Journal of Plankton Research Vol.22 no.3 pp.447–472, 2000
Modelling Si–N-limited growth of diatoms
Kevin J.Flynn and Veronique Martin-Jézéquel1
Ecology Research Unit, School of Biological Sciences, University of Wales
Swansea, Singleton Park, Swansea SA2 8PP, UK and 1UMR 6539, CNRS,
Université de Bretagne Occidentale, Institut Universitaire Européen de la Mer,
Technopole Brest-Iroise, Place Nicolas Copernic, F-29280 Plouzane, France
Abstract. A mechanistic model for silicon (Si) physiology is developed, interfaced with a model of
nitrogen (N) physiology, which is capable of simulating the major documented facets of Si–N physiology in diatoms. The model contains a cell cycle component that is involved in regulating the timing
of the synthesis of valves, girdles and setae. In addition to reproducing the timing of diatom cell
division within a light–dark cycle, the model simulates the following features seen in real diatoms.
Synthesis of valves only occurs during G2 interphase and M, while the girdles and (if appropriate)
setae are synthesized during G1. Si stress alone results in a loss of setae, followed by a thinning of the
valves in successive generations until a minimum Si cell quota is attained. After this point, the
duration of G2 increases and growth is Si limited. Concurrently, the carbon (C) cell quota increases,
offering the capability to simulate the documented increase in sinking rates with Si stress. N stress
alone results in an increase in the duration of G1 and G2 interphases, and high Si cell quotas. From
this complex model, which must be run for arrays of subpopulations to simulate non-synchronous
growth, a simpler model is developed. This is capable of reproducing similar growth dynamics,
although with no reference to component parts of the frustule. When allied to a photoacclimative
submodel, a prediction of the model is that diatoms starved of Si will release increased amounts of
dissolved organic C because cell growth is halted more rapidly than the photosystems can be
degraded.
Introduction
Silicate metabolism in diatoms is inextricably linked to the regulation of cell
growth and division (Volcani, 1978; Martin-Jézéquel et al., 2000), and critically
the formation of new valves in the daughter cells (Crawford, 1981; Volcani, 1981;
Pickett-Heaps et al., 1990; Martin-Jézéquel et al., 1997). Orthosilicic acid uptake
as well as silica deposition into the valves are mainly confined to part of the cell
cycle, just before (G2 interphase) and during mitosis (Brzezinski, 1992; Brzezinski and Conley, 1994; Schmid, 1994). Only the formation of girdle bands and setae
occurs during the G1 interphase.
Silicon (Si) incorporation into the frustule follows several steps involving a free
Si pool and a few proteins; a major regulatory step is the uptake phase (MartinJézéquel et al., 2000). The silicification process is not energetically expensive, and
does not require concurrent photosynthesis (Raven, 1983; Sullivan, 1986; MartinJézéquel et al., 2000). Silicon metabolism is thus very different from nitrogen (N)
metabolism. The latter is closely coupled to carbon (C) metabolism and, especially during nitrate assimilation, to the provision of reductant either directly as
photoreductant or via C catabolism.
The cellular content of Si and the rate of entry of Si vary with the stage of the
cell cycle. Under limitations that prolong G2 [e.g. light, temperature, N, iron (Fe)],
the increased period available for Si uptake enables a heavier silicification of the
cells [(Martin-Jézéquel et al., 2000) and references therein]. Si limitation itself
© Oxford University Press 2000
447
K.J.Flynn and V.Martin-Jézéquel
seems to be the only condition of growth directly related to a decreased silicification of the frustules (Paasche, 1973a; Brzezinski et al., 1990).
Within diatoms, Si is required in broadly equimolar amounts to N. As the
concentrations of nitrate and silicate in sea water at the start of the temperate
water production cycle are often also similar, there is clearly a potential for conutrient limitation of diatom growth. Further, because the regeneration of Si is a
slow process in comparison with the rapid cycling of N and phosphorus (P), dominated by dissolution of Si from frustules, post-spring bloom growth of diatoms is
likely to become rate limited by Si. Hence non-diatom species may come to dominate the phytoplankton population (Officer and Ryther, 1980; Egge and Aksnes,
1992; Conley et al., 1993). Modelling Si–N interactions adequately would thus aid
the simulation of phytoplankton succession.
Attempts to model Si-limited growth have been hampered by the fact that Si
nutrition is rather different to that of other nutrients. As the nutrient is primarily
required for a critical phase of growth, the amount of Si in the cell does not relate
simply to growth rate (Davis et al., 1978). The model of Brzezinski employed cell
cycle events to demonstrate the steady-state interactions between silicate and cell
growth (Brzezinski, 1992). However, that model is not suited for placement
within a dynamic modelling scenario, especially one that should ideally also be
capable of simulating Si–N co-limitation within a light–dark cycle as would
usually occur in nature. To date only one model, that of Davidson and Gurney,
has attempted to simulate Si–N co-limitation (Davidson and Gurney, 1999). In
their model, the individual nutrient limitations were handled using quota models.
It was found that using threshold, additive or multiplicative mechanisms to
combine the individual quota controls to give a final control of growth were
unsuitable; Davidson and Gurney developed a more complex, sigmoidal,
mechanism (Davidson and Gurney, 1999). While their model is compact, it
cannot simulate changes in the C cell quota that occur during Si stress and affect
the important process of diatom sedimentation (Bienfang et al., 1982).
A series of mechanistic models has now been developed that can simulate
interactions between ammonium and nitrate (Flynn and Fasham, 1997; Flynn et
al., 1997), N nutrition and light (Flynn and Flynn, 1998; Geider et al., 1998), and
iron–light–ammonium–nitrate (Flynn and Hipkin, 1999). These models all
contain elements of metabolic regulation with recognizable parallels in cell physiology. As a consequence, they offer dynamic reviews of our knowledge of algal
physiology and may thus be useful tools in hypothesis setting and experiment
design.
The aim of this work was to develop a mechanistic model for Si physiology that
is compatible with the previously developed mechanistic models, specifically with
the ammonium–nitrate interaction model (ANIM) of Flynn et al. (Flynn et al.,
1997). This new dynamic model simulates the synthesis of frustules, girdles and
setae and, like the steady-state model of Brzezinski (Brzezinski, 1992), employs
a cell cycle component. We then present a simplified alternative model that, while
it does not reproduce all the subtleties of the full model, offers a compromise suitable for more routine use, still reproducing the important facets of Si assimilation
and interaction with N.
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Modelling Si–N-limited diatom growth
Construction of the complex model
The main model consists of six levels (state variables), describing the C biomass
and cell numbers in a volume of water, and the cell-specific content of four internal Si pools. The latter are dissolved (DSi) and particulate pools for Si in the
valves (VSi), girdles (GSi) and setae (SSi). DSi describes all Si within the cell not
currently part of the structure. Definitions and values for constants are given in
Table I.
Various processes require feedback operations. These employ rectangular
hyperbolic (Michaelis–Menten type) or sigmoidal functions that are normalized
to maximum pool sizes allowing ready modification of those pool sizes. Halfsaturation constants in normalized rectangular hyperbolic equations, typically
0.05, are chosen to enable rate processes to continue at optimal (often maximal)
rates until near the limit of pool size without having to resort to using very small
integration steps in the simulation to control the final feedback action. The use
of sigmoidal functions to regulate complex feedback processes is explained in
Flynn et al. (Flynn et al., 1997). Generally, these functions allow smoother feedback operations to be achieved than if simple rectangular hyperbolic functions
are employed. Sigmoidal functions are also more appropriate where allosteric or
other complex interactive processes are being summarized by a single equation.
Table I. Values of constants for the complex (C) and simple (S) models
Constant
Model
Value
Unit
Comment
cell–1
Ccellmax
Ccellmin
DSidiv
C
C
C
75
25
2.5
pg C
pg C cell–1
pg Si cell–1
DSimax
C
12.5
pg Si cell–1
DSimax2
C
0.6
pg Si cell–1
G9max
G9min
G2Tmin
GSimax
MT
SiKt
C
C
C
C
C
C
0.0132
0
0.0833
1
0.0625
4
Si C–1
Si C–1
day
pg Si cell–1
day
µM Si
SiKt2
C
8
µM Si
SSimax
µmax
C
C
5
2.9
pg Si cell–1
day–1
maximum possible cell size
minimum possible cell size
minimum size of DSi to allow valve construction
(set as the minimum possible Si cell quota)
size of DSi that transinhibits transport via the
primary Si transporter (set at the maximum
possible value for valve Si, VSi)
size of DSi that transinhibits transport via the
G1Si transporter (set at the maximum possible
DSi value during G1 interphase). Must be less
than GSimax
maximum girdle Si:C
minimum girdle Si:C
minimum period of G2 interphase (= 2 h)
maximum amount of Si in girdles
period of M phase (= 1.5 h)
half-saturation constant for Si transport via the
primary Si transporter
half-saturation constant for Si transport via the
G1 Si transporter
maximum amount of Si in setae
maximum theoretical growth rate
Ccellmax
Ccellmin
SiKg
SiCellmax
SiCellmin
µmax
S
S
S
S
S
S
75
35
0.345
30
7
2.9
pg C cell–1
pg C cell–1
µM
pg Si cell–1
pg Si cell–1
day–1
maximum possible cell size
average minimum cell size of population
half-saturation constant for Si-dependent growth
maximum Si quota
minimum Si quota
maximum theoretical growth rate
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K.J.Flynn and V.Martin-Jézéquel
The power and constants used in these sigmoidal functions were chosen by
inspection to achieve a robust operation of the model consistent with the behaviour of diatoms as recorded in the literature.
Several of the following equations contain conditional statements operating by
Boolean logic (i.e. if the condition is true then the value is 1, else it is 0).
The assimilation of Si is critically dependent on the cell cycle. In the model, as
in reality [(Brzezinski and Conley, 1994) and references therein], the regulatory
phase for valve formation is considered to be G2, and that for the setae and the
girdle bands is G1. The actual construction of the cell cycle simulator will differ
depending on the simulation platform; however, in essence it contains two timers:
one for the duration of G2 and one for M. The operation of the model cycle is
described below.
Following cell division, the cell is in G1 phase, in which the bulk of cell growth
occurs. The duration of G1 depends primarily on the availability of nutrients other
than Si, including light for phototrophs. S phase, when DNA is synthesized, is not
modelled per se but is included with G1. In the model, the trigger for the next
division is set by the C cell quota (Ccell); thus, the cell cycle cannot exit G1 and
S phases unless the cell size is at least twice the minimum size of a new daughter
cell (i.e. 2 · Ccellmin). In Boolean terms:
exit G1S = (Ccell ≥ 2 · Ccellmin)
(1)
Under the influence of nutrient limitation, the cell cycle may be prolonged or
arrested prior to this trigger point. (The trigger could be co-controlled by critical
levels of other nutrients such as N and/or P.) Once this stage is passed, the cell is
in G2 and the G2 timer is started. The minimum period of this phase in the model
is described by the parameter G2T. During this period, the most rapid phase of
Si transport is initiated, with Si accumulated to support the synthesis of new
valves.
The size of DSi varies up to a maximum level DSimax. At the point in time just
before the use of DSi in the synthesis of VSi, DSi must contain sufficient Si to
make new VSi for the next cell generation. The relative size of DSi (RelP) is
described by:
DSi
RelP = ––––––
DSimax
(2)
Si transport (SiT) for the support of valve synthesis is described by:
ExtSi
(1 – RelP) · 1.05
SiT = µmax · 3 · DSimax · ––––––––––– · –––––––––––––– · (cell cycle = (G2,M)) (3)
(ExtSi + SiKt) (1 – RelP) + 0.05
The maximum rate of transport is normalized to µmax, the maximum theoretical rate of cell growth, and is sufficient to enable the transport of the required Si
within a portion of the whole cell cycle (hence reference to 3 · DSimax). As µmax
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Modelling Si–N-limited diatom growth
and DSimax are used as scalars, the model can be reconfigured readily for different maximum growth rates and Si content for the valves. Transport is a function
of a standard Michaelis–Menten equation with reference to the external Si
concentration (ExtSi) and the half-saturation constant for transport, SiKt.
Brzezinski and Martin-Jézéquel et al. argue that the value of SiKt is considerably
higher than may be estimated from experimental studies of transport into asynchronous cell suspensions because transport is confined to only a portion of the
cell cycle (Brzezinski, 1992; Martin-Jézéquel et al., 2000). Following the data in
Brzezinski (Brzezinski, 1992), we set SiKt at 4 µM. The transport rate is moderated (transinhibited) by the size of DSi, via the value of RelP [which varies
between 0 and 1; equation (2)], using a hyperbolic function with a half-saturation
constant of 0.05. Thus, as DSi approaches DSimax, transport is halted, while cells
that are Si stressed will maintain a higher transport potential for a greater period.
The Boolean term in equation (3) allows transport of Si for valve synthesis only
when the cell is in the G2 or M phases of the cell cycle.
During growth limitation by all nutrients, the period of cell cycle phases
increases, some proportionately more so than others. Thus, while the major
change during N stress is the prolonging of G1, the duration of G2 also lengthens
(Olson et al., 1986). A result of this is that there is a longer period of Si transport
in cells subjected to N stress. Such cells have higher Si cell quotas than N-replete
cells (Harrison et al., 1976; Martin-Jézéquel et al., in preparation). To simulate this
event, the period of G2T can be made a variable, related to the N:C status. In the
absence of experimental data sets relating the period of the G2 interphase to the
N status, we use a simple linear function. Thus:
G2T = G2Tmin · (1 + f · (1 – Qmu))
(4)
where G2Tmin is the minimum period of G2T (set here to 2 h), Qmu is an index
of the N:C status with a value between 0 and 1 for poor to high values of cellular
N · C–1 [see equation (2) in (Flynn et al., 1999)], and f is a scalar (set here to 3 to
enable the period of G2T to vary over a range of 1–4, in this instance giving
periods of 2–8 h). More complex, non-linear equations could be substituted.
Once the period of time defined by G2T has elapsed the cycle still cannot exit
G2, to enter M phase, unless a critical amount of Si is available within the cell for
synthesis of new valves. This critical value is set by DSidiv, which describes the
minimum amount of Si required for the synthesis of two new valves (in practice,
the value of DSidiv will be close to the minimum Si cell quota). In Boolean terms:
exit G2 = (G2 timer > G2T) · (DSi ≥ DSidiv)
(5)
If the external level of Si is high, then the amount of Si accumulated after the
period set by G2T may be significantly more than this minimum level. Valves will
then be relatively thick. However, if there is insufficient Si available, then the cell
cycle is suspended at this stage (i.e. in G2) and C growth must be slowed or halted;
this aspect is considered below. Assuming that the condition is met and equation
(5) is satisfied, the final part of the cycle (M phase) is entered and a timer started
451
K.J.Flynn and V.Martin-Jézéquel
of duration MT. During this time, the synthesis of valves (VSis) occurs [see the
text for equation (8)] and cells divide.
Cell division occurs over a fraction of the cell cycle. If division occurred over
1 day, the rate of increase would be 0.693 day–1, or Ln(2). When division occurs
over a fraction of a day, MT, the cell-specific growth rate (cellµ) is given by:
cellµ = Ln(21/MT) · (cell cycle = M)
(6)
The increase in cell numbers, which only occurs during the M phase of the cell
cycle, is:
d
— · cell = cell · cellµ
dt
(7)
VSis occurs concurrently with cell division, during M phase. We argue that the
maximum amount of Si taken in during G2 must be capable of being converted
into VSi during M phase. Accordingly, the maximum possible rate of VSis has
been set as two times that for transport as the duration of M phase (MT) is
approximately half of the minimum duration of G2 phase (G2T). The value of
VSis (g Si cell–1) is, like transport, made a function of the size of DSi via RelP
[equation (2)], normalizing it to DSimax using a half-saturation constant of 0.05.
Thus:
RelP · 1.05
VSis = µmax · 6 · DSimax · ————— · (cellµ > 0)
RelP + 0.05
(8)
The cell now enters G1 phase again, the major period of cell growth. During
the growth of diatoms in G1, girdles of additional Si are deposited to allow for
the increase in cell volume. Insufficient Si for girdles must restrict C growth (Cµ).
In those species of diatom that have setae, these are also synthesized during G1.
However, setae are of secondary importance, their synthesis having no known
direct implications for cell or C growth. The loss of setae is one of the first symptoms of Si stress (Paasche, 1980a; Brzezinski et al., 1990).
There are several transporters for Si in diatoms (Hildebrand et al., 1997, 1998)
and the Kt for Si transport is reported to change during the cell cycle (Sullivan,
1977). Thus, a high-affinity (low Kt) system operates during G2, with a lower affinity system present at other times. If the enhancement of Si transport that occurs
during G2 and M phases was due solely to the synthesis of additional transport
proteins, then Kt would not alter. We thus have an additional input into DSi to
support synthesis of girdles and setae. The Si transporter that operates during G1
could be set only to operate during the G1 phase, or (as here) it could operate all
the time, although transinhibition would come into force at a level of DSi above
a certain level (DSimax2). The rate of Si transport via the G1 Si transporter, G1SiT,
is set as:
452
Modelling Si–N-limited diatom growth
ExtSi
G1SiT = µmax · (GSimax + SSimax) · ––––––––––––
ExtSi + SiKt2
(9)
DSi
11 – 1 –––––––
22 · 1.05
DSi
· ––––––––––––––––––– · (DSi ≤ DSimax2)
DSi
1 – ––––––– + 0.05
DSimax2
max2
1
2
where GSimax and SSimax are the maximum cell quotas of Si associated with girdles
and setae, respectively, and SiKt2 is the half-saturation constant for this transporter. The latter half of the equation is a transinhibition function using a normalized hyperbolic function (K of 0.05) limiting transport to fill DSi to a maximum
level as set by DSimax2. If DSi is greater than DSimax2, as will occur during rapid
Si transport in G2/M phases, then G1SiT is zero.
The synthesis of girdles must relate to the changing cell size. As the synthesis
of girdles essentially results in a pro rata increase in cell volume, we assume that
the amount of Si in girdles is proportional to the C cell quota (Ccell) and hence
that we can use the ratio of GSi:Ccell (G9; with units of g Si g C–1) to regulate
the synthesis of girdles in response to increasing Ccell. This ratio is also used to
restrict Cµ in the absence of Si for the synthesis of those girdles. For simplicity,
we assume that the thickness of Si in girdles is invariable (as a function of Si
stress) in contrast to the thickness of the valves. The constants, G9min and G9max
describe the minimum and maximum values for G9. If G9 is near G9min, then the
need for the synthesis of girdles must be maximal while Cµ must be restricted.
If G9 is near G9max, then Cµ is unrestricted. Again, to provide flexibility in the
model structure we normalize this control, so the relative value of G9 (relG9) is
given by:
(G9max – G9)
relG9 = –––––––––––––
(G9max – G9min)
(10)
This returns a value between 0 (halts GSi synthesis) and 1 (promotes maximal
GSi synthesis).
The synthesis of girdles (GSis) is:
DSi
–––––– 2 · (1 + K
1 DSi
girdle)
max2
GSis = µmax · GSimax · –––––––––––––––––––––––
·
DSi
–––––– + Kgirdle
DSimax2
1
2
relG9 · (1 + KG9)
––––––––––––––– · (cell cycle = G1)
relG9 + KG9
(11)
where the first half of the equation describes the control of GSis as a function of
DSi and is scaled to µmax and GSimax. The latter part uses a normalized hyperbolic
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K.J.Flynn and V.Martin-Jézéquel
function to control GSis as a function of G9. Kgirdle and KG9 are constants in the
hyperbolic functions, set at 0.05. This synthesis is only allowed when the cell cycle
is in G1 phase (which in the model includes S phase).
If Si transport is insufficient to enable the synthesis of valves at cell division
or of girdles during G1, C growth (Cµ) must be slowed and finally halted. The
control of Cµ, confined within the Si-delimited space in the cell, assumes that
feedback processes within the cell (such as the increasing concentration of key
metabolites) must halt the synthesis of new biomass. For simplicity, that feedback is assumed to follow a rectangular hyperbolic relationship, although a
sigmoidal function (which is more appropriate for describing allosteric regulation) may be employed.
Cell size (Ccell; g C cell–1) is used to determine a control factor (RelSize); thus:
(Ccellmax – Ccell)
RelSize = (Ccell < Ccellmax) · –––––––––––––––––
(Ccellmax – Ccellmin)
(12)
RelSize tends to zero as the value of Ccell becomes maximal. Here we assumed
that Ccellmax is three times Ccellmin; the range can be readily altered as
required. The potential rate of C-specific growth, {Cµ}, is computed elsewhere
with reference to illumination or other nutrient limitations such as N, e.g. (and
as used in the simulations given here) from ANIM (Flynn et al., 1997). If the
potential rate of C growth computed elsewhere is {Cµ}, then the operational
value of Cµ is:
RelSize · 1.05 (1 – relG9) · (1.05)
Cµ = {Cµ} · ––––––––––––– · ––––––––––––––––
(RelSize + 1.05) 1 – relG9 + 0.05
(13)
This equation limits Cµ if the cell size approaches the maximum possible
(controlled via RelSize and a normalized hyperbolic curve with a constant of 0.05)
and also if the amount of Si for girdles is restrictive [controlled via relG9, equation (10), with a normalized hyperbolic curve again with a constant of 0.05]. The
change in C biomass is:
d
— · C = C · Cµ
dt
(14)
The above equations thus link the availability of Si to the regulation of C-specific
growth. As Cµ is slowed, the N status alters, thus controlling N physiology
through the feedback paths shown in Flynn et al. (Flynn et al., 1997).
Availability of Si for the synthesis of setae does not affect Cµ. The control of
the synthesis of setae (SSis) is:
454
Modelling Si–N-limited diatom growth
DSi
–––––– 2
1 DSi
3
max2
SSis = µmax · SSimax · –––––––––––––––––
DSi 3
–––––– + Ksetae
DSimax2
1
2
(15)
SSi
11 – 1–––––––
22 · 1.05
SSi
max
· ––––––––––––––––––– · (cell cycle = G1)
SSi
1 – ––––––– + 0.05
SSimax
1
2
The first part of the control uses a sigmoidal function to relate SSis to DSi, with
a constant of Ksetae and a power of three. This ensures that setae are only synthesized when DSi is relatively full and thus synthesis of setae decreases most rapidly
when Si stress develops. Altering the value of the power and of Ksetae changes the
effect of Si stress on the synthesis of setae. The second part of the equation limits
the synthesis of setae when the cell has a maximum coverage of setae (SSimax),
using a normalized hyperbolic function with a half-saturation constant of 0.05. If
the simulation is for a species that lacks setae, this equation is deleted and reference to SSimax removed from equation (9).
As DSi, VSi, GSi and SSi are all cell quotas (pg Si cell–1), the values of these
halve when the cell divides. As division is not instantaneous, the division of these
pools between the daughter cells is also not instantaneous, but occurs concurrently with the transfer of Si between DSi and VSi. The changes in the size of the
dissolved pool, valves, girdles and setae are, respectively:
d
— · DSi = SiT + G1SiT – VSis – GSis – SSis – DSi · cellµ
dt
(16)
d
— · VSi = VSis – VSi · cellµ
dt
(17)
d
— · GSi = GSis – GSi · cellµ
dt
(18)
d
— · SSi = SSis – SSi · cellµ
dt
(19)
and
The model as described above generates synchronous division. In order to
generate a representation of asynchronous growth, the model has been set to
simulate the growth of 20 subpopulations with random initialization of cell size
and position in the cell cycle. This number of subpopulations is normally
sufficient, especially given that the cell division in each population is not instantaneous, but occurs over the period set by MT.
455
K.J.Flynn and V.Martin-Jézéquel
The model has been configured to simulate the growth of a diatom with setae
approximating to the size of Thalassiosira weissflogii (note that this species does
not have setae), with reference to the data for various diatom species in Paasche
(Paasche, 1973a, b, 1975) and Harrison et al. (Harrison et al., 1976, 1977). The
typical cell size is set at 50 pg C cell–1, with the total Si content of a cell of this
size varying between 5 and 25 pg Si cell–1. Parameterization for different species
is hindered by the fact that most studies of diatom Si physiology do not report Si
content relative to both cell and C, while the C content per cell rises with Si stress
(Harrison et al., 1977). In addition, data from chemostat studies do not report the
Si content of Si-replete cells. There are also few estimates of the size of dissolved
and particulate pools of Si. However, because the model equations are normalized, the values of minimum and maximum pool sizes can be readily altered. The
model was constructed and run using Powersim Constructor (Isdalstø, Norway)
using the Euler integration method with a step size of 5.625 min.
The behaviour of the complex model under steady state
Transport rates for Si in diatoms are related to both the concentration of the
substrate, by Michaelis–Menten kinetics, and the number of transporters, which
varies with the cell cycle phase (Martin-Jézéquel et al., 2000). With Si limitation,
the incorporation of Si is related to substrate availability and correlates with Silimited growth rates [Figure 1a; cf. (Guillard et al., 1973; Conway et al., 1976;
Conway and Harrison, 1977; Davis et al., 1978; Paasche, 1980a)]. The cell quota
of biogenic silica under Si limitation (Figure 2a) is in accordance with the literature (Paasche, 1973a; Harrison et al., 1976; Brzezinski et al., 1990), falling with
little or no change in µ until a critical quota is attained, when µ then declines with
little subsequent change in the Si quota.
In general, maximum Si cell quotas are obtained at lower (non-Si limited)
growth rates (Sullivan and Volcani, 1981; Brzezinski, 1992; Martin-Jézéquel et al.,
2000). Thus light and temperature limitation of growth leads to an increase in
Si cell–1 (Paasche, 1980b; Taylor, 1985; Martin-Jézéquel et al., 2000). Under N
limitation, the major regulation of Si incorporation is not by Si availability, but
related to the cell cycle. As the duration of G2 increases with N stress, so more Si
can be incorporated into the values until a maximum level is attained; Si cell–1 is
inversely correlated to the growth rate [(Harrison et al., 1976); V.MartinJézéquel, P.Claquin, J.Kromkamp, G.Kraay and M.Veldhuis, unpublished). In
the model, the Si transport curve (Figure 1b) does not show significantly elevated
Si transport, in part also because the level of Si cell–1 was actually near maximal
throughout N stress (Figure 2b). However, Si:C does rise with this treatment
(Figure 3b), as it should. In view of constraints imposed by the modelling software, it was not possible to prolong the period of G2 indefinitely with decreasing
growth rates associated with increasing N stress. In addition, this particular issue
requires clarification of the relationship between N stress and the period of G2;
at present, we have used a very simple assumption [equation (4)] for this control.
The range of variation in Si quota obtained by the model is in accordance with
the literature. Depending on the nutritional status, the amount of Si per cell can
456
Modelling Si–N-limited diatom growth
Fig. 1. Simulated silicate and nitrate transport rates at different steady-state growth rates in
continuous light with growth being limited by silicate (a), nitrate (b) or both nutrients (c).
vary with a range of 1–3 in Thalassiosira species (Binder and Chisholm, 1980;
Brzezinski, 1985; Harrison et al., 1990). However, very few data are available for
comparison of the various Si pools relative to cell C and N. The only data available for continuous cultures were generated by Davis (Davis, 1976) and
Harrison et al. (Harrison et al., 1976, 1977). The extensive review of Brzezinski
gives a mean mass ratio of 0.3 for Si:C and 2.24 for Si:N (Brzezinski, 1985). The
same review indicates a variability within clones due to the growth irradiance of
the order of 44% for Si:C and 50% for Si:N. Unfortunately, these data were
acquired for one light condition; different light–dark regimes may give rise to
different variations.
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K.J.Flynn and V.Martin-Jézéquel
Fig. 2. Simulated silicate and carbon cell quotas at different steady-state growth rates in continuous
light with growth being limited by silicate (a), nitrate (b) or both nutrients (c). The spread of data is
due to varying degrees of synchrony between subpopulations in different model runs.
In T.weissflogii, the range of Si:C is around 0.25 and N:C is 0.125 (mass ratios)
(Brzezinski, 1985); model output is consistent with these values (Figure 3).
Under Si limitation, Si:C is driven by the Si cell quota (Figure 3a), in accordance
with Harrison et al. (Harrison et al., 1977). On the contrary, under N limitation
(Figure 3b), the changes in Si:C are driven by both Si and C cell quotas, again
in accordance with Harrison et al. (Harrison et al., 1977). Paasche shows that with
temperature Si:C increases with the growth rate, due to decreasing cellular C
content at high temperature (Paasche, 1973b). In view of the minimal interference between Si metabolism with N and C metabolism (Martin-Jézéquel et al.,
2000), N:C is quite stable under Si limitation, as also simulated by the model
(Figure 3a). The positive correlation of N:C with the growth rate under N limitation from the model as expected in accordance with Goldman (Goldman,
1976) and Harrison et al. (Harrison et al., 1977), (Figure 3b). No data exist in the
literature to confirm the model performance under dual limitation (Figure 3c),
458
Modelling Si–N-limited diatom growth
Fig. 3. Simulated silicate and nitrogen carbon quotas (mass ratios) at different steady-state growth
rates in continuous light with growth being limited by silicate (a), nitrate (b) or both nutrients (c).
although the output appears logical given the responses seen to single nutrient
limitations.
Estimates of the variation of the length of the G1 and G2 interphases are based
on the analysis of Olson et al. (Olson et al., 1986), Vaulot et al. (Vaulot et al., 1987)
and Brzezinski et al. (Brzezinski et al., 1990) for T.weissflogii. Under Si limitation,
the major influence in the model was a marked increase in G2, and under N limitation by an increase in G1. Distributions of cellular Si in valves, girdle bands and
setae are directly linked to the regulation of these phases (Figure 4). No specific
data set is available to confirm the values obtained in the model. Timing of Si
deposition was shown in T.weissflogii, but by an indirect approach (Brzezinski
and Conley, 1994). These authors noticed ~40% of the Si in girdle bands. In our
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K.J.Flynn and V.Martin-Jézéquel
model, the Si percentage in the girdles is lower than this value, around 10%. The
percentage value in the model will also reflect the contribution of Si in the setae
(which are not present in T.weissflogii), while the absolute content is a function
of the value of the constant GSimax in the model and can be readily altered.
Silicification of the setae was analysed by Brzezinski et al. (Brzezinski et al., 1990)
and is linked to the length of G1; thus N-stressed cells have more Si in their setae
(Figure 4b). In the model, the process is associated with both the period of G1
and the availability of Si (Figure 4a). Thus, there is a marked difference in the
amount of Si in the setae between Si- and N-limited situations (Figure 4a and b).
Valve silicification, which is mostly dependent on the duration of G2, mirrors
Fig. 4. Simulated distribution of cellular silicate between values, girdles and setae at different steadystate growth rates in continuous light with growth being limited by silicate (a), nitrate (b) or both
nutrients (c).
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Modelling Si–N-limited diatom growth
closely the total cellular Si, as expected from the literature (Martin-Jézéquel et
al., 2000).
Figure 5a shows that the relationship between growth rates and Si cell quota
does not suggest that a very satisfactory mode of model control is possible using
the quota approach as it lacks a progressive curvilinear relationship as obtained
for N and P [(Harrison et al., 1976; Tilman and Kilham, 1976); but see (Paasche,
1973b; Davidson and Gurney, 1999)]. In principle, the quota model is inappropriate because the current amount of Si in the frustules is of no consequence for
the growth of the next cell generation. If Si is exhausted suddenly, the Si cell quota
may be high, but no further cell division is possible. Thus, only DSi should be
considered as forming a usable cell quota. However, the relationship between
Fig. 5. Steady-state model output during Si-limited growth under continuous light and in a light–dark
cycle. Relationships between Si cell quota and growth rate (a), external concentration of silicate and
growth rate (b), and cell-specific transport and growth rate (c).
461
K.J.Flynn and V.Martin-Jézéquel
growth rates and the size of DSi is no more promising (not shown). DSi may actually be higher at very low growth rates as a consequence of the extension of G2
and hence of the increase in the average DSi content across the whole population
during Si-limited growth.
Figure 5b shows that Monod kinetics describe the control of steady-state Silimited growth. The half-saturation constant for Si-limited growth, Kg, for
continuous light is 0.345 µM, and 0.1847 µM in the light–dark cycle (note that SiKt
was set at 4 µM and SiKt2 at 8 µM). The two points at a value of µ of 0.693 day–1
(1 division day–1) in the light–dark simulation reflect the fact that cell division and
Si acquisition in diatoms are not restricted to one or other of the illumination
phases. Thus, once µ falls below 0.693 day–1 there is more time to accumulate Si
at lower concentrations of Si (over the dark phase) with no effect on the acquisition of C (hence, also, Kg being lower for the simulation in the light–dark cycle).
Figure 5c shows the linear relationship between µ and Si transport until the
maximum µ is attained for a given irradiance regime, after which the Si content
per cell rises with no change in µ (this event may also be deduced from Figure 5a).
Construction of the simple model
The simple model was developed from an analysis of steady-state simulations
generated by the complex model (Figure 5). The critical process that limits cell
growth (i.e. division, cellµ, rather than C growth) is the rate of transport of Si into
the cell. The basis of the simplified model is thus to make cellµ a function of Si
transport (SiT) (Figure 5c), which in turn essentially follows Monod kinetics
(Figure 5b) now using a half-saturation constant for growth (Kg) rather than for
transport (Kt). However, to link this in with C-specific growth (Cµ), we need to
consider the value of Ccell. If the increase in Ccell halts, because of N stress for
example, then cell division is not required and so neither is Si transport. If Ccell
is high, because a lack of Si prevents cell division, then Cµ and other processes
such as N assimilation must be halted. In steady-state growth, cellµ and Cµ are
the same. Accordingly, two control links are introduced related to the value of
Ccell: one to regulate SiT and the other to regulate Cµ and hence also N assimilation.
The transport rate for Si is now given by:
ExtSi
(1 – RelSize)2
SiT = µmax · SiCellmax · –––––––––––– · –––––––––––––––––
(ExtSi + SiKg) (1 – RelSize)2 + 0.01
(20)
SiT is scaled to µmax and to the maximum amount of Si per cell (SiCellmax), and
through a Michaelis–Menten equation to the external substrate concentration
using the half-saturation constant for growth (SiKg). The form of Si in the cell
(dissolved, valves, girdles, setae) is not considered. RelSize is as described in
equation (12). However, while individual cells may attain the minimum cell size
(Ccellmin), it is very unlikely that all cells in a population would do so simultaneously. Ccellmin was thus set at a higher level (the lowest average value
attained by the population) than in the complex model (Table I). The form of the
462
Modelling Si–N-limited diatom growth
latter part of equation (20), referring to RelSize, is important for the performance
of the model; it must halt SiT when cells are small, and allow it to operate at a
maximum rate when cells are large. We use a sigmoidal function with a power of
2 and a constant of 0.01, allowing very little transport when Ccell is close to
Ccellmin. The value of SiCell can exceed SiCellmax when growth is limited by N.
The degree to which this happens is a function of the part of the equation relating cell size to transport and also to cellµ in equation (21), which is described
below.
The control of Cµ by RelSize is the same as the first half of equation (13), with
omission of the part referring to girdles. The control of cellµ is different, being a
linear increasing function of SiT as long as SiT remains less than a critical value
(Figure 5c). That function is set by the minimum cell quota for Si (SiCellmin); thus,
cellµ = SiT/SiCellmin. Once the value of this, cellµ, reaches the maximum possible
growth rate, µmax, this link is severed and cellµ = µmax. As SiT is now faster than
it need be, the value of SiCell increases towards SiCellmax with no further increase
in µ (Figure 5c). The value of cellµ is also moderated by a sigmoidal curve referring to RelSize. Thus:
cellµ =
SiT
SiT
––––––– < µ
–––––––
11 SiCell
2 · SiCell
max
min
+
min
1
2
2
(21)
SiT
(1 – RelSize)2 · 1.005
––––––– ≥ µmax · µmax · ––––––––––––––––––
(1 – RelSize)2 · 0.005
SiCellmin
The values of the power function and constant are important for the simulation
of Ccell and Sicell. The change in SiCell is given by:
d
— · SiCell = SiT – SiCell · cellµ
dt
(22)
The simple Si model thus contains only five equations, in comparison with 19
equations running as multiple arrays for each subpopulation in the complex
model.
Batch culture simulations: a comparison of the complex and simple models
In these simulations (Figures 6–9), there was a continual low input of fresh media
(with a dilution rate equivalent to 5% of maximum growth rates) that was either
low in Si (15:50 µM Si:N) or low in N (50:15 µM Si:N). Thus, the simulated cells
were not suddenly starved of nutrients, as may be implied from the external
concentrations (Figure 6). The low-Si system slowly consumed most of the nitrate
(Figure 6) with only the Si:C ratio falling (Figure 7). The low-N system was also
subjected to a level of Si limitation, with both the Si:C and N:C ratios falling
(Figure 7).
It is apparent from the comparative plots of data from the complex and simple
models in Figures 6–8 that the performance of the simple model under different
463
K.J.Flynn and V.Martin-Jézéquel
Fig. 6. Simulated batch cultures supplied with a slow input of fresh medium containing 15:50 or
50:15 µM of Si and nitrate N (low Si and low N, respectively). The plots show the model output for
both the complex and simple model structures.
Fig. 7. Changes in the mass ratio of Si:C and N:C in simulated batch cultures; as for Figure 6, these
show the model output for both the complex and simple model structures.
combinations of Si and/or N stress mirrors closely that of the complex models. To
a large extent, differences between the model output were because of the increasing level of synchronous growth developing between the arrayed subpopulations
in the complex model as they became increasingly nutrient limited.
Under nutrient-replete conditions, the Si:N mass ratio is around two (Brzezinski, 1985; Harrison et al., 1990). Si limitation has a rapid severe effect on cellular
Si (Figure 8, low Si) with a concurrent decrease of Si:C (Figure 7) as the C cell
quota increases. In contrast, N limitation affects both C and N metabolism with
significant decreases in the N cell quota and N:C, but relatively little change in
the C quota [Figure 8, low N; cf. (Harrison et al., 1977)].
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Modelling Si–N-limited diatom growth
Fig. 8. Changes in the cell quota for Si and C in simulated batch cultures; as for Figure 6, these show
the model output for both the complex and simple model structures. Also shown are the growth rates
for the complex simulations with the inset in the low-Si plot giving an example of the diel variation
in population cell division (the dark phase was the latter half of each day).
In contrast to the situation with N (Syrett, 1981; Huppe and Turpin, 1994),
transport and incorporation of Si are not affected by the light–dark cycle (Blank
et al., 1986; Martin-Jézéquel et al., 1997, 2000). Accordingly, the incorporation of
Si with cell division is not limited to the light phase; diatom division typically
shows activity in light and dark phases, as shown by the model [inset of Figure 8,
low Si; cf. (Nelson and Brand, 1979)]. Patterns of silicification into the different
parts of the cells are related to the cell cycle (Pickett-Heaps, 1991; Pickett-Heaps
et al., 1994), as simulated by the complex model in Figure 9c and d, in accordance
with the timing of deposition during each interphase (Brzezinski and Conley,
1994; Martin-Jézéquel et al., 2000). The formation of valves occupies ~10% of the
total cell cycle, with the deposition of girdle bands and setae continuing during
most of the cycle (Paasche, 1980a). Si limitation induces a progressive Si incorporation into the cells (Figure 9c) due to the increase in the absolute and relative
period of the G2 phase. However, this is associated with a decrease in the total
cellular Si with declines in the Si content of valves (Paasche, 1973a) and setae
(Brzezinski et al., 1990) (Figure 9c). These simulated results also follow the observations of a decreased silicification of the frustule, without an immediate impact
on cell growth rates under Si limitation. In contrast, N limitation primarily affects
the duration of the G1 phase [Figure 9d; (Vaulot et al., 1987)]. In particular, this
enables a maximal synthesis of setae (Figure 9d), without direct impact on the
total silicification (cf. Figure 2b).
With Si limitation, and its associated decrease in Si transport, there is a change
in the regulation of Si incorporation with a decreasing silicification of the frustule
465
K.J.Flynn and V.Martin-Jézéquel
Fig. 9. As Figure 6, but for cells belonging to a single subpopulation in the complex model. Transport
of nitrate and silicate into cells grown in low-Si (a) or low-N (b) systems; peaks of Si transport correspond with the period before and during cell division, with the shoulder after the peak the period of
Si transport for synthesis of girdles and setae. During N limitation, nitrate transport was continuous.
The distribution of Si within the same population and the duration of cell interphases are shown in
(c) and (d). The difference between the total and the sum of valves, girdles and setae is the dissolved
Si pool that accumulates during G2 phase prior to the synthesis of valves during cell division.
and a complete cessation of the cell division unless, as here, there is a slow input
of Si to the system [Figure 9a; cf. (Paasche, 1980a; Martin-Jézéquel et al., 2000)].
The inset in Figure 9a shows the different timing of Si and N transport; the latter
is coupled with C fixation in the illuminated first half of each day. Only during N
stress does nitrate transport become continuous (Figure 9b).
466
Modelling Si–N-limited diatom growth
Conclusions
We have presented two models capable of simulating the documented behaviour
of Si-dependant dynamics of diatom growth. By placing these models with a
submodel to simulate N interactions, we have generated a Si–N interaction model
complete (in the complex Si model) with a cell cycle simulation. The performance of both models under different types of stress is in keeping with data from
the literature, including correct simulations of changes in cell and C quotas for
both Si and N under different conditions of Si and/or N stress. Davidson and
Gurney used a sigmoidal function to relate cell size, as defined by N cell–1, to the
interaction between the Si and N quotas (Davidson and Gurney, 1999). This could
be viewed as analogous to the sigmoidal relationship between the need for Si and
cell size [indexed in our model to C cell–1 rather than to N cell–1 as in (Davidson
and Gurney, 1999)] in equation (20). However, while the model of Davidson and
Gurney (Davidson and Gurney, 1999), hereafter the DG model, gives a pseudomechanistic simulation for the Si–N interaction where N is the more limiting
nutrient (as was the instance in the data set they modelled), it does not do so
where Si the more limiting. The latter situation is more likely in nature as N is
regenerated continuously within the photic zone.
A comparison of the simple model with the DG model for Thalassiosira
pseudonana is given in Figure 10. The models were configured as described in the
figure legend, with the simple model tuned (using Powersim Solver v2 software)
to match the DG output for cell number and external nutrients under the low-N
scenario for which the DG model was developed (Davidson and Gurney, 1999;
Davidson et al., 1999). This matching of the simple model output for the low-N
scenario was achieved using a lower cell quota for Si than used by the DG model,
part way between the value used by Davidson and Gurney (Davidson and
Gurney, 1999) and the values of Paasche (Paasche, 1973b) for this diatom. While
the N cell quotas, nutrient usage and cell growth dynamics are similar between
both models for the N-limiting scenario (Figure 10a and e), and the C cell–1 values
from the simple model are as expected from the experimental data set (Davidson et al., 1999; Figure 10e), the Si cell quotas are contrary (Figure 10c). The
behaviour of the DG model is not in keeping with the literature (Sullivan and
Volcani, 1981; Brzezinski, 1992; Martin-Jézéquel et al., 2000) in that the simulated
Si cell quota does not increase on N deprivation, but rather falls (Figure 10c).
During exponential growth, the simple model gave SI:C and N:C mass ratios of
around 0.18; on N exhaustion, the former increased to 0.28, while the latter
decreased to around 0.06. These are the expected responses. The DG model does
not simulate C quotas, only cell quotas, so no comparison can be drawn between
the models on this point. Using the same parameters, except with a different
nutrient scenario, a low-Si system for which the DG model was not developed,
the differences in the cell quotas for Si and N are just as obvious. Again, the DG
model does not reproduce the expected shift with nutrient exhaustion, in this case
of N cell–1 rising. This is because while cell growth halts, the DG model cannot
simulate the continuing C growth with the increase in C cell and N cell quotas
(Figure 10d and f) that the literature says should occur in Si-limited cells. If the
467
K.J.Flynn and V.Martin-Jézéquel
468
Modelling Si–N-limited diatom growth
Si quota in the simple model is increased to enable a matching with the DG model
in the low-Si scenario, the Si is exhausted in the low-N scenario (which it is not
by the DG model). Until the appropriate experiments are conducted with
adequate parameterization of both model structures, it will not be possible to say
for sure whether the simplicity of the DG model structure outweighs the apparent technical inaccuracies in its output.
Inclusion of a photoacclimative submodel (Flynn and Flynn, 1998; Geider et
al., 1998) allows the consideration of the interaction between Si stress and photosynthesis. Unfortunately, published data sets are not adequate for a rigorous
testing of the models; in this context, the models should be seen as offering a route
to formulate hypotheses for testing with experiments. This is certainly the case
for the complex model because it contains a high level of detail. One prediction
from the model is that during Si stress dissolved organic carbon (DOC) must be
released. With Si limitation, Cµ must be halted [equation (13)]. However, because
the photosystems may well not degrade rapidly enough, the fixation of C
continues out of step with Cµ. Either the cell must burst or this surplus C must
be lost as DOC.
Finally, the model may be combined with the Fe-submodel of Flynn and Hipkin
(Flynn and Hipkin, 1999), enabling a consideration of the growth of diatoms in
Fe-deplete areas. Fe stress slows all growth processes, but would not be expected
to affect silicification itself. With a prolonging of G2 [requiring a modification of
equation (4) to include Fe stress], the prediction is that Fe-stressed cells, like Nstressed cells (Figures 2–4), will have thick valves. Recently, we have linked the
simple Si model with the Fe–light–ammonium–nitrate model of Flynn and Hipkin
(Flynn and Hipkin, 1999); simulations show an increase [two to sixfold depending on the values of the powers in equations (20) and (21)] in Si cell–1 with Fe
stress. This is in keeping with the observation that diatoms from low-Fe waters
do indeed have high Si cell quotas (Boyle, 1998; Hutchins and Bruland, 1998;
Takeda, 1998).
Acknowledgements
This work was supported by the Natural Environment Research Council (UK)
and Centre National de la Recherche Scientifique and Université de Bretagne
Occidentale (France).
Fig. 10. Comparison between the performance of the model of Davidson and Gurney (Davidson and
Gurney, 1999), the DG model, and the simple Si–N model. The DG model was constructed using the
‘summer’ parameters for T.pseudonana of Davidson and Gurney (Davidson and Gurney, 1999) and
run in batch simulations using two alternative nutrient regimes: low N and low Si. Values for the
simple model of Ccellmin (12.28 pg C cell–1), Sicellmin (1.483 pg Si cell–1), Sicellmax (2.39 pg Si cell–1)
and the theoretical maximum growth rate (6.15 day–1, giving 1.5 day–1 in a 12 h:12 h light–dark cycle)
were obtained from tuning the simple model to the DG model output in the low-N system for cell
numbers and external nutrients. Ccellmax was set at 20 pg C cell–1 [see (Davidson et al., 1999)].
469
K.J.Flynn and V.Martin-Jézéquel
References
Bienfang,P.K., Harrison,P.J. and Quarmby,L.M. (1982) Sinking rate response to depletion of nitrate,
phosphate and silicate in four marine diatoms. Mar. Biol., 67, 295–302.
Binder,B.J. and Chisholm,S.W. (1980) Changes in the soluble silicon pool size in the marine diatom
Thalassiosira weisflogii. Mar. Biol. Lett., 1, 205–212.
Blank,G.S., Robinson,D.H. and Sullivan,C.W. (1986) Diatom mineralization of silicic acid. VIII.
Metabolic requirements and the timing of protein synthesis. J. Phycol., 22, 382–389.
Boyle,E. (1998) Pumping iron makes thinner diatoms. Nature, 393, 733–734.
Brzezinski,M.A. (1985) The Si:C:N ratio of the marine diatoms: interspecific variability and the effect
of some environmental variables. J. Phycol., 21, 347–357.
Brzezinski,M.A. (1992) Cell-cycle effects on the kinetics of silicic acid uptake and resource competition among diatoms. J. Plankton Res., 14, 1511–1539.
Brzezinski,M.A. and Conley,D.J. (1994) Silicon deposition during the cell cycle of Thalassiosira weisflogii (Bacillariophyceae) determined using 123rhodamine and propidium iodide staining. J.
Phycol., 30, 45–55.
Brzezinski,M.A., Olson,R.J. and Chisholm,S.W. (1990) Silicon availability and cell-cycle progression
in marine diatoms. Mar. Ecol. Prog. Ser., 67, 83–96.
Conley,D.J., Schelske,C.L. and Stoermer,E.F. (1993) Modification of the biogeochemical cycle of
silica with eutrophication. Mar. Ecol. Prog. Ser., 101, 179–192.
Conway,H.L. and Harrison,P.J. (1977) Marine diatoms grown in chemostats under silicate or
ammonium limitation. IV Transient response of Chaetoceros debilis, Skeletonema costatum and
Thalassiosira gravida to a single addition of the limiting nutrient. Mar. Biol., 43, 33–43.
Conway,H.L., Harrison,P.J. and Davis,C.O. (1976) Marine diatoms grown in chemostats under silicate
or ammonium limitation. II. Transient response of Skeletonema costatum to a single addition of the
limiting nutrient. Mar. Biol., 35, 187–199.
Crawford,R.M. (1981) The siliceous components of the diatom cell wall and their morphological
variation.. In Simpson,T.L. and Volcani,B.E. (eds), Silicon and Siliceous Structures in Biological
Systems. Springer-Verlag, New York, pp. 129–156.
Davidson,K. and Gurney,W.S.C. (1999) An investigation of non-steady-state algal growth. II. Mathematical modelling of co-nutrient-limited algal growth. J. Plankton Res., 21, 839–858.
Davidson,K., Wood,G., John,E.H. and Flynn,K.J. (1999) An investigation of non-steady-state algal
growth. I. An experimental model ecosystem. J. Plankton Res., 21, 811–837.
Davis,C.O. (1976) Continuous culture of marine diatoms under silicate limitation. II. Effect of light
intensity on growth and nutrient uptake of Skeletonema costatum. J. Phycol., 12, 291–300.
Davis,C.O., Breitner,N.F. and Harrison,P.J. (1978) Continuous culture of marine diatoms under
silicon limitation. 3. A model of Si-limited diatom growth. Limnol. Oceanogr., 23, 41–52.
Egge,J.K. and Aksnes,D.L. (1992) Silicate as regulating nutrient in phytoplankton competition. Mar.
Ecol. Prog. Ser., 83, 281–289.
Flynn,K.J. and Fasham,M.J.R. (1997) A short version of the ammonium-nitrate interaction model. J.
Plankton Res., 19, 1881–1897.
Flynn,K.J. and Flynn,K. (1998) The release of nitrite by marine dinoflagellates—development of a
mathematical simulation. Mar. Biol., 130, 455–470.
Flynn,K.J. and Hipkin,C.R. (1999) Interactions between iron, light, ammonium and nitrate: insights
from the construction of a dynamic model of algal physiology. J. Phycol., in press.
Flynn,K.J., Fasham,M.J.R. and Hipkin,C.R. (1997) Modelling the interaction between ammonium and
nitrate uptake in marine phytoplankton. Phil. Trans. R. Soc. Lond., 352, 1625–1645.
Flynn,K.J., Page,S., Wood,G. and Hipkin,C.R. (1999) Variations in the maximum transport rates for
ammonium and nitrate in the prymnesiophyte Emiliania huxleyi and the raphidophyte Heterosigma
carterae. J. Plankton Res., 21, 355–371.
Geider,R.J., MacIntyre,H.L. and Kana,T.M (1998) A dynamic regulatory model of phytoplankton
acclimation to light, nutrients and temperature. Limnol. Oceanogr., 43, 679–694.
Goldman,J.C. (1976) Phytoplankton response to wastewater nutrient enrichment in continuous
culture. J. Exp. Mar. Biol. Ecol., 23, 31–43.
Guillard,R.R.L., Kilham,P. and Jackson,T.A. (1973) Kinetics of silicon-limited growth in the marine
diatom Thalassiosira pseudonana Hasle and Heimdal (=Cyclotella nana Hustedt). J. Phycol., 9,
233–237.
Harrison,P.J., Conway,H.L. and Dugdale,R.C. (1976) Marine diatoms grown in chemostats under
silicate or ammonium limitation. I. Cellular chemical composition and steady-state growth kinetics
of Skeletonema costatum. Mar. Biol., 35, 177–186.
Harrison,P.J., Conway,H.L., Holmes,R.W. and Davis,C.O. (1977) Marine diatoms grown in
470
Modelling Si–N-limited diatom growth
chemostats under silicate or ammonium limitation. III. Cellular chemical composition and
morphology of Chaetoceros debilis, Skeletonema costatum and Thalassiosira gravida. Mar. Biol., 43,
19–31.
Harrison,P.J., Thompson,P.A. and Calderwood,G.S. (1990) Effects of nutrient and light limitation on
the biochemical composition of phytoplankton. J. Appl. Phycol., 2, 45–56.
Hildebrand,M., Volcani,B.E., Gassman,W. and Schroeder,J.I. (1997) A gene family of silicon transporters. Nature, 385, 688–689.
Hildebrand,M., Dahlin,K. and Volcani,B.E. (1998) Characterization of a silicon transporter gene
family in Cylindrotheca fusiformis: sequences, expression analysis and identification of homologs
in other diatoms. Mol. Gen. Genet., 260, 480–486.
Huppe,H.C. and Turpin,D.H. (1994) Integration of carbon and nitrogen metabolism in plant and algal
cells. Ann. Rev. Plant Physiol. Plant Mol. Biol., 45, 577–607.
Hutchins,D.A. and Bruland,K.W. (1998) Iron-limited diatom growth and Si/N uptake ratios in a
coastal upwelling regime. Nature, 393, 561–564.
Martin-Jézéquel,V., Daoud,N. and Queguiner,B. (1997) Coupling of silicon, carbon and nitrogen
metabolisms in marine diatoms. In Dehairs,F., Elskens,M. and Goyens,L. (eds), Integrated Marine
System Analysis. Vriej University of Brussels, pp. 65–83.
Martin-Jézéquel,V., Hildebrand,M. and Brzezinski,M. (2000) Silicification in diatoms. Implication for
growth. J. Phycol., submitted.
Nelson,D.M. and Brand,L.E. (1979) Cell division periodicity in 13 species of marine phytoplankton
on a light-dark cycle. J. Phycol., 15, 67–75.
Officer,C.B. and Ryther,J.H. (1980) The possible importance of silicon in marine eutrophication. Mar.
Ecol. Prog. Ser., 3, 83–91.
Olson,R.J., Vaulot,D. and Chisholm,S.W. (1986) Effects of environmental stresses on the cell cycle of
two marine phytoplankton species. Plant Physiol., 80, 918–925.
Paasche,E. (1973a) Silicon and the ecology of marine plankton diatoms. I. Thalassiosira pseudonana
(Cyclotella nana) growth in a chemostat with silicate as limiting nutrient. Mar. Biol., 19, 117–126.
Paasche,E. (1973b) Silicon and the ecology of marine plankton diatoms. II. Silicate-uptake kinetics
in five diatom species. Mar. Biol., 19, 262–269.
Paasche,E. (1975) Growth of the plankton diatom Thalassiosira nordenskioldii Cleve at low silicate
concentrations. J. Exp. Mar. Biol. Ecol., 18, 173–183.
Paasche,E. (1980a) Silicon. In Morris,I. (ed.), The Physiological Ecology of Phytoplankton. Studies in
Ecology, Vol. 7. Blackwell, Oxford, pp. 259–284.
Paasche,E. (1980b) Silicon content of five marine plankton diatom species measured with a rapid filter
method. Limnol. Oceanogr., 25, 474–480.
Pickett-Heaps,J. (1991) Cell division in diatoms. Int. Rev. Cytol., 128, 63–108.
Pickett-Heaps,J., Schmid,A.M.M. and Edgar,L.A. (1990) The cell biology of diatom valve formation.
Prog. Phycol. Res., 7, 1–168.
Pickett-Heaps,J.D., Carpenter,J. and Koutoulis,A. (1994) Valve and seta (spine) morphogenesis in the
centric diatom Chaetoceros peruvianus Brightwell. Protoplasma, 181, 269–282.
Raven,J.A. (1983) The transport and function of silicon in plants. Biol. Rev., 58, 179–207.
Schmid,A.M.M. (1994) Aspects of morphogenesis and function of diatom cell walls with implication
for taxonomy. Protoplasma, 181, 43–60.
Sullivan,C.W. (1977) Diatom mineralization of silicic acid. II. Regulation of Si(OH)4 transport rates
during the cell cycle of Navicula pelliculosa. J. Phycol., 13, 86–91.
Sullivan,C.W. (1986) Silicification by diatoms. In Silicon Biochemistry. Ciba Foundation Symposium
121. Wiley, Chichester, pp. 59–89.
Sullivan,C.W. and Volcani,B.E. (1981) Silicon in the cellular metabolism of diatoms. In Simpson,T.L.
and Volcani,B.E. (eds), Silicon and Siliceous Structures in Biological Systems. Springer-Verlag, New
York, pp. 15–42.
Syrett,P.J. (1981) Nitrogen metabolism of microalgae. Can. Bull. Fish. Aquat. Sci., 210, 182–210.
Takeda,S. (1998) Influence of iron availability on nutrient consumption ratios of diatoms in oceanic
waters. Nature, 393, 774–777.
Taylor,N.J. (1985) Silica incorporation in the diatom Coscinodiscus granii as affected by light intensity.
Br. Phycol. J., 20, 365–374.
Tilman,D. and Kilham,S.S. (1976) Phosphate and silicate growth and uptake kinetics of the diatoms
Asterionella formosa and Cyclotella meneghiniana in batch and semicontinuous culture. J. Phycol.,
12, 375–383.
Vaulot,D., Olson,R.J., Merkel,S. and Chisholm,S.W. (1987) Cell-cycle response to nutrient starvation
in two phytoplankton species Thalassiosira weissflogii and Hymenomonas carterae. Mar. Biol., 95,
625–630.
471
K.J.Flynn and V.Martin-Jézéquel
Volcani,B.E. (1978) Rôle of silicon in diatoms metabolism and silicification. In Bendz,G. and
Lindqvist,I. (eds), Biochemistry of Silicon and Related Problems. Plenum Press, New York, pp.
177–204.
Volcani,B.E. (1981) Cell wall formation in diatoms: morphogenesis and biochemistry. In Simpson,T.L.
and Volcani,B.E. (eds), Silicon and Siliceous Structures in Biological Systems. Springer-Verlag, New
York, pp. 157–200.
Received on July 21, 1999; accepted on October 6, 1999
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