidwell (1977) concluded that 'No (respiratory) 14CO2 appears because none is produced, not because it is recycled', it is
difficult to ignore the contemporaneous study of Lloyd et al. (1977) which demonstrated the release of 14CO2 by respiration. Models incorporating the loss of photosynthetically fixed 14CO2 by respiration were written by Hobson et al. (1976).
They, and Dring and Jewson (1982) who extended their work to consider the effect
of differential respiration rates for 'old' and 'new' carbon, restricted attention to
the relationship between net 14C fixation and net photosynthesis.
Recycling of respiratory CO2
Lloyd's work (Lloyd et al., 1977) leads to the conclusion that recycling of respiratory CO2 must occur to some degree. At present, there is no general model for
recycling of CO2 by an algal cell and the signs are (Raven, 1993) that we presently
lack the necessary physiological knowledge to write exact models that fully incorporate internal CO2 cycling. Thus, so far we have only been able to deal with the
physiological processes associated with the recycling of respiratory CO2 in a very
generalized manner. If it is assumed that recycled respiratory CO2 is used in preference to external CO2, then at high photosynthetic rates (where the assumption is
most likely to be met) the resultant reduction in specific activity of the photosynthetic pool would mean that the initial rates of 14C fixation would approximate to
photosynthesis minus the rate of respiration (i.e. net photosynthesis).
Smith and Platt (1984) correctly point out that in analysing 14C observations,
researchers inevitably rely upon some conceptual model, explicit or otherwise, of
what the method measures. However, the empirical basis for such conceptual
models is currently poor. With the availability of high-precision techniques to
measure total CO2 concentration (e.g. Johnson et al., 1987; Robinson and
Williams, 1991), this can now be remedied and accurate comparisons of net 14CO2
uptake with gross and net CO2 fixation can be made. The design and interpretation of such comparisons is best based on explicit hypotheses of net 14CO2
assimilation. Models exist in the literature (Hobson et al., 1976; Dring and Jewson,
1982; Smith and Platt, 1984; Smith et al., 1984; Williams, 1993), but the theory is
incomplete, terminology inconsistent, implicit assumptions are not always fully
spelt out and a number of the time-dependent integrals are lacking. Models to
explain or interpret field measurements of 14C uptake (e.g. Marra et al., 1981;
Smith and Platt, 1984; Smith et al., 1984) comprise an ecological model in which is
embedded a physiological model—characteristically that of Hobson et al. (1976).
Obviously, the ecological model can only be as good as the physiological model it
contains, and until we know the validity of the physiological model the usefulness
of ecological models must remain in question.
The objective of the present work has been to write a set of models that take
account of the physiological processes of carbon respiration and recycling within
the algal cell using a consistent set of concepts and formulations and, where
1942
I4
C uptake—models to account for respiration and recycling
Rate constants
P = photosynthesis (T"1)
R = respiration (T^1)
RIccy = recycled part of respiration (T"1)
Time-dependent properties
Co - biomass at time = 0 (M)
COM (0 = residual C o at time = t (M)
C new (t) = residual newly formed biomass at time = t (M),
C^ew (t) = residual newly formed biomass at time = /, differential respiration
of new carbon (M)
Octal (0 = total biomass = Cold (t) + Cmvi (r) or Cold (/) + Clcv/ (t) (M)
F(t) = <14C biomass' = C n e w (t) (M)
At t = 0, Cnew, C^ew and F are all equal to zero.
Integral properties
G = gross photosynthesis (M)
G*1 = gross photosynthesis: differential respiration of new carbon (M)
N = net photosynthesis (M)
iVq = net photosynthesis: differential respiration of new carbon (M)
F r e s p = net 14C fixation: simple respiration model (M)
F ^ p = net 14C fixation: differential respiration of new carbon (M)
Fliecy and F j ^ = net 14C fixation: recycling models I and II (M)
Moderators
W(t) = 14C specific activity at time = t (dimensionless)
q = relative preference for newly fixed carbon (dimensionless)
co = extent of recycling of respiratory CO 2 :0 < w < 1 (dimensionless)
Frame 1. Definition of parameters in the models.
possible, to obtain time-dependent integrals. Once achieved, these models lay a
theoretical basis for the design and analysis of experiments, and the potential for
the development of algorithms for the interpretation of field observations of the
14
C uptake by phytoplankton.
Theory
Formal representation of I2carbon and net
14
carbon fixation
Embedded to varying degrees in existing models of 14C uptake is the concept of
'new' and 'old' carbon. 'Old' carbon being that present at the beginning of the incubation, 'new' being that synthesized during the incubation. [For the sake of
1943
PJJe B. Williams and D.Leftvre
consistency and conciseness, we have adopted Dring and Jewson's terminology for
newly formed ('new') and pre-existing ('old') cell carbon, whilst recognizing that
the term 'new' is also used to describe nitrate-based photosynthesis. Strictly, the
latter use should be restricted to nitrogen assimilation and this would reduce the
potential for confusion.] The 14C technique at its simplest is taken to measure the
net accumulation of 'new' carbon, whereas chemically determined net photosynthesis measures the accumulation of'new' carbon minus the respiratory loss of'old'
carbon. This gives rise to net photosynthesis as one formal lower boundary for 14Cdetennined carbon fixation. Hobson et al. (1976) made the implicit simplifying
assumption that 'new' and 'old' carbon were respired at the same rate, so the two
forms of carbon are not formally separated in their model. Dring and Jewson
(1982) introduced a coefficient (q) which allowed for differential rates of respiration of these two carbon pools. They assumed that these two pools remained separate physiological entities throughout the time course with no progressive
conversion from 'new' to 'old' carbon. We have written the concept of new and 'old'
carbon explicitly into all models as we find the formalization clarifies the assumptions being made.
Hobson et al. (1976) and also Dring and Jewson (1982) undertook calculations
of the accumulation of the 14C isotope itself by considering its specific activity. This
required introducing an extra coefficient into the equations. Although this is
strictly correct, it achieves little, as experimenters routinely convert 14C uptake
rates to total carbon rates. As Williams (1993), we have treated the 14C as a tracer
without mass.
Equations for carbon metabolism incorporating respiratory losses of carbon
The first level equations for net 12C and 14C [equations (l)-(3)] metabolisms were
presented by Hobson et al. (1976). They are given here in an expanded form incorporating the formalization of 'new' and 'old' carbon. The equations for overall
gross and net photosynthesis are as follows:
where
d/
(0 - RCtotti (0 = (p- R)(CM (0 + c new (o)
The equation for net 14C fixation is:
1944
(2)
14
C uptake—modeb to account for respiration and recycling
= PCtotel (0 - RCaev (0 = P (C old (r) + C MW (0) - KC new (0
(3)
If it is assumed that at / = 0, G{t), N(t) and Fiap(t) are equal to zero, then their
time-dependent integrals [from Hobson et al. (1976) and Williams (1993)] are:
(4)
(5)
(6)
In deriving these solutions, it is assumed that 'old' and 'new' carbon are respired
at the same rates. As this is improbable, the implicit assumption in equation (2)
is unlikely to be correct.
Equations to account for the differential respiratory loss of either new or old
carbon
The problem of the differential respiration of 'old' and 'new' carbon was addressed
by Dring and Jewson (1982) who used a coefficient q (the ratio of the respiration
of 'new' to 'old' carbon; where q < 1.0 then the 'old' carbon is used in preference,
where q > 1.0 then the 'new' carbon is used in preference) to describe the selective
utilization of newly formed organic material. Using this formulation, the equivalent equations for gross and net photosynthesis and 14C fixation become:
}
where:
= PCtotBi (t) = p(cold (t) + cu (o)
C2 ew (t) =
PCn
^
P-(q-l)R
d/
(7)
F
1
\e(P-gR)i-e-Ri\
[
^d (0 + C^cw W) " R{C0Ui (0 + qC^
J
(0)
PC** (0 - <7*c2cW (0 = /'(Co.d (r) + c2eW (0) - ^ ^ c ^ (0
(8)
(9)
1945
PJ.Ie B. Williams and D.Leftvre
Dring and Jewson used an analog computer to study the time-dependent
behaviour of the equations. We have found all three time-dependent integrals:
P-(q-l)R
^}
(11)
fp n J
\.r — \q— l j n J
At t = 0, G%t) Ni(t) and F ^ p (f) are equal to zero.
These equations make the assumption that the 'new' and 'old' carbon pools are
homogeneous and remain separate, with no transfer from 'new' to 'old'. Williams
(1993) used a four-compartment model with a periodic block transfer of material
from 'new' to 'old' to simulate this. There is also an implicit assumption that there
is no recycling of respiratory CO2 within the cell and that specific activity of the
14
CO2 at the site of CO2 is the same as the external CO2. Steemann Nielsen (1963)
drew attention to this problem and the signs are (see Discussion) that it plays an
important role in determining net 14CO2 fixation. Thus, if we plan to analyse
curves of 14C uptake, then we will need to develop models to take account of the
recycling of respiratory CO2.
Equations to account for the recycling of respiratory CO2
Recycling of respiratory CO2 (initially essentially 14C-free) potentially gives rise to
a reduction in the specific activity of 14CO2 at the site of the enzymatic fixation. At
its simplest, if 100% of the respiratory CO2 were recycled and used in preference
to external 14C-labelled CO2, then the rate of 14C fixation could be modelled as
P-R at the onset of the incubation, i.e. equal to net photosynthesis. There is an
obvious problem with this simple P-R representation: it would give negative rates
of net photosynthesis when R > P, which cannot be the case for the 14C-determined
rates. If we wish to use such a model, then one operational solution is to ignore this
part of the relationship (the shaded area in Figure 1) and to simply set the 14C rate
to zero when R > P. We would not see this as a satisfactory solution either on
physiological or computational grounds. We have thus explored two alternative
solutions which are based on the following expectations: (i) that at high photosynthetic rates, when there will be a high demand on internal CO2, a large percentage
of the respiratory CO? will be recycled into photosynthesis; (ii) the extent of recycling is reduced as photosynthesis approaches and becomes less than respiration,
resulting in 14C uptake moving towards a measure of gross photosynthesis as the
photosynthetic rate decreases. This provides two 'fixed' points (shown in Figure 1)
1946
14
C uptake—models to account for respiration and recycling
in the gross production/net production/14C relationship. One may expect that the
transition of net 14C uptake as a measure of gross rather than net photosynthesis
(i.e. the reduction of the extent of recycling) occurs as the demand on the internal
CO2 is reduced (where R > P), giving an oblique J-shaped curve (see Figure 1). It
also implies the only at the point when the gross photosynthetic rates become zero
do the 14C rates also become zero, i.e. there is no threshold of photosynthesis for
net 14C fixation. This is the general experience with 14C-determined P versus I
curves. This implies a progressive rather than an abrupt decrease in CO2 recycling
as the photosynthetic rate decreases.
The first attempt to model this was by Williams (1993), who modelled the
specific activity (W) of the 14C in the photosynthetic CO2 pool by assuming it to
be determined by the relative proportions (and rates of supply) of internally (/)
and externally (E) derived CO2, i.e.
(13)
W= —
E+I
The factor W (0 < W < 1), i.e. the reduction in specific activity of the external
CO^ can then be used as a moderator to gross photosynthesis to give the rate of
14
C fixation:
U
dF'
^>
(0
dG(t)
J±
(14)
where G is gross photosynthesis and Frecyis net 14C fixation.
We return to the problem of how to represent the change in W without obtaining negative values in 14C-determined photosynthetic rates when R> P. Our aim
has been to do this using only measurable properties, i.e. the physiological rate
constants for photosynthesis (P) and respiration (R). In seeking a solution to the
problem, we are hampered by the lack of detailed knowledge of the physiology
surrounding the utilization of internal and external CO2—there is no accepted
model for this and how it varies as the balance of respiration and photosynthesis
changes. The following general solution was suggested by Williams (1993):
In the present study, we have used this expression and modelled the time
dependency of W as:
1947
PJJe B.WUHams and D.Lcftvre
where the exponent (-Rt) allows for the general increase in the specific activity of
respiratory CO2 as that of respiratory substrate increases with time. The initial
value of W (i.e. when t = 0) from the above equations gives a pattern of the recycling effect on the rate of 14C fixation and its relationship to gross and net photosynthesis which is intuitively correct (see Figure 1, Case III). There is an implicit
assumption in the above expression that the system is dynamic inasmuch that respiratory CO2 does not accumulate and have an integrated effect, an assumption
which is probably not unreasonable at high photosynthetic rates when the
turnover of internal CO2 is very rapid. Almost certainly, the true relationship is
more complex; however, since in experimental work we are mainly concerned
with initial rates, this level of simplification is probably regarded to be acceptable
and prevents the mathematics becoming unnecessarily cumbersome.
The differential equations for the overall rate of 14CO2 fixation would be:
en)
dr
where:
P + Re-Rt
then:
^ ^
= [PQe<'-*>] [
I
1 (18)
We have been unable to find an analytical integral solution for the above equation, the problem being the exponent in the denominator, and it is thought that it
is unlikely that a simple integral exists (A.G.Davies, personal communication). We
have had to resort to numerical solutions for this formulation.
The second representation of the physiology is a simple extension of the P-R
solution, in which the effect of recycling is reduced as the photosynthetic rate
decreases in relation to respiration. This is achieved by modelling the rate constant of the recycled part of respiration (/?recy) as a fraction of total respiration,
using a simple moderator of R:
P+ R
(19)
Thus the rate of 14C fixation would be P-R^y, this reduces the extent of recycling of respiratory CO2 as the photosynthetic rate decreases when the relation1948
14
C uptake—models to account for respiration and recycling
ship asymptotes to zero. We would anticipate this to be a fair representation of the
physiological events at low photosynthetic rates.
So far we have assumed that when P> R, 100% recycling occurs. A fuller representation to include the maximum intrinsic extent of recycling as well as its
change as the P/R ratio changes. This would require introducing a factor o> (the
extent of recycling /"ma,, where 1 £ w ;> 0) such that R^^ = u>R. We have not
pursued the mathematics further, because the signs from concurrent experimental
studies (see Williams et ai, 1996) were that it would not be called for.
Although the P-R^y approach is conceptually different, it turns out to be
arithmetically identical to the simple form of the previous solution [equations (17)
and (18)]. The time-dependent solution, taking into account the progressive
incorporation of newly fixed 14C into the respiratory CO2, would take the following form:
(20)
Only the respiration constant in the primary expression needs to be given a time
dependency, that in the second part of the equation simply acts as a moderator.
This moves the time dependency of R into the numerator and, in contrast with the
first model [see equation (18)], an integral can be found:
(21)
(P + R)(P-2R)
The integration assumes that at t = 0, F}^y(f) = 0.
Discussion
Initial rates
Figure 1 shows the initial rates given by the various models and as such is a statement of the various hypotheses at zero time. As there is an interplay in the equations between P and R, normalizing the rates (of net and gross photosynthesis
and the rate of 14C fixation) by respiration gives plots of very general nature; this
form of presentation has been adopted for Figure 1. The respiration model
[equation (3)] gives 14C rates equal to gross photosynthesis (Curve I). The
simple lP—R' model would run along the net photosynthesis line into the forbidden area (for the 14C) when P-R<0 (Curve II). The recycling models [equations (18) and (21)] comply with the requirement that when P » R, the initial
rates of 14C fixation approach net photosynthesis, whereas when P approaches
and becomes smaller than R, the 14C rate moves progressively towards the gross
photosynthesis line (Curve III), i.e they should link the two 'fixed points' placed
on the graph.
1949
P-Me B. Williams and D.Lefevre
o-
5-
Curve I
y/^
^ ^
4-
Curv« II
y r
^ ^ ^
3Curve III
2-
1 0I
•i
L
S
2
3
4
5
6
-
Fig. L Initial rates of gross and net photosynthesis, along with net I4C uptake from the various models,
normalized against respiration. Curve I: Gross photosynthesis [equation (1)] and respiration model
[equation (3)]. Curve II: Net photosynthesis [equation (2)] and the simple P—R model of recycling.
Curve III: Recycling models F 1 ^ and F 1 ^ [equations (18) and (20)]. The two dots placed on the
graph represent the 'fixed' point for the recycling model, details are given in the text.
Time-dependent behaviour
We have chosen to study the time dependency of the equations using a selected
set of values for P and R. The review of Langdon (1993) provides a range of rate
constants for Pmax and R, characteristic of various microalgal groups (Table I); the
modal values from his paper have been used as case examples. In addition, we
have also studied a case with values of P and R taken to be appropriate to the compensation point for the diatoms (Case IV in Table I). In addition to the conventional time, we have also displayed physiological time, based on the intrinsic
generation time, calculated from the photosynthetic constant (intrinsic generation
time = /*-> In2).
The two recycling (F^ and F1^) models give essentially identical results,
despite the differences in form of the equations and in the integration procedures.
The data presented in Figures 3 and 4 come from the second model (F]^) for
which we have an analytical integral solution.
Over the initial part of the time course (i.e. the first 3-6 h) with P as P ^ (i.e. 1
day 1 ), the 'respiration' model runs close to gross photosynthesis (see Figure
Table L Rate constants for photosynthesis (P) and respiration (R) used in the model runs (the units
of the rate constants are day 1 )
Case
RIP
Category
Cyanophyceae
Diatoms and Prymnesiophyceae
Dinophyceae
Compensation point for diatoms
I
II
III
1.0
1.0
1.0
0.1
0.1
0.16
0.16
03
rv
0.16
0.16
0J
1.0
1950
14
C uptake—models to account for respiration and recycling
;eoo
400 -
200
2
4
6
8
10
0
12
2
TIME(hr)
4
6
8
10
12
TIME (hr)
2
4
6
8
10
12
'-20 -
O
TIME (hr)
2
4
6
8
10
12
-40 -I
TIME(hr)
Fig. 2. Tune courses of gross and net photosynthesis and outputs from the respiration models ( f " ^ ) .
Over the period 0-6 h, the value of q is 1; from 8 to 12 h, q has values 0.1,1 and 10. All rates are calculated with Q set at 1000 u.g. (A) Case l:P=\,R
= 0.1 (Cyanophyceae). (B) Case l\:P=\,R
= 0.16
(diatoms and Prymnesiophyceae). (C) Case III: P=l,R = 03 (Dinophyceae). (D) Case IV: P = R =
0.16 (compensation point for diatoms). *G^(t) is the time-dependent integral for gross photosynthesis;
equation (10)]; *N«(t) is that for net photosynthesis [equation (11)]; *F«(0 is that for net I4 C fixation
[equation (12)]. All allow for the differential respiration of new carbon. The vertical arrows indentified the spread (at (= 12h) of the predictions over the range q = 0.1 to q = 10; the horizontal cross-bar
gives the position of the q = 1.0 line.
2A-D), whereas the recycling models give rates approximating to net photosynthesis at high P/R ratios (see Figure 3A and B) moving in the direction of gross
photosynthesis as the P/R ratio falls (cf. Figure 3C and D). Neither of these findings are unexpected. The earlier study of Williams (1993) demonstrated the
former and the latter is imposed by the form of the model.
In the latter parts of the time course (6-12 h), a comparison of Case I—III (i.e.
1951
PJJe B. Williams and D.Leftvre
B
600-
600-
400
400-
200-
200
3
12
8
6
9
12
9
12
TIME flu)
TIMEflir)
80-,
600
s
60
a
2*"
2003
6
TIME flu)
-20
3
6
8
12
-40
T!ME(hr)
Fig. 3. Tune courses of gross and net photosynthesis and outputs from the recycling model ( F ^ ) . (A)
Case I. (B) Case D. (C) Case III. (D) Case IV. For details, see Figure 2. *G(t) and *N(t) refer to the
time-dependent integrals for gross and net photosynthesis [equations (4) and (5), respectively];
*FII_recy(t) is that of the recycling model [equation (21)].
models of various algal groups) shows that the effect of varying the rate constant
for respiration on the predicted rates of 14C uptake is generally greater for the
recycling than respiration models (cf. Figures 2A-C and 3A-C). The details are
better seen in Rgure 4, in which the rates have been scaled between the limits of
gross and net photosynthesis [i.e. (F-N)/(G-N)],as used by Williams (1993). This
effectively scales out the effect of the size of respiration in the presentation, but
1952
I4
C uptake—modeb to account for respiration and recycling
leaves it in the calculation. The effects then become very clear—there are only
minor effects due to changes in the respiration rate on the scaled values for Fresp,
but there is a progressive reduction in the normalized rates of predicted 14C
uptake with time. Figure 4 shows very clearly the more substantial effect of the
respiration rate in the case of the recycling models. The normalized slope of the
progress curve in the recycling model remains essentially constant with time. The
positive slope of the line is an expression of the aim in the model to allow the
increase in the specific activity of 14C in the respiratory CO2 as time progresses.
The general shift upwards of the recycling lines, as the constant for respiration is
increased, is a consequence of the asymptotic nature of equations (15) and (19).
The effect of variations in q (i.e. the differential rates of respiration of'new' and
'old' carbon) is complex because it affects the modelled rates of 14C fixation as
well as both gross and net photosynthesis [see equations (10)-(12) and Figure 2].
The approach used by Dring and Jewson (1982), and followed in the present work,
scales the respiration of'new' carbon in relation to 'old' (i.e. Cnew = q*Coid). As
time progresses, there is an accumulation of Cnew and a removal of Coid,
consequently unless q = 1, the cell carbon-specific respiration rate will accordingly
change (increasing when q > 1, decreasing when q < 1) as the ratio of CneJCoid
increases. Because of this, the resultant plots of varying q also have in them a
secondary effect due to a varying total carbon-based respiration rate constant.
The consequence of this on the behaviour of the model was not considered by
either Dring and Jewson (1982) or by Williams (1993). Thus, before discussing the
effect of variations in q on the behaviour of 14Cfixation,one needs to consider the
consequences of its concurrent effect on overall respiration, to establish whether
it is possible to separate the two effects. Studies with models with differences in
respiration rates (i.e. Cases I—III) give a basis for separating the direct and indirect (i.e. via changes in the overall respiration rate constant) effects of q on the
modelled rates of 14C uptake. In the case of the respiration models, the curves for
Case I, II and III are much the same (see the upper part of Figure 4A), there are
only small (i.e. <10%) changes in the slope of the curves with a 3-fold change in
the respiration rate. The general conclusion we would draw is that the response of
the normalized rates in the models to variations in q are not seriously aliased by
consequential variations in the overall respiration rate constant and thus one can
proceed with a discussion of the direct effect of q on the modelled rates.
The effect of q is seen to be neither linear nor symmetrical, the plots in Figure 4
show that this is not a simple consequence of scaling due to the boundaries. The
effect of decreasing q is to move the lines towards gross photosynthesis, i.e. the
upper formal boundary. Thus, values of q less than unity (i.e. preferential utilization
of'old' carbon) have little effect on the outcome of the model. In part this is due to
the constraint set by the upper boundary (gross photosynthesis) allowing little scope
for low values of q to have much effect. The effect of values of q above unity is difficult to show in Figure 2, but is clearly seen in Figure 4. The time dependency of the
effect of q is seen in Figure 4; as time progresses, the drift of the F^p curve from
gross to net photosynthesis slows down, constrained by the lower formal boundary.
Within the time scales offieldmeasurements of production (e.g. 12 h), values of q >
3 would have a marked bearing on the interpretation using the F%^, models.
1953
P
9
0.4
6
9
TIME (hr)
6
TIME (hr)
12
Fig. 4. Data used in Figure 2, replolted as (F-N)/(G —N), where N = net photosynthesis, G = gross photosynthesis and F = modelled 14C
net uptake. The vertical dotted lines give the percentage of the intrinsic doubling time (i.e. P'1 In2). (A) Cases I, II and III. (B) Case IV.
For details, see Figure 2. In (A) the lower three (thicker) lines are the outputs from the recycling model (Cases I—III); the upper five sets
of lines are from the respiration model (Cases I—III), shown for values of q = 0.1,0.3,1.0,3 and 10. (B) Case IV only; in this case, the
single thick line refers to the output from the recycling model.
14
C uptake—modeb to account for respiration and recycling
In summary, the two recycling models give the same results, i.e. at high photosynthetic rates the modelled values of net 14C fixation are close to net photosynthesis, as the photosynthetic rates approach that of respiration so the modelled 14C
rates move away in the direction of gross photosynthesis. The simple Hobson et
al. type respiration model at all photosynthetic rates gives 14C rates close to gross
photosynthesis.
The effect of the differential utilization of'new' and 'old' carbon (i.e. Dring and
Jewson's q) is complex because of a secondary effect of the value of the overall
carbon-based respiration constant. An analysis suggests that this effect can be
ignored. The value of q affects net and gross photosynthesis as well as net 14C
uptake. The effect of q is not symmetrical, values of q below unity only having a
minor effect on the outcome of the model. Although the behaviour of the recycling models does seem to accord with observations when P » R (see Richardson et al., 1984; Jespersen, 1994; Williams et al., 1996), the behaviour at low P/R
ratios is less secure. The form used allows a continuous change as photosynthesis
approaches zero, which we would see to be physiologically probable, some supporting evidence does exist from comparisons of 14C and 12C uptake (CalvarioMartinez, 1989). However, there are reports that show increasing apparent PQ
values (i.e. +AO2/-A14CO2) at low irradiances (Andersen and Sand-Jensen, 1980;
Megard et al., 1985). Whether this is due to a change in the true PQ (i.e.
+AO2/-ACO2) or the 14C/12C relationship as the photosynthetic rate approaches
zero is not clear. If it were the latter, then the behaviour of the net 14CO2 uptake
in relation to total CO2 uptake would be quite different from that modelled in the
present work by equations (18) and (20). There is a very evident need to establish
the A14C/A12C relationships at low P/R ratios, especially if we are to attempt to calculate properties such as critical depth in aquatic systems from field measurements of 14C uptake.
Combined respiration and recycling models
The above discussion has not considered the effect of q on the performance of the
recycling model. One can anticipate in a qualitative manner that increasing q will
move the modelled curves in the direction of gross photosynthesis, the reverse of
the effect of q on the respiration model. Thus, with time, the two models would
tend to converge; the greater the value of q the sooner this occurs. Clearly, since
recycling cannot occur without respiration, any model that aspires to describe the
latter part of the progress curves (i.e. 6-12 h, >25% of the intrinsic generation
time) needs to incorporate the combined effects of respiration and recycling. One
could attempt to model the situation as a product of the two simple models [e.g.
as a product of equations (2) and (17) or (21)]. Whereas this might be acceptable
for the initial rates (which is an uninteresting case), it would not for the later parts
of the time course. The reason being that the equations used do not respect mass
balance, leaving a potential for major errors as the extent of recycled 14CO2
becomes substantial. We have come to the view that such a combined model is not
best handled with the type of differential equations used in the present study, but
would be better simulated by a four-box model, based on Figure 1 in Williams
1955
PJJe B.Williams and D.Leffevre
(1993). This approach would also provide a means to circumvent the problem of
q affecting the overall respiration rate constant.
Consequences of the extracellular release of photosynthetate
Algae release dissolved organic material (DOM) and in pure culture (essentially
the problem being considered in the present study) the material may undergo no
further respiration and recycling. This effect has not been written into the models,
but may be anticipated in a general way. Release of DOM will reduce the extent
of respiration of new carbon and as such will have the same effect as reducing the
value of q, i.e. moving the observed 14C rates towards gross photosynthesis. The
effect on the recycling model is likely to be more complex. If, as a consequence of
DOM release, there is less respiration of organic material, then there will be less
CO2 (initially 12C rich) recycled. This will reduce the extent of dilution of the
specific activity of the 14C at the site of CO2 fixation. This will have the effect of
increasing the rate of net 14CO2 fixation and moving the 14C rates towards gross
photosynthesis. The scale of the effect will be dependent on the value of q and
initially the fractional shift will be equal to the fraction of the photosynthetate
released as DOM. All the foregoing argument assumed that the extent of DOM
release is included in the 14C measurement. In order to anticipate the scale of the
effect, separate measurements of particulate and soluble production need to be
made.
Implied strategy for experimental studies
The objective of this study has been to provide explicit models of the physiology
of net isotopic carbon uptake to facilitate the design and analysis of experiments
to establish what processes determine the extent of net 14C accumulation, with
the eventual aim of providing algorithms for the analysis of 14C observations. This
objective has been realized and the results of the study define a fairly clear
experimental strategy and very importantly they also define the experimental
time scales in terms of photosynthetic rates (expressed as intrinsic doubling
times).
In constructing a generic physiological model, we need to know with some
urgency whether we need simply to model the respiration of the fixed material
or the combined effect of respiration and recycling. The first priority is to know
whether recycling of respiratory CO2 is occurring to any great extent under the
conditions used. The study of Lloyd et al. (1977) on algae supported on nets, the
work of Richardson et al. (1984), Jespersen (1994) and Williams et al. (1996) with
cultures, and the view of a number of ecologists that the 14C technique measures
net primary production, would suggest this to be the case. This can be confirmed
by a study of initial rates (e.g. 3-4 h, i.e. <20% of the intrinsic generation time)
where the difference in behaviour between the respiration and the recycling
models is clearest and least confused by uncertainties over the value of q. The
differences in behaviour of the two models can be seen by comparing the time
periods up to 6 h in Figures 2C and D and 3C and D. If over such a period the
1956
14
C uptake—models to account for respiration and recycling
observed initial rates of 14C turn out to be comparable to gross photosynthesis,
then one can conclude that recycling of respiratory CO2 is not important. This
part of the time course gives no information on the importance of losses of 14C
due to respiration, because it has little discernible effect over short time scales.
In this situation, where there is no evidence for recycling, then the later part of
the progress curve (i.e. 6-12 h) would yield information both on the respiration
effect and (assuming the respiration rate was known) on the value of q.
If on the other hand, the rates approach those of net photosynthesis, then one
would conclude that recycling of respiratory CO2 was a major determinant on the
net accumulation of 14C. With the present set of equations, if recycling is occurring to any extent, then it is questionable whether one can gain information on the
physiology from the second part of the progress curve. To do this one would need
to separate the effect of recycling from the combined effects of recycling and respiration, and for this models that combine the two are needed. The behaviour over
long time periods is critical in view of the 12 h incubations for 14C measurements
recommended in the JGOFS protocols.
Conclusions
(i) We have written a consistent set of models that describe the separate effects
of respiration and recycling on the net 14C uptake. The models incorporate
explicitly the formalization of 'new' and 'old' carbon. We have found integrals for
all but one of the equations.
(ii) We have worked with two models to describe the effect of recycled respiratory CO2 on the uptake of 14C tracer. Although the two models are based on
fundamentally different rationalizations of the physiology, they give identical
results. We have found an integral for one of these models,
(iii) We have a very incomplete understanding of how to model recycling at low
P/R values.
(iv) Interpreting the effect of the differential respiration of 'new' and 'old' carbon
(i.e. the value of q) is complex because it not only affects all three processes (gross
and net photosynthesis as well as net 14C uptake), but also the value of q has a
time-dependent secondary effect on the overall carbon-based respiratory constant. An analysis of the consequence of this secondary effect suggests that it is of
minor importance. The primary effect of q is found not to be symmetrical about
unity; values of q above one having a greater effect than those below. To an extent,
but not wholly, this reflects the closer proximity of the q = 1 line to the upper as
opposed to the lower boundary, thus limiting the scope for response to values of
q below one. The consequence is that values of q below unity have very little effect
on the outcome of the model, whereas values of three and above have a pronounced effect.
(v) The rate constant for respiration has a greater effect on the behaviour of the
recycling than the respiration model.
(vi) Although it is obvious that recycling cannot occur without respiration, we
have been unable to produce a model that satisfactorily combines these two
effects. We have come to the view that because the equations used in the present
1957
PJJe B. Williams and D.Lefevre
study are not constrained by mass balance, they are not the appropriate solution
to the problem. A four-compartment box model would seem to offer a better
approach.
(vii) Because we have no model that combines respiration and recycling, we are
unable to establish the quantitative effect of the value of q on the recycling. Qualitatively, we can anticipate that the first-order effect is the reverse of its effect on
respiration, i.e. values of q above one will move the modelled lines towards gross
photosynthesis with time, thus causing the predicted lines of the respiration and
the combined respiration/recycling to converge with time,
(viii) The consequence of the above is that if recycling occurs to any extent (and
the signs are that it does), then uncertainties as to how to model the combined
effect of respiration and recycling, and a lack of information on the value of q,
mean that the interpretation of incubations longer than 30-50% of the intrinsic
generation time is uncertain. This is probably of little practical consequence when
photosyntheticrates approach Pmax,for then P» R and net and gross photosynthesis will have much the same value. The problems of interpretation become
serious when photosynthesis approaches and becomes less than respiration, i.e. at
and below the photosynthetic compensation point. The consequence is that we are
presently left with a very incomplete understanding of how to interpret 14C-determined rates for calculations of critical depth.
(ix) The models have been used to define a strategy to develop experiments to
establish the separate effects of respiration and recycling on the time course of net
14
C uptake. The initial rates give the clearest resolution of the two processes. It
would appear that with photosynthetic rate constants in the region of 1 day 1 , incubation periods of up to 3-6 h would be suitable to determine the importance of
recycling in controlling net 14C uptake. With the present models, only in the
absence of recycling can one study the effect of respiration and determine the value
of q.
Acknowledgements
D.L. was supported on MAST II project MAS2-CT92-0031 (MEICE) awarded to
B.Riemann, F.Thingstad and P.J.LW. We would like to thank Dr Ian Joint for
helpful discussions over the manuscript, Dr Paul Tett for his suggestion for the
form of equation (19) and we are indebted to Dr Alan Davies for confirming our
integrations. During the period the paper was prepared, PJ.LW. was the recipient
of a Higher Educational Funding Council for Wales Professorship.
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Received on January 30,1996; accepted on June 7,1996
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