Transport limited nutrient uptake rates in Ditylum
brightweZW
Walter J. Pasciak2 and Jerome Gads
Department
Maryland
of Geography
21218
and Environrnental
Engineering,
The Johns Hopkins
University,
Baltimore,
Abstract
The analysis of diffusional
transport limitation
of nutrient uptake rates by phytoplankton,
derived earlier for spherical cells, is extended to include,
in an approximate
manner,
cylindrical
and disk-shaped cells. Transport limitation
of nntrient (NOs and NO;)
uptake
rates by the marine diatom, Ditylum
brightwdlii,
was demonstrated
experimentally
at low
nutrient concentrations
in quiescent media. The effect of transport limitation
was decreased
by mixing and elirninated completely when the organism was cultured in a medium being
sheared at a high enough rate. It also disappeared at high enough nutrient concentrations.
In a previous article we described quantitatively how the rate of uptake of a nutrient by a single-celled,
spherical phytoplankter can be limited by the inability of
the nutrient to diffuse to the cell as rapidly
as the cell can assimilate the nutrient at its
ambient concentration. We presented plots
of the nutrient uptake rate as a function of
ambient nutrient concentration for different values of a parameter of the cell relative to the particular nutrient (Pasciak and
Gavis 1974). The parameter, a dimensionless number, has the form
P = 14.4~ RDK/V,,
(1)
where R ( cm) is the cell radius, K ( PM )
and V, ( pmole cell-l h-l) are the Michaelis-Menten constants of the cell for a particular nutrient, D ( cm2 s-l) is the nutrient
diffusivity, and 14.4 is a constant necessary
to maintain consistent units. Small values
of P indicate that diffusional transport will
limit the uptake rate of the nutrient when
its concentration is low.
Our purpose here is to extend the analysis to nonspherical cells and to describe an
experimental
investigation
in which we
demonstrated
that diffusional
transport
does limit the rate of uptake of nutrient by
an organism that has a low value of P.
1 This work was supported by NSF grant GA36282.
Reactor
’ Present address : Office of Nuclear
Regulation,
U.S. Nuclear Regulatory
Commission,
Washington,
D.C.
20555.
LIMNOLOGY
AND
OCEANOGRAPIIY
Application
to nonspherical
cells
Many phytoplankton
are spherical or
may be considered approximately spherical
without introducing
serious error. Others
are far enough from spherical that application of the results derived for spherical
cells to them gives erroneous descriptions
of transport limited uptake rates. We tabulatcd values of P in our previous article
for several nonspherical
cells, assuming
that half of their major dimension was
equivalent to a spherical radius. Solution
of the equations for transport limited nutrient uptake, like those given in our previous article, for nonspherical cells should
yield results that differ from those for
spherical cells by constant, dimensionless
shape factors, smaller than unity, that depend on cell geometry. Thus, our tabulated
values of P are erroneously high for those
organisms, as would be the corresponding
transport limited uptake rates.
It is unfortunate that the equations cannot be solved in terms of known functions
for most cell geometries. They can be
solved, however, for prolate and oblate
spheroids that absorb nutrient uniformly
on their surfaces. A prolate spheroid (a
sphere elongated at the poles, squeezed at
the equator) of high eccentricity resembles
a cylinder, and an oblate spheroid (a
sphere squeezed at the poles, bulging at
the equator) of high eccentricity resembles
a disk. Thus if we can obtain shape factors
for prolate and oblate spheroids we can ex604
JULY
1975,
V.
20(4)
Transport
Table 1.
Prolate
Solution
limited
of corresponding
electrostatic
-.-_ ___ _ _._--------
605
uptake rates
problem.
-----.
. _.-.- - =---=__=.=--.- - z : ==_z-zi-7 -_.-_
-- -
spheroid
Oblate
spheroid
$I E potential
where
q z charge
(volts)
,
(coulombs)
,
c E permittivity
R
R2 z major
of
the
and minor
(s/ohm/cm)
medium
axes
of
spheroid,
,
respectively,
1’
and EJ is
roid
a spheroidal
parameter
defined
for
any point,
(a,
b,
c),
external
to
the
prolate
sphe-
by
-a-_2
+ b2-t 5’
5 + R2
1
and to
the
oblate
spheroid
a?+ b'
=
.
2
+
C; + R2
1
from
,
2
by
---..
When F2 -* ~0 , far
= 1
?; +R2
a spheroid,
C/J+ 0 .
c--2
5 = 0 is
tend our results to cells whose shapes may
be approximated by cylinders and disks.
The problem of diffusion to a uniformly
absorbing spheroid immersed in an infinite
medium in which the concentration of the
diffusing nutrient being absorbed is given
is formally identical to the problem of the
potential external to a conducting spheroid
of given total charge immersed in a dielectric medium. The solution to the latter
problem is well known and is given, for
example, by Stratton ( 1941). We can
adapt it to the diffusion problem as follows.
The potential, +, is given by the equa-
1
5 + R2
the
surface
of
a spheroid
.
tions of Table 1 for prolate and oblate
spheroids. In the diffusion problem the potential, 4, corresponds to the concentration,
c (PM); the c h arge, 4, corresponds to the
mass transfer rate, Q ( pmole h-l) ; and the
medium pcrmittivity,
E, corresponds to the
diffusivity,
D. This may be seen on comparison, in pairs, of the electrostatic equations and the diffusion equations listed in
Table 2.
Then, the concentration, c, is given by
Eq. 2 and 3 (Table 3). Because it is necessary to express concentrations relative to
C, the concentration at large distances from
Pasciak and Gavis
606
Comparison
Table 2.
of electrostatic
and diffusion
Electrostatic
-
equations.
Diffusion
. --- --- --- - --- - - - - -- -- -. ._I_-
PC/J = 0
PC
if = - vcl,
ds
q = - t: D/c).
tial,
two equations
@, or
the
operator).
substance,
are
Gauss'
surface
are
J,
theorem
across
area).
equivalent
Laplace's
concentration
The second
ing
substance
are
two are
(both
vector
of
electrostatics,
a surface
The last
to
is
the
two
equation
n- ds
for
definitions
are
for
$I and c from
V is
unit
normal
the
third
pair
Table
the
the
electric
vector
with
1;
the
(V”
gradient
to
the
pairs
electrostatic
represents
E,
rate
and the
of
of
the
represent
Laplacian
flux
of
The third
transport
ds is
poten-
the
operator).
a surface;
substitution
other
the
field,
for
vector
which
position
expression
the
9 in
of
of
and the
ds
Q = - D /Vc 9 i- ds
as a function
quantities;
(n is
= 0
Q = /Jo;
equation
found
the
..- __-_-._--_-_._
J = - DVc
q = i: 1r.n
The first
- -_ _.- ---_---.
of
the
unit
second
pair.
steps
in
diffustwo
diffusing
element
of
The last
their
deri-
vation.
.- -- - -_ ---___
I_--
_-_____--I__-__----.----___I
a cell, instead of relative to zero at large
distances as in the electrostatic case, we
subtracted c from C in the left-hand sides
of the equations. We have also divided
the right-hand side by 3.6 [ (3,600 s h-’ )/
( 1,000 cm3 liter-l) ] to maintain consistent
units.
When we introduce the eccentricity, e,
dcfincd by Eq. 4, Eq. 2 and 3 become 5
and 6 at the spheroid surfaces, [ = 0. The
surface concentration, Co, differs from C
since, when transport limited, a cell creates
a region around it in which the nutrient
concentration is depleted, with the lowest
concentration at its surface.
In order to determine C we may equate
Q, obtained by rearrangement of Eq. 5 or
6, to V in the Michaelis-Menten
cxprcssion
based on concentration at the cell surface,
v = VmCcl/( K + Co),
(7)
for the transport limited uptake rate of a
cell, This yields Eq. 8 and 9 (Table 4) on
- ----
-------
---___
rearrangement. These are quadratic equations by which Co can be found as a Function of any given C and of known other parame ters.
It is, however, more convenient to introduce the reduced, dimensionless concentrations, C” and Co* by means of Eq. 10 and
11. Then 8 and 9 reduce to Eq. 12 with P
redefined by 13 and a, a shape factor, defined for prolate spheroids by Eq. 14 and
for oblate spheroids by 15. Because Eq.
12 is identical to 10 of our previous article,
Eq. 14 and 15 define the desired shape factors. We should note, however, that the
shape factors arc not to be applied to the
parameter P’ derived, as described in our
previous article, for motile organisms. The
term by which P is multiplied to give P’
was derived for spherical organisms and
should be applied only to such organisms.
The shape factors may, however, be applied to the factor F, derived by Gavis and
Transport
Table 3.
Solution
for the tmnsport
limited
607
uptake rates
problem.
Prolate
Oblate
(3)
Eccentricity
(4)
With
equation
4 equations
2 and 3 become,
at
5 = 0
,
(5)
C
- Co = i&flR~
Ferguson ( 1975)) for transport limited CO2
uptake by phytoplankton.
We have plotted a for prolate and oblatc
spheroids from Eq. 14 and 15 as a function
of e in Fig. 1. When e = 0 both spheroids
are spheres and a = 1. The shape factors
decrease monotonically with increasing eccentricity.
That for the prolate spheroid
decreases rapidly toward zero as ecccntricity approaches unity and the spheroid degenerates into an infinitesimally
thin cylinder. The oblatc spheroid degenerates into
an infinitesimally
thin disk at e = 1, and a
has the limiting value 2/7c ( = 0.64).
As an illustration
of the application of
shape factors we may use the example of
Ditylum brightwellii.
This is an approximately cylindrical cell with an almost triangular cross section, about 150 p long and
about 50 ,X in diameter. Thus R1 = 75 p
and R2 = 25 E-C,so that e = 0.94, and a =
0.55.
tan-l
(6)
__-
We had previously
calculated P = 2.4
for nitrogen uptake by this organism, based
on R1 = 75 p”, K = 0.6 PM, V,, = 12 X lOA
pmole h-’ cell-‘, and D = 1.5 X lo-‘) cm2 s-l.
The value of P for a = 0.55 should be 1.3.
Thus D. brightwellii
is more influenced by
transport limitation than we had originally
indicated. It should be understood, how-
dol,
01
02
03
0.4,
05,
06
0.7,
06,
0.9
IO1
ECCENTRICITY
I?&. 1. Shape factors for prolate
spheroids as a function of eccentricity.
and oblate
608
Pasciak and Gavis
Table 4.
Derivation
of the shape factors.
- - - L=GI=?77 177: _Y_=Z.-72 =.=z
Prolate
“m c:+(--------28.3
vDRe
In
lte+K
l-e
-C)Co-
KC
=0
(8)
.
Oblate
-1
+K-
VI "e2
Reduced
C ) Co - KC
(9)
concentrations
C* I: C /K
(‘0)
C; : Co/K
Reduced
=0 .
(‘1)
.
quadratic
Co*’ + (l/P
t
1 - c*)
c;
- c*
= 0
(12)
.
where
P G (14.4
ITDRK/V,,,)
a
(13)
(Prolate)
(14)
and
2e
a = ri--;
1 - e
a : -----ce--tan-'
(Oblate)
_______.___.-._.-__-______.--------------------ever, that, because a prolate spheroid
whose e = 0.94 does not perfectly describe
the shape of this organism, this value of P
still only approximates the true value, although it is a better approximation
than
the one based on the spherical cell.
Corrected P values for the remaining
nonspherical organisms listed in our earlier paper are given in Table 5. In all cases,
cell shapes were approximated by appropriate prolate or oblate spheroids.
Recause shape factors are less than unity,
transport limitation affects uptake rates of
nonspherical cells more than those of spherical cells of the same volume and the same
values of K/Vm.
.
(15)
n-G?-
---
Experimental
The organism-We
chose D. brightwellii
as the organism to be investigated because
it should exhibit transport limited uptake
rates at low concentrations of NOS- and
NOa-, because it is easily cultured, and because its NO3- and NOa- uptake rates have
been extensively
investigated
previously
(Eppley and Coatsworth 1968; Eppley et
al. 1969). The original inoculum was obtained from the Food Chain Research
Group at Scripps Institution
of Oceanography. In culture it grew almost entirely
as single, nonmotile, roughly cylindrical
cells with triangular cross section, about
150 p long and 50 p in diameter.
Transport
Table 5.
:_7 _ 7
Organism
limited
P and I?’ values for nonspherical organism.
: -z;_:z.
-.-:=_=.=z-_._._. z=.
--z-Y-.
Nutrient
NO;
NO,
Original
P
609
uptake rates
1.__
Shape
i__l.
Eccentricity
_.
--__._____:
_ -__.- . .._- LY$__
Shape
factor
Correct
P
2.4
Prolate
0.94
0.54
1.3
12.0
Prolate
0.94
0.54
6.5
Prolate
0.97
0.46
1
0.64
0.34
220
(disk)
101
NO;
0.53
Oblate
NO;
1.2
Prolate
0.97
0.46
0.55
NO;
3.8
Prolate
0.87
0.65
2.5
NO,
0.51
Oblate
(disk)
0.64
0.33
Oblate
(disk)
0.64
NO,
50
Cultures-The
culture
medium
was
made up of a mixture of half seawater and
half 2.5% sodium chloride solution, supplemented with .trace metals and vitamins according to the IMR formula (Eppley et al.
1967). It was filtered through a Whatman
glass fiber filter, type GF/C, and autoclaved for 40 min at 2 atm in e-liter volumes in 2.Sliter Fernbach culture flasks.
When cooled, the medium was enriched to
100 PM KHZP04 and 25 ,uM KN03 by addition of sterile 10 mM KHgP04 and 2.5
mM KN03 solutions.
Stock cultures were prepared by inoculation of the e-liter volumes of sterile enriched culture medium with 100-300 ml of
nitrate starved culture in the plateau
growth phase. These were allowed to grow
into the plateau growth phase upon consumption of all the NOa-. Stock cultures
were never left in the plateau growth phase
for more than 3 days, however, to avoid excessive bacterial growth. Before the 3-day
period was over the stock cultures were renewed by repetition of this procedure.
The “working cultures” in the experiments were prepared from the stock cultures in the same way as the stock cultures
were prepared.
The organisms were cultured and the
experiments were carried out in a 18-20°C
constant temperature room. Both stock
32
and working
cultures were illuminated
with “cool-white”
fluorescent lighting on
a 12/12-h light-dark cycle with an intensity
of 0.007 ly min-l (3,300 lux).
Analyses-In
most of the experiments
NOg- was the limiting
nutrient; in the
others it was NOa-. We used the B (sensitive) procedure ( Strickland and Parsons
1972) for analysis of N03- and the same
method for analysis of NOa-, omitting the
reducing step.
We performed the analyses with a Technicon Au toAnalyzer, determining
concentrations by comparison with percent light
transmittance from standard solutions of
known concentration.
The correlation coefficient for data used to prepare a standard curve of concentration against percent
light transmittance was at least 0.999.
We counted cells of working cultures
with a Palmer-Maloney counting chamber;
cell counts ranged between 700 and 1,000
cells ml-l in the experiments.
Flask experiments-In
two sets of experiments we measured the rates of NOs- uptake by cells in quiescent media in stationary flasks and in agitated media in flasks
placed on a shaker table with a lo-cm
movement at 80 cycles min-l. In both sets
increasing amounts of a 250 PM KNOs solution and decreasing amounts of deionized water were added to a series of six-
610
Pasciak and Gavis
I
Fig. 2. Cross section of 8 culture vessel used
for producing controlled shear in growth medium.
teen 250-ml flasks in amounts such that
when 100 ml of working culture was added
to each flask the NO:%-concentration ranged
from 0 to 7.5 PM in the series, The deionized water was added in such amounts
that the total volume was identical in all
the flasks.
The NOa- concentration in the working
culture was monitored periodically
before
the start of the experiment. Immediately
upon, or slightly before, depletion of NORthe experiment was started when 100 ml of
working culture was added to each of the
sixteen flasks. The procedure was done in
this way to avoid the lag phase that nitrate
starved cells of D. hrighttoellii exhibit (Eppley et al. 1969). The sixteen flasks were
divided into groups of four. As soon as the
culture was added to the first four flasks
they were shaken, and an initial 15-ml
sample was taken from each and poured
into a 16-ml centrifuge tube. The inoculatcd flasks were placed on an illuminated
table, at an intensity of 0.018 ly min-l
(9,000 lux). The centrifuge tubes were
then spun at 2,000 g for 1 min, and the supcrnatant was poured into a sample cup
and put aside for analysis of NOa-. The entire process was repeated with the next four
flasks, and so on, until culture was added
to and samples taken from all sixteen. The
interval between each succeeding group of
four flasks was 4 min. A second and final
15-ml sample was taken from each flask
after 30 min and processed as described for
the initial samples. The final samples were
taken from each group of four flasks at 4min intervals to match the intervals between the initial
samples, Cells were
counted in the working culture near the
midpoint of each experiment. It was not
necessary to count more often, because in
the 30-min duration of each experiment
the increase in cells was less than the error
inherent in making the counts.
Shear field experimnts-We
also did experiments in a controlled shear Eield in
which the rate of shear was kept temporally
constant and approximately
spatially uniform. We did these in part to ascertain at
what shear rate .transport limitation
was
overcome by motion in this nonspherical
cell and in part to have a second method of
investigating transport limited uptake rates.
In the flask experiments we could study
uptake rates at several NO:s- concentrations
simultaneously,
lessening time-dependent
systematic variation that may bc caused by
properties, like V,,, that could change with
time. In the shear field experiments we
could directly compare uptake rates between quiescent and moving cells, but, because of equipment limitations, could not
study uptake rates or different NOy- conccntrations simultaneously.
Experiments in the shear field were performed with an apparatus designed to allow control over the shear rate. Seven culture vessels were constructed, each of two
concentric Plexiglas cylinders with the inner one covered and sealed at the top and
bottom, the outer one covered and sealed
at the bottom only (Fig. 2). The outer
cylinder was placed on a turntable while
the inner one was held fast by a clamp on
its center shaft. Culture medium (400 ml)
was placed between the two cylinders of
each vessel, and the outer one was rotated
at a speed necessary to give the desired
shear rate in the medium through control
of the turntabIc speed.
Trunsport
limited
The shear rate (s-l) is approximately the
product of the outer cylinder radius and its
angular speed divided by the width of the
gap between the cylinders. For our apparatus the shear rate is 0.5 times the rotational speed in rpm. Although the shear
rate across the gap between two concentric
rotating cylinders of the sizes we used is
not entirely uniform, the variation across
the gap is small enough that we may assume the shear rate to be approximately
uniform. Taylor (1923) showed that vortex flow will be avoided if the inner cylinder is held stationary and the outer one is
turned, whereas turning the inner cylinder and holding the outer one stationary
produces instabilities,
even at low rpm.
Ry placing a culture of large visible cells
in the vessel we could observe the flow
field. Under all experimental conditions
the flow appeared uniform and nonturbulent.
Six of the turntables and their culture
vessels were mounted side-by-side on a
specially constructed stand on which they
could be turned separately or together by
pulleys and V-belts powered by an adjustable-speed DC motor. Two motors drove
three turntables each. Any desired combination of turntable
speeds could be
achieved with different combinations
of
pulley sizes and motor speeds. The turning
speeds ranged between 0 and 30 rpm, depending on the experiment. The seventh
vessel was not turned and served as a control. The vessels were illuminated
with
“cool-white”
fluorescent lighting of 0.018
ly min-l (9,000 lux) intensity,
Two different types of experiments were
done with this “controlled-shear”
apparatus. In one the uptake in two vessels turning at a shear rate of 13.5 s-l was compared
with uptake in two stationary vcsscls. Some
experiments were performed in duplicate,
where the initial concentration was always
the same in all four vessels, while in other
experiments one pair of vessels had a diffcrent initial concentration from the other
pair. The latter enabled the collection of
twice as many data points in the same pcriod. In the second type of experiment up-
uptake rates
611
take was measured in six vessels, each
turning at a different speed, and compared
with uptake in one stationary vessel.
Expcrimcnts in the controlled-shear apparatus were conducted as follows: A nitrogen depicted working culture that had
been held in the NOa- depleted, plateau
growth stage for less than 1 day was spiked
with either NOs- or NOZ- to a concentration between 5 and 7 ,uM and set aside for
l-3 h for preincubation.
In the first type
of expcrimcnt the period of preincubation
was until the nutrient concentration had
decreased l-3 PM from the initial value.
The culture was then distributed
among
the vessels. An initial sample was taken
from one of the vessels for nutrient analysis, and the turntables were started; 30-80
min later a final sample was taken from
each of the vcsscls. As before, the samples
were prepared for nutrient analysis by centrifugation.
The culture was then collected from the
vessels, mixed together to ensure homogeneity, and redistributed
among them. An
initial sample was taken from one of the
culture vessels, and later again a final sample from each of them. The cultures were
again combined, mixed, and redistributed.
This entire process was repeated until the
nutrient was completely depleted.
In the second type of expcrimcnt the
working cul turc, spiked with NOs-, was
preincubatcd
until the nitrate concentration was ca. 2 ,uM. The rest of the procedure
was similar to that described above except
that after the working culture was redistributcd among the vessels it was respiked
with NOs- so that its concentration was
equal to that when the cultures were first
distributed
among the vessels (about 2
,uM); in addition, the turntable speeds were
changed between runs. The procedure was
repeated until a sufficient number of data
points wcrc collected.
Results
Single reciprocal plots-Reciprocal
of the Michaelis-Menten
expression
V=V,,,C/(K+C)
forms
(16)
612
Pasciak and Gavis
I
I
0
1 I
1
2
3
4
I
5
A
6
7
C (PM)
-
I
0
I I I
I
2
3
4
I
5
B
6
7
C (PM)
Fig. 3. Single reciprocal plot of nitrate uptake
by Ditylum brightwdii
under quiescent conditions.
A. K = 0.7 PM and V,,%= 3.7 x 10” pmole cell-l
h-l. B. K = 0.7 PM and V, = 3.2 x 10-O pmole
cell” h-l.
are most conveniently used to ascertain Vvn
and K from data. According to Dowd and
Riggs (1965) the best method is based on
rearrangement of Eq. 16 into the form
c/v
= (l/Vm)C
+ K/Vm.
(17)
Then if the data are plotted as C/V against
C, the result is a straight line of positive
slope equal to l/V, that crosses the negative C axis at K and the C/V axis at K/V,
when cells grow at rates given by Eq. 16.
0
I
I
2
I
3
4
C (PM)
5
6
7
Fig. 4. Single reciprocal plot of nitrate uptake
by Ditylum
br-ightwellii
consuming nutrient
in a
medium
undergoing
irregular
mixing.
K = 0.7
PM and V,, = 5.4 x lo-’ pmole cell-l h-l.
If the uptake rate does not obey Michaelis-Menten
kinetics the line becomes a
curve. In particular, when transport limitation occurs and P is small, the line curves
upward as it descends toward lower values
of C, crossing the C/V axis at a point higher
than that at which the line would cross for
a nontransport limited organism with the
same values of K and V,n. Examples of
such plots are those of Figs. 3 and 4.
Because transport limitation
of uptake
rate decreases with increasing nutrient
concentration
and becomes negligible at
high concentrations, the data at higher concentrations should fall along the straight
line. Extrapolation of that line to the concentration axis enables calculation of V,n
and K for organisms that are transport limited at lower concentrations.
When C, the ambient nutrient concentration, is very small, approaching, zero, we
may derive simply from Eq. 12 (Table 4)
that the ratio of Co, the nutrient concentration at the cell surface, to C is
co/c
= P/( P + 1).
08)
Substituting Co in terms of C from Eq. 18
into 7 we get, when C + 0,
c
-=V
- P
VT?% ( P+1 >-F
or
c-+0,
(19)
Transport
limited
Hence, transport limitation reduces the uptake rate by the factor P/ ( P -t- 1) in this
limit; the intercept of the curve on the C/V
axis of a reciprocal plot of C/V against C
is (P + 1)/P higher than the intercept of
the straight line extrapolated from the data
at higher concentrations.
Conversely, P is given by
P=p/(l-PI,
(21)
where p is the ratio of the intcrccpts on the
C/V axis of the straight line and of the
curve for transport limited uptake.
Uptake in nonagitated flasks-Figure
3
shows reciprocal plots made from data for
under
uptake of Non- by D. brightwellii
nonmixed conditions. Experimental conditions were the same for both A and 13.
The uptake rate for each point is the difference between two successive measured
concentrations in an experiment divided
by the time interval between mcasuremcnts and by the cell count. The concentration against which each point is plotted
is the average concentration between the
measurements. Each point represents approximately an average uptake rate at its
concentration,
The asymptotes in Fig. 3A and B were
drawn through the data points at high concentrations and extended to low concentrations. It is from them that the values of
K and V, are determined.
For Fig. 3A,
K = 0.7 PM and VW = 3.7 X 1Om6 pmole
cell-’ h-l, and for Fig. 3B, K = 0.7 PM and
v, = 3.2 X lOL6 pmole cell-l h-l.
The values of P, determined from these,
are 0.51 for the data of Fig. 3A and 0.60
for the data of Fig. 3B for D = 1.5 X 10e6
cm2 s-l, and the organism properties R =
75 p and e = 0.94 listed in Table 1. The
lower curves of both figures are plots of C
times the inverse of Eq. 7:
WV = C(K + co)/cov7nlc
(22)
with Co given as a function of C by Eq. 12
(Table 4) for these values of P. This,
rather than Eq. 17, relates measured C/V
values to measured C values, when transport limitation occurs. The data are better
fit by curves for lower values of P.
613
uptake rates
0
2
4
SHEAR
6
8
RATE
IO
I2
14
(s-1)
Fig. 5. Relationship
between the relative uptake rate and shear rate for NO,- uptake by
Ditylum brightwellii.
The different symbols represent different
successive runs with the same working culture.
Uptake in agitated flasks-Figure
4
shows a reciprocal plot made from data for
underuptake of NOs- by D. brightwellii
going irregular mixing. The points were
plotted as described for Fig. 3. The
straight line was again drawn through the
points at high concentration
rather than
through all the data points, for two reasons.
First, the value of K obtained is 0.7 PM,
which is the same as that from Fig. 3. If
the straight line were drawn through all
the data points, the K value would have
been about 1.5 ,uM. Second, it was not
known a priori if the mixing was sufficient
to remove fully the effect of diffusion limitation.
From the straight line the value of K is
0.7 ,uM and that of V,. is 5.4 X 10dGpmole
cell-l 11-l. The value of P is 0.35. The
curved line represents the solution of Eq.
22 for this P value. It is an estimate of what
the uptake rate of the cells would be as a
function of nutrient concentration if they
were consuming nutrient under quiescent
conditions. It also illustrates the effect of
mixing in changing the shape of the curve.
Uptake in the controlled-shear apparatus
-It is convenient, in plotting data from the
controlled-shear
apparatus, to express the
ordinates as a ratio of uptake rate at a given
shear ra tc, VEchcar,
to that at zero shear rate
in a stationary cylinder, V,,,, shear( Fig. 5))
or as the reciprocal of that ratio (Fig. 6).
There are two reasons for doing this.
The first has to do with the variability of
V,, a phenomenon described by Dugdale
(1967) and discussed further by Eppley
614
Pasciak and Gavis
nor to count cells. Because the ordinate is
a ratio of uptake rates, cell counts and time
intervals cancel out; they are the same for
numerator and denominator.
Figure 5 is a plot of rate of uptake of
NOS- by D. brightzuellii as a function of
shear rate determined with the controlledshear apparatus. The ordinate shows the
uptake rate at the various shear rates divided by the uptake rate at zero shear rate;
0
1.0
2.0
30
40
50
the abscissa shows shear rate. Each value
of
VsllcXzlrand of V,,,, Sllc,nrwas determined
C (WM)
as the average between the initial and final
measurements of concentration in an experiment.
Figure 6A shows data for NOs- uptake
by D. brightweZlii in the controlled-shear
apparatus as a function of concentration.
Each point represents the fraction that the
* 0
uptake rate was decreased at any nutrient
:
:: Cu
concentration in a stationary cylinder com>
0
pared to that in cylinders in which the
0,
shear rate was maintained at 13.5 s-l, large
0
1.0 20
30
40
50
60
7.0
8.0 enough to eliminate transport limitation
C (pM)
completely.
The curved lines represent
Fig. 6. A. Nitrate uptake by Ditylum
btightmathematical solutions of Eq. 7 divided by
w&i
in the controlled-shear
apparatus.
R. Nitrite
Eq. 16, with Co given by 12 (Table 4), as
uptake by D. brightwdii
in the controlled-shear
a function of C for K = 0.7 PM, and for the
apparatus.
Each point represents the fraction upindicated P values, The intercept at C = 0
take was decreased in the stationary vessels comnared to the moving vessels. The different symbols
is easily shown to be qua1 to P/( P + 1).
rcprcscnt
experiments
performed
on succeeding
Most of the data points fall between the
days with different
working cultures.
lines P = 0.7 and P = 1.5.
Figure 613 is a similar plot for NOz- uptake, The curved lines represent solutions,
et al. (1969). B ecause the experiments in
the controlled-shear
apparatus lasted for
noted above, for a value of K = 4 PM de3-5 h, time variation of V,, has to be con- termined by Epplcy and Coatsworth (1968)
sidercd. A change in V,, oE a factor of 2, and the indicated values of P. Most of the
say, changes P by a factor of 2. If the data data points fall between the lines P = 1.5
were plotted in reciprocal form, it would
and P = 4.0.
bc difficult
to distinguish
the effect of
transport limitation from the efEect of a Discussion
char& of Vqjz.during the experiment since
The data show conclusively that transthe latter would cause large shifts in the
port
limitation of the rates of N03- and
positions of plot ted points, When plotted
NOauptake by D. brightwellii
does occur
as a ratio of uptake rates, differences in
at
low
concentrations
and
that
when the
v,n are similar for numerator and dcnomiorganism
is
placed
in
a
sufficiently
high
nator, and the differences do not cause as
shear
field
transport
limitation
is
overcome.
much change in the positions of points as
Figure 3 depicts NOs- uptake data for
in the reciprocal plots.
D.
brightzoellii from two identical experiThe other reason is that there is no riced
to measure time intervals between samples, ments for cells stationary with respect to
B
Transport
limited
their surrounding medium. There is a discrepancy between the values of P determined for these data, as shown in the figure, and the value 1.3, determined from
previously published data. The discrcpancy is due to the difference between the
values of V, determined in these cxpcrimcnts and the value of V, determined by
Epplcy and Coatsworth ( 1968). Our values of V,, are almost three times as large
as the value they determined.
In both Fig. 3A and B the curves gencrated from the equations with the P values
( corrected for shape), determined from
the measured values of V,, and K, are lower
at low concentration than the experimental
points. Better fits to the data are obtained
with lower values of P: 0.31 in Fig. 3A
and 0.33 in Fig. 3B.
There are several possible explanations
for why the data fall on a curve with a
lower value of P than is predicted.
The
first is that the equations we have developed are based on the assumption that absorbing sites are uniformly distributed over
the entire cell surface. In cells like D.
brightzveZZii that are elongated, however,
we can imagine that the absorbing sites
are located around the surface on the long
axis of the cell and not at the ends. If the
sites are not uniformly distributed the values of P should be smaller than if they
were. The equations defining P must then
have additional shape factors to correct
for the nonuniform distribution of absorbing sites on the cell surface.
A second explanation is that the values
of P calculated from K and VPjzfor a prolate spheroidal ccl1 ovcrestimatc the true
values for a cell that is only approximated
by a prolate spheroid. If this is so, then
determination of P by curve-fitting through
data, as was done in the case of the higher
curves of Fig. 3, gives truer values than
those determined from K and V,,, as in the
lower curves.
Another explanation may be that not all
cells in a culture absorb nutrient at the
same rate. Some cells, in a healthy culture
at nontransport
limiting nutrient concentrations, absorb nutrient at a slower rate
uptake rates
615
than average. Many of these may be in less
than perfect condition.
As nutrient concentrations fall to transport limiting lcvcls,
these cells may become so debilitated that
their absorption rate drops off to a smaller
fraction of their original uptake rate than
does that of healthier cells. The average
uptake rate for the culture may decrease
more than would be observed if only
healthy cells were transport limited. Data
points on Fig. 3 would then fall higher
than they otherwise should, and the P value
of the curve that fits the data would be
lower than expcctcd from high nutrient
concentration data. Moreover, uptake rates
should decline for this reason to abnormally
low values as nutrient concentration
declines to very low levels regardless of
whether the cells arc transport limited.
Deviations from the straight line on a reciprocal plot should be seen in such culturcs at low concentrations.
Specifically,
they should occur for cultures in well
mixed media.
Figure 4 depicts NOs- uptake in D.
brightzoellii obtained from an experiment
in which the cultures were irregularly
mixed on a shaker table. In Fig. 3 the data
points were above the curved line generated from Eq. 22 with the P values (0.51
in Fig. 3A and 0.60 in 3B) obtained by
means of 13 (Table 4). IIowever, in Fig.
4 this is not so: the data points fall below
the curved line. If the cells were consuming nutrient in quiescent conditions the
data points would have been on or above
the curved line, as in Fig. 3. This illustrates
the effect mixing has on increasing the UPtake rate for diffusion limited cells. An cxplanation for why some points fell above
the line is that, as noted above, some of the
cells became debilitated
at low nutrient
concentrations and ceased to absorb nutricnt. An alternative explanation, however,
is that the medium was not completely or
well enough mixed.
The standard deviations of V,,, determined from the data in Figs. 3 and 4 are
estimated at 7, 6, and 10% of the respective
V,, values. These were determined from ’
the data points at high concentration in the
616
Pas&k
figures. For each datum point located beyond 3.5 PM, the slope of the line between
the point and the point (0.7 PM, 0) was
detcrmincd.
From these slopes values of
V, were determined, and the standard deviation of V, was computed. This was
done for each figure. IIowever, these standard deviations represent only the scatter
of the data points. The accuracy of the
value of V, is limited also by the accuracy
by which the cell concentration was determined. This can be seen as follows: The
value of the ordinate of each point is computed by means of
c/v=
(CI+C@A~/~(C~-C~),
where C1 and Ca are the initial and final
nutrient concentrations, 2 (cells liter-l) is
the cell concentration, and At (h) is the
time between the initial and final measuremcnts.
At high concentrations the value of ( C1
- C, ) is weakly dependent on concentration; hence, the slope of the data points is
proportional to At times the cell concentraThen,
tion. Errors in At are negligible.
since V,, is determined from the reciprocal
slope, the accuracy of the determination of
V, can be no better than the accuracy of
the determination of the cell concentration.
The standard deviations of the measurements of the cell concentrations were 33,
19, and 27% of the average value in the experimcnts depicted by Figs. 3 and 4. If
both standard deviations are simply added
the sums are 40, 25, and 37%.
Errors of this magnitude may be invoked
as a fourth explanation of the discrepancies
between data and determined curves in
Fig. 3 and for the difference in V, values
of the data in Fig. 4 and of the data in Fig.
3.
It is more difficult to obtain an accurate
value of K than of V, for D. brightwellii.
Epplcy et al. ( 1969) determined a value of
of K of 0.6 PM with a 1.7 PM spread at the
95% confidence level. The effect of diffusion may be in part responsible for their
large spread, since they wcrc not aware of
its effect on single reciprocal plots. If a
straight line were fitted to all the data of
and Gavis
plots like those of Fig. 3, the value of K
would be overestimated ( about 2.5 PM),
In fact, in an earlier publication,
Eppley
and Coatsworth ( 1968) determined a value
of K of 2.1 PM.
Another difficulty in measuring K for D.
brightwellii
is that its value is very small.
This requires that the data points at high
concentrations have very little scatter, In
Figs. 3 and 4, straight lines could have
been drawn through the points at high concentration
to give slightly
smaller or
slightly larger values of K. Of them, Fig.
3A would show the least spread in K. Values of K between 1.2 PM and 0.5 PM
could have been obtained. In Fig. 3B a
value as large as 1.5 PM and as small as 0.1
,uM could have been obtained. In Fig. 4,
if a straight line were drawn through all of
the data points, the value of K would be
about 1.5 PM. As previously noted, part of
the reason for not drawing the straight line
in Fig. 4 through all of the data points was
to have the K values of Figs. 3 and 4 agree.
The value of 0.7 PM gives a good fit for
the data in all of the figures, and it is also
close to the value determined by Eppley
et al. ( 1969).
The values of P that fit the controlledshear data shown in Fig. 6A are high relative to those of Fig. 3; an average value,
P = 1, is two to three times the values in
the earlier figure. The discrepancy may
exist because of the difference in experimental conditions between the experiments.
It is frequently
observed (Caperon and
Meyer 1972; Eppley et al. 1969) that the
environmental
conditions of cells before
an experiment in which uptake rates are
determined have an important effect on the
value of V, obtained.
The data in Figs. 3 and 4 were collected
during the first 30 min of uptake, soon after
the cells had experienced a very low nutrient concentration,
In experiments in the
controlled-shear apparatus the cultures had
been preincubated
at high concentration
( 5-7 PM ) for l-3 h before the experiment
began. Therefore, the value of V, could
have been different in the flask experiments
from that in experiments in the controlled-
Transport
limited
shear apparatus, leading to the different
values of P observed.
Figure 6B shows data collected in the
controlled-shear
apparatus for NOZ- uptake. The curved lines are drawn for K =
4 PM and for the indicated P values. This
plot is in contrast to that of Fig. 6A where
the value of K is much smaller. Transport
limitation of the NOa- uptake rate is less
severe than that of the NOS- uptake rate,
as indicated by the higher value of P for
NOa- uptake.
Finally, we wish to emphasize that V,
and K can be determined in the usual way
for a species with small P if the experiments
are performed in suitably mixed systems.
Although
reciprocal
plots
should
be
straight lines for a low P species at high
enough nutrient concentrations,
the data
points curve more or less upward as low
concentrations are reached, depending on
P and the rate of mixing. Attempts to pass
straight lines through data in which a few
of the (lower) points curve upward may
succeed, with the appearance of somewhat
greater scatter. Such lines produce an inordinately large value of K. In fact, we
have already noted that a straight line
could have been fit to the data of Fig. 4,
with overestimated values of K resulting.
Transport limitation may thus lead to erroneous measured values of K (C axis intcrcept) and V, (inverse of slope) for
some species unless properly accounted for
or eliminated by sufficient mixing.
We plan to discuss some implications
for phytoplankton
in nature later. What
we have demonstrated here is that transport limitation of nutrient uptake rate can
occur for at least one species of single-
uptake
617
rates
celled phytoplankton and can be presumed
to occur for other species. It may, therefore, be an important factor for phytoplankton growth in nature.
References
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growth of marine phytoplankton.
1.
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l&7.
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Submitted:
Accepted:
17 October 1974
20 March 1975