Photosynth Res (2007) 93:223–234
DOI 10.1007/s11120-007-9178-9
REVIEW
E-photosynthesis: a comprehensive modeling approach
to understand chlorophyll fluorescence transients and other
complex dynamic features of photosynthesis in fluctuating light
Ladislav Nedbal Æ Jan Červený Æ Uwe Rascher Æ
Henning Schmidt
Received: 24 September 2006 / Accepted: 16 April 2007 / Published online: 11 May 2007
Springer Science+Business Media B.V. 2007
Abstract Plants are exposed to a temporally and spatially
heterogeneous environment, and photosynthesis is well
adapted to these fluctuations. Understanding of the complex, non-linear dynamics of photosynthesis in fluctuating
light requires novel-modeling approaches that involve not
only the primary light and dark biochemical reactions, but
also networks of regulatory interactions. This requirement
exceeds the capacity of the existing molecular models that
are typically reduced to describe a partial process,
dynamics of a specific complex or its particular dynamic
feature. We propose a concept of comprehensive model
that would represent an internally consistent, integral
framework combining information on the reduced models
that led to its construction. This review explores approaches and tools that exist in engineering, mathematics,
and in other domains of biology that can be used to develop
a comprehensive model of photosynthesis. Equally
important, we investigated techniques by which one can
rigorously reduce such a comprehensive model to models
of low dimensionality, which preserve dynamic features of
interest and, thus, contribute to a better understanding of
photosynthesis under natural and thus fluctuating conditions. The web-based platform www.e-photosynthesis.org
is introduced as an arena where these concepts and tools
are being introduced and tested.
Keywords Chlorophyll fluorescence emission Forced oscillations Non-linearity Photosystem II System biology System decomposition Model reduction
J. Červený
Centre of Applied Cybernetics, Czech Technical University,
16607 Prague 6, Czech Republic
Abbreviations
CAM
Crassulacean acid metabolism
Chl
Chlorophyll
F m¢
Maximum fluorescence yield measured in lightadapted organism during saturating flash of light
DF
Difference between the maximum fluorescence
Fm¢ and steady state fluorescence yields
ODE
Ordinary differential equations
PQ
Plastoquinone
PSI
Photosystem I
PSII
Photosystem II
QA
Primary quinone acceptor of Photosystem II
QSSA Quasi steady-state approximation
SBML Systems Biology Markup Language
U. Rascher
Institute of Chemistry and Dynamics of the Geosphere ICG-III:
Phytosphere, Forschungszentrum Jülich, Stetternicher Forst,
52425 Juelich, Germany
Introduction
H. Schmidt
Fraunhofer-Chalmers Research Centre, Sven-Hultins Gata 9D,
41288 Gothenborg, Sweden
Photosynthesis has evolved and occurs in an environment
in which light is continuously changing over a wide range
L. Nedbal (&) J. Červený
Institute of Systems Biology and Ecology ASCR, Zámek 136,
37333 Nove Hrady, Czech Republic
e-mail: [email protected]
L. Nedbal J. Červený
Institute of Physical Biology, University of South Bohemia,
Zámek 136, 37333 Nove Hrady, Czech Republic
123
224
of time scales. Light intensity changes over seasons and
days and rapid fluctuations occur due to moving clouds,
during sun flecks in the canopy understory (Sims and Pearcy 1993) or in ocean waves (Sabbah and Shashar 2006).
Responses to fluctuating light are observed from the level
of chloroplasts up to organs and whole plants (reviewed in
Hütt and Lüttge 2002). The regulatory mechanisms that lie
behind this immense adaptability of photosynthesis are yet
not fully understood (Rascher and Nedbal 2006). Fietz and
Nicklisch (2002) proposed that photosynthesis in fluctuating light is not a mere interplay of mechanisms acclimating
to constant light at its extremes. Rather, it is the dynamics
of regulation that is the dominant adaptive mechanism. The
highly non-linear forced oscillations of plant photosynthesis confirm this notion (Nedbal et al. 2003). In another
remarkable demonstration of the behavior of plant photosynthesis, Kulheim et al. (2002) showed that Arabidopsis
thaliana mutants lacking the qE-type or DpH-dependent
non-photochemical quenching exhibit reduced fitness when
grown in fluctuating light regime. No such difference was
seen when the mutants and wild-type plants were compared
in static laboratory conditions including high-light stress.
These highly relevant dynamic features that occur in fluctuating light are poorly understood.
A widely used approach to elucidate mechanisms behind
dynamic features of complex systems is to combine
experimental studies and mathematical modeling (see e.g.,
Schmidt and Jacobsen 2004). Recently, this approach has
been integrated in a distinct scientific discipline of systems
biology (Kitano 2001). The main advantage of the systems
biology approach is that after biological knowledge is
encapsulated into models, and validated by experimental
data, predictions can be made that reveal the dynamics of
the investigated system.
Using mathematical models is not new in photosynthesis
research. Biochemical models have been built for simulating dynamic behavior of various subsystems: Photosystems II (reviewed in Wydrzynski and Satoh 2005),
Photosystem I (reviewed in Golbeck 2006) and the CalvinBenson cycle (Farquhar et al. 2001; Poolman et al. 2000,
2004; BIOMD0000000013 in http://www.biomodels.net/).
Measuring and modeling chlorophyll fluorescence
transients
Detailed time-series required for construction and validation of photosynthetic models are frequently obtained by
measuring kinetics of chlorophyll a fluorescence emission.
Fluorescence experiments have proven to be one of the
most powerful techniques to quantify photosynthetic
efficiency and non-photochemical energy dissipation
(Govindjee 1995; Papageorgiou and Govindjee 2004). The
most common of these techniques is the pulse-amplitude-
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Photosynth Res (2007) 93:223–234
modulation (Schreiber et al. 1986; Schreiber 2004) combined with saturating light pulse method (Genty et al.
1989). The method relates the ratio of steady-state variable
to maximum fluorescence (DF/Fm¢) to quantum yield of
Photosystem II photochemistry and to light use efficiency
of plants. The relatively long saturating flashes of light
(~1 s) limit application of this technique to investigation of
relatively slow processes. The fast fluorescence transients
occurring during dark-to-light transitions have also been
frequently studied. Experimental results and models relevant in physiological irradiance levels (Kautsky and Hirsch
1931; Duysens and Sweers 1963) and in high-irradiance
(Neubauer and Schreiber 1987; Strasser et al. 1995) have
been summarized in numerous reviews (e.g., Dau 1994;
Lavergne and Trissl 1995; Lazár 1999, 2006 and several
chapters in Papageorgiou and Govindjee 2004). In spite of
this enormous effort, a large degree of uncertainty persists.
Bernhardt and Trissl (1999) formulated this uncertainty for
the models of PSII connectivity and fluorescence sigmoidicity in the following words: ‘‘we conclude that it is
hardly possible to distinguish experimentally between
different models by any current method’’.
Photosynthesis is a tremendously complex system but
models are usually very small and, in our opinion, selected
often on the Occam’s razor principle rather than on
hypothesis falsification of Karl Popper (Popper 1990). The
largely simplified photosynthesis models are frequently not
able to capture complex dynamics features. For example,
the characteristic upper-harmonic modes, described in
Nedbal et al. (2003) cannot be interpreted using any of the
earlier photosynthetic models. Nedbal et al. (2005) showed
that by improving the mathematical model and by introducing a time delay, the improved model was able to
reproduce the newly observed dynamic features reasonably
well; however, the model failed when the frequency of the
light forcing was changed (L. Nedbal, unpublished).
Circadian oscillations of Crassulacean Acid
Metabolism (CAM) photosynthesis
Circadian rhythms of CAM photosynthesis represent
another type of complexity that occurs on a timescale that
is largely different from the chlorophyll fluorescence
transients described above. CAM photosynthesis shows
distinct diurnal changes in the day–night cycle and also
exhibits an endogenous, circadian rhythm in continuous
light and darkness (Warren and Wilkins 1961; Lüttge and
Ball 1978; Black and Osmond 2003). The endogenous
rhythm in the CAM plant Kalanchoe¨ daigremontiana,
however, only persists at medium temperature; above and
below thresholds, rhythmicity changes reversibly to nonstochastic arrhythmic behavior, which shows a complex
intrinsic structure (Grams et al. 1997; Lüttge and Beck
Photosynth Res (2007) 93:223–234
1992). Temperature profiles for the expression of endogenous rhythmicity and arrhythmicity of CO2 exchange can
be shifted by slow temperature changes (Rascher et al.
1998) and the endogenous, diurnal rhythm can be entrained
by external light rhythms (Lüttge et al. 1996; Bohn et al.
2001) or temperature regimes (Bohn et al. 2002). During
entrainment, CO2 gas-exchange was found to be complex,
showing non-stochastic fluctuations with a complex harmonic spectrum. In addition, spatio-temporal patterns of
dynamically changing photosynthetic efficiency were
observed in the anatomically homogeneous leaves of
K. daigremontiana under constant external conditions.
Spatial heterogeneity was greatest at the transition between
different phases of CAM. Rascher et al. (2001) and
Rascher and Lüttge (2002) concluded that these spatial
patterns are due to the underlying non-linear properties of
CAM metabolism. Blasius et al. (1998, 1999) developed a
skeleton model, which could be reduced to only four pools
that were coupled by rate equations. This simplified model
was able to fully reconstruct the diurnal oscillations of
CAM photosynthesis in the day–night rhythm as well as
the endogenous oscillations in continuous light. Furthermore, when spatial decoupling of single leaf areas was
included, the model was readily extrapolated to a spatial
network and the mechanistic understanding of this novel
spatial phenomenon became available (Beck et al. 2001;
Rascher et al. 2001).
Modeling complexity of photosynthesis
In this overview, we discuss a systematic approach with the
goal of gaining an increased understanding of photosynthesis, in terms of complex dynamic features of photosynthesis that occur in fluctuating light. We propose that
the dynamic properties of photosynthesis, such as complex
transients in fluctuating light, can be understood at a higher
systemic level that includes not only the primary
photochemical and biochemical reactions but also their
regulation, the molecular dynamics of protein complexes,
and of the thylakoid membrane in a changing environment.
Mathematical models of photosynthesis should thus contain all the necessary elements. A major challenge in
modeling photosynthesis is its high complexity and the
potentially huge dimensionality of subsystems, such as that
of the Photosystem II (PS II), which, for instance, can
contain several thousands of combinatorial dynamic states.
A systematic modeling approach will thus need to be
hierarchical. Practically solvable models cannot include all
known photosynthetic components and their dimensionality must be reduced while preserving their capacity to
simulate dynamic behavior of interest.
Small and compactly intervened subsystems, such as
PSI and PSII cores, and their dynamics can be modeled in
225
great molecular details (see e.g., Müller et al. 2003;
Novoderezhkin et al. 2005). At the same time, more
peripheral photosynthetic components that turnover on a
slower time-scale can be ignored in most of these detailed
molecular models without compromising their capacity to
describe ultra-fast phenomena. This model reduction is
based on insights into the modeled system.
In another example of model reduction, Duysens and
Sweers (1963) were able to account for the dynamics of
photochemical quenching and Joliot and Joliot (1964) for
the sigmoidicity of the fluorescence transients by simple
models. These empiric, highly reduced models could not
include then unknown details of PSII photochemistry and,
yet, performed well in simulating particular dynamic
features.
The modelers of photosynthesis have generated a wide
spectrum of small and medium size models that are
reduced either empirically or by molecular insight. These
reduced models can be mapped along the two-light-reaction Z-scheme of the electron transfer chain and the CalvinBenson cycle of CO2 assimilation forming a mosaic of
modeling domains (for basics of photosynthesis, see
Blankenship 2002). Some of the subsystems and dynamic
features are addressed in this mosaic by multiple models
(e.g., Lazár 2006; Farquhar et al. 2001). In other segments
of the photosynthetic reaction chains, such as cytochrome
b6f or mobile redox pools, the detailed kinetic models are
scarce or absent. Filling the gaps in the mosaic of photosynthetic models as well as validating contradictory models
requires systematic approach of constructing comprehensive models, model standardization, mathematically rigorous model reduction, and experimental validation of the
models—all advocated in this overview.
Systems biology approach
The traditional analytic approach within biology has been
to study the characteristics of isolated parts of a cell or
organism. In contrast, the systems biology approach
(Kitano 2002) focuses on elucidating the functions of
components within a system by studying how the different
involved components interact with each other. It is the
interactions between components in networks that actually
are the basis for many properties of life, and very often the
behavior of a system cannot be explained by its constituents alone. Foundations of systems biology were built,
among others, by Nobel laureates Erwin Schrödinger
(1945) and Ilya Prigogine (1969), and by one of the
founders of cybernetics Norbert Wiener (1948). In the
1940s and 1950s, these revolutionary concepts were hard to
apply in the mainstream biology because the level of
understanding of the material basis of the information,
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Photosynth Res (2007) 93:223–234
mass and energy flows in the biological systems was
inadequate.
It was not before the first years of the 21st century, soon
after the first genomes were deciphered, that systems
biology could take off and since then has become a buzzword. System biologists initially focused mostly on genetic
networks, and the link between genomics and systems
biology is still important in the field (Kitano 2001). However, numerous examples now show that the focus has
shifted towards proteomics and metabolomics and as far as
to stress physiology (Csete and Doyle 2002). Recently
substantial activity has emerged in systems biology of
plants (Minorsky 2003).
the original small models. It is important to point out that
the reduced models cannot be constructed by simply
neglecting the parts of the system that seem not to be
interesting. Instead, it is important to translate the significant dynamic features across scales. The determination of
smaller models that preserve a certain behavior of interest
is called model reduction and is a commonly used procedure in engineering sciences for which several standard
approaches have been developed. A drawback of any
model reduction is that each reduction of complexity will
naturally lead to a loss of certain properties of the model
(Korn 2005) and the technique of reduction must be chosen
to preserve dynamic features of interest.
Comprehensive model and model reduction
Mathematical framework
Photosynthetic processes occur on widely diverse timescales: from the ultra-fast time domain (Müller et al. 2003;
Holzwarth et al. 2006a; van Grondelle and Novoderezhkin
2006) up to relatively slow domains, such as of the CalvinBenson cycle reactions (reviewed in Poolman et al. 2000;
Farquhar et al. 2001). Furthermore, the stimuli to which
photosynthesis is exposed in natural and in experimental
environments range over wide time-scales, for instance,
from picosecond laser flashes, light-flecks in moving plant
canopies to seasonal variations. To include dynamic processes ranging from sub-picoseconds to years in a single
comprehensive mathematical model is not practical. If the
interest lies on the systems behavior on fast time-scales,
the slow dynamics will not affect the systems behavior and
the corresponding dynamic states can be considered
constant. On the other hand, if one is interested in slow
dynamic processes, the more rapidly reacting components
can be considered to be in a steady state. Including out-offocus components explicitly in the model would make
model simulation computationally impossible while
revealing no new information on the systems dynamics.
Thus, the comprehensive model of photosynthesis
advocated in this review is valuable not as a practically
solvable model but, rather as a framework in which the
earlier small models, each focusing on a different
time-scale of interest and/or behavior(s) of interest, are
integrated and synchronized. The notion of comprehensive
model should thus not be understood in terms of a model
that captures all possible behaviors of photosynthesis.
Rather, it represents internally consistent, integral
information on all the smaller models that led to its
construction.
In a reverse direction, information contained in the
comprehensive model can be used to generate, in a
consistent manner, reduced models to describe particular
dynamic features of interest in areas that were peripheral to
Systematic model integration must start with setting up a
shared mathematical framework. Mathematical models of
dynamical systems are most commonly represented by the
means of ordinary differential equations1. In physics and
engineering sciences, a special representation of these
equations is often used, the state-space representation.
Using this representation general non-linear model can be
expressed in the following form:
123
d~
xðtÞ ~
¼ f ð~
xðtÞ; ~
uðtÞÞ
dt
ð1Þ
~
yðtÞ ¼ ~
gð~
xðtÞ; ~
uðtÞÞ
ð2Þ
In the equations above, ~
xðtÞ is an n-dimensional vector
of the state variables, describing, e.g., concentrations of
proteins, gene expression levels, membrane potentials, or
probabilities of certain states (as we will see later in the
example of PS II).
~
uðtÞ is a q-dimensional vector of model inputs. Model
inputs are external stimuli to which the model is going to
react. The typical input of photosynthesis models is
irradiance.
~
yðtÞ is a p-dimensional vector of outputs. Model outputs
are usually states or functions of states that are relevant to
experiments. Typical outputs of photosynthesis models are
fluorescence emission, oxygen evolution, carbon dioxide
uptake, or concentrations of states that can be directly
measured, e.g., by characteristic light absorption. These
model outputs are compared to measured data and a model
is chosen that achieves the best agreement.
1
It should be mentioned that, if components are distributed heterogeneously in space, effects, such as diffusion, are important within a
system, and other mathematical formalisms, such as partial differential equations, should be used. However, this is outside the scope of
this paper.
Photosynth Res (2007) 93:223–234
227
~
f is a vector valued, often non-linear, function of the
same dimension as the state vector ~
xðtÞ. For given state ~
xðtÞ
and input ~
uðtÞ at time t, the function ~
f determines the time
derivatives of the state variables, and thus determines the
dynamical properties of the model. Using numerical
integration methods, present in standard software packages
(e.g., Hindmarsh et al. 2005), the differential equation
(Eq. 1) can be simulated and the time dependent trajectories of the states can be determined.
Finally, ~
g is a vector-valued function that determines the
outputs at time t for given states and inputs at that specific
time. The dynamics of the model are independent of ~
g.
Most of the relevant photosynthetic reactions are of
higher order and the corresponding differential equations
are non-linear, as shown in Eq. 1. Sometimes, the
non-linear nature is essential for modeling of particular
dynamic features (reviewed e.g., in Lazár 2006).
However, it is often an advantage to consider linear
models of non-linear systems as, for example, when one of
the reacting species in a second-order reaction can be
considered to be available approximately at a constant
concentration. Then, the behavior of non-linear systems
can be analyzed sufficiently well by studying a linear
model that has been obtained by linearization around a
certain equilibrium/steady-state of interest. Furthermore, a
highly developed theory and a large amount of powerful
analysis methods exist for linear systems. The direct
analysis of non-linear systems is often impractical or even
impossible. An example of successful application of linear
methods to the analysis of complex non-linear behavior can
be found in Schmidt and Jacobsen (2004).
A linear model can be described by the following linear
state-space description:
$
d~
xðtÞ $
¼ A~
xðtÞ þ B~
uðtÞ
dt
$
$
~
yðtÞ ¼ C~
xðtÞ þ D~
uðtÞ
ð3Þ
ð4Þ
The vectors ~
xðtÞ, ~
uðtÞ, and ~
yðtÞ have the same interpretation
as
above.
$
AðtÞ is the n · n dimensional state matrix. The offdiagonal ijth element Aij can be interpreted as the rate
constant of the reaction converting the jth state to the ith
state. The diagonal elements Aii represents all pathways by
which
the ith state can be converted.
$
B is the n · q dimensional input matrix. The ijth element Bij(t) determines the effect that the jth input has on
the$rate of the ith state.
C is the p · n dimensional output matrix. The ijth element Cij(t) determines how much the jth state participates
$
in the ith output variable. It should be noted that the C
matrix is often chosen to be identity matrix, meaning that
each state of the model is considered an output variable of
the model. $
Finally, D is the p · q dimensional feed-through matrix.
The ijth element Dij (t) determines how much the jth input
contributes to the ith output. A trivial example of a feedthrough is with orange exciting light as an input and with
red fluorescence detector recording the output. A fraction
of the input light can contribute to the measured output
when the optical filters are not perfect. In order to simulate
result of such an ‘imperfect’
measurement, one can use the
$
feed-through matrix D.
State-space model of Photosystem II
We will now show how a system can be mapped onto a
model in the state-space description, discussed above. As
an example system, showing a high degree of complexity,
we will use PSII. The complexity of PSII originates from
its non-linearity under far from equilibrium experimental
conditions. More elemental, the PSII complexity lies in the
very high dimensionality of the state-space.
Each state of the PSII core is made up of at least nine
components, even in an idealized case of a homogeneous
PSII population (see Govindjee 1990; Lavergne and Briantais 1996 for discussions on PSII heterogeneity),
neglecting cytochrome b559 and the Fe atom. Each of
these nine components themselves can be in different states
(see Table 1).
When translating a system to a model, we need to decide
what the state variables ~
xðtÞ of the model should be. In the
case of PSII, each state in the model will correspond to a
unique combination of the states in which the nine different
components can reside. Since we have six components that
can be in two states, one component with three states, and
two components with five different states we obtain a total
of 4,800 theoretically possible, combinatorial states ~
xðtÞ in
the model for PSII. Each state variable xi(t) corresponds to
the probability that the system is momentarily in the ith
state.
It is the high
dimensionality of the state-space and of the
$
state matrix AðtÞ that make the solution and the interpretation of the highly comprehensive PSII model in Table 1
practically impossible. A total of 4,800 states in the model
correspond to 4,800 · 4,800 = 23.04 million theoretically
possible PSII reactions. The combinatorial complexity
would increase even further if such a huge model of PSII
would be combined with other models of similar complexity to form a more comprehensive model. It is at this
point that model reduction plays an important role in
modeling. The challenge is to determine reduced models of
lower complexity that still show the behavior of interest.
123
228
Table 1 States of the principal
components of Photosystem II
Photosynth Res (2007) 93:223–234
Antenna
PSII donor side
PSII acceptor side
PSII/PQ
Peripheral antenna
Ground
Excited
Core antenna
Ground
Excited
Mn Cluster
S0
S1
YZ
P680
Neutral
Neutral
Oxidized
Oxidized
ChlD
Ground
Excited
Pheo
Neutral
Reduced
QA
Neutral
Reduced
QB
Neutral
Reduced
Model reduction—insight based approach of PSII
model reduction
Models of biological systems are frequently reduced using
insight into the inner workings of the system. We note that
there is no unique reduced model, but that the model
reduction and its result are directly dependent on the
behavior of interest. In this example of model reduction,
we chose as behavior of interest the effect that the exposure
of PSII particles to an ultra-short saturating flash of light, at
time t = 0, has on the fluorescence emission. This means
that the single input to our model is the incident flux of
photons (light), and the output is a linear combination of all
states in which chlorophyll is excited and that give rise to
fluorescence emission. The model that we are going to
reduce is based on Prokhorenko and Holzwarth (2000) and
Holzwarth et al. (2006b).
Due to our choice of fluorescence emission as output
variable, the peripheral components of the electron
transport chain in PSII (in Table 1: the Mn cluster and YZ
on the donor side and the QA and QB on the acceptor
side) do not need to be considered in the reduced model.
The reason for this is that the excited chlorophyll that
gives rise to fluorescence cannot live longer than several
nanoseconds after the flash and that the peripheral components of the electron transport chain do not change their
state during the excited chlorophyll lifetime2. Thus, the
reduced model does not need to include details at slower
time-scales.
Also, we can assume that the fluorescence detectors,
used in these measurements, are not fast enough to capture
processes that happen during the ultra-fast laser flash. Thus,
the reduced model does not need to include dynamics that
occur on time-scales comparable to the laser flash duration.
This means that the reduced model can represent the effect
of excitation by an initial condition3 in which the antenna is
2
This argument is not valid for delayed fluorescence (also called
delayed light emission).
3
Here, we also implicitly assume flash intensity is strong enough to
excite all the antenna systems in the sample, but is not so strong to
generate double-excited antenna systems.
123
S2
S3
S4
PQH2
Empty
Oxidized
2-reduced
excited at the time t = 0 s. Demonstrating another type of
reduction, the peripheral antenna can be removed from the
PSII particles biochemically, leaving only core antenna and
the reaction center chlorophyll RC ChlD to hold the
excitation. The resulting reduced model includes only five
components with a theoretical maximum of 48 combinatorial states (Table 2).
Another insight-based reduction is possible, when
considering which states cannot be reached from the
chosen initial state (Antenna*.ChlD.P680.Pheo.QA). These
unreachable states include double-excited antenna and
states that do not preserve electro-neutrality of the entire
complex (more positive charges than negative or vice
versa) or that would require multiple charge separation.
Eventually, only the six combinatorial states shown in
the Table 3 can be generated from the particular initial
state.
The final reduced model thus has only six states that
interact with each other by 10 reactions for which the rate
constants are different from zero. The reduced model, in
terms of its states and rate constants, is represented in
Table 4 (Holzwarth et al. 2006b).
$
In Table 5,$ the reduced model is represented by the AðtÞ
matrix. The B matrix has no meaning for the model, since
the input u(t) is zero for all times. Instead, the excitation of
the model for simulation purposes is obtained by translating the considered input signal (saturating flash of light at
time t = 0) to the initial, non-equilibrium, conditions of the
model: x2(t = 0s) = 1 and xi(t = 0s) = 0 for i „ 2. The
output variable of the model ~
yðtÞ consists of a single element, the photon flux of the fluorescence emission that is
proportional to probability of the excited states multiplied
by the rates of the fluorescence de-excitation:
$
y(t) = k2,1(fluor.) . x2(t) + k3,1(fluor.) . x3(t). Thus, the C
matrix is of 1 · 6 dimension: (0, k2,1(fluor.), k3,1(fluor.), 0,
0, 0). The model $does not contain any feed-through,
resulting in a zero D matrix.
An advantage of this insight-based approach to model
reduction is that it is relatively straightforward. However, a
detailed insight into the system is required and many
assumptions have to be made that, even if they seem
obvious, might not be valid.
Photosynth Res (2007) 93:223–234
Table 2 Reduced set of states
of the principal components of
Photosystem II in a ‘subnanosecond’ experiment with
removed peripheral antenna
(compare with Table 3)
229
Antenna
Core antenna
Ground
PSII donor side
P680
Neutral
Oxidized
ChlD
Ground
Excited
Pheo
QA
Neutral
Neutral
Reduced
Reduced
PSII acceptor side
Excited
Oxidized
Table 3 Reduced set of PSII principal components and states in a ‘sub-nanosecond’ experiment with removed peripheral antenna
PSII components
P680
ChlD
Core antenna
Pheo
QA
PSII state ID
(reduced set)
PSII state annotation
1
Neutral
Ground
Ground
Neutral
Neutral
Ground state in dark (terminal state)
2
Neutral
Ground
Excited
Neutral
Neutral
Excited antenna state (initial state)
3
Neutral
Excited
Ground
Neutral
Neutral
Excited reaction center primary donor
4
Neutral
Oxidized
Ground
Reduced
Neutral
RC with the primary charge separation
5
Oxidized
Neutral
Ground
Reduced
Neutral
RC with the secondary charge separation
6
Oxidized
Neutral
Ground
Neutral
Reduced
RC with the tertiary charge separation
Table 4 Reduced set of PSII reactions and reaction rates (‘sub-nanosecond’ experiment with removed peripheral antenna)
Rate constant name,
Initial state (from)
Final state (to)
#
#
Rate constant value, s–1
Annotation/comment
kb2,1
2
P680/ChlD/A*/Pheo/QA
1
P680/ChlD/A/Pheo/QA
2e + 5 (heat)
6.7e + 7 (fluorescence)
Antenna
de-excitation
kb2,3
2
P680/ChlD/A*/Pheo/QA
3
P680/ChlD*/A/Pheo /QA
1.92e + 10
Energy transfer
from A to ChlD
kb3,2
3
P680/ChlD*/A/Pheo/QA
2
P680/ChlD/A*/Pheo /QA
2.5e + 10
Energy transfer
from ChlD to A
kb3,1
3
P680/ChlD*/A/Pheo /QA
1
P680/ChlD/A/Pheo /QA
2e + 5 (heat)
6.7e + 7 (fluorescence)
Reaction center
de-excitation
kb3,4
3
P680 ChlD*/A/Pheo/QA
4
P680/ChlD+/A/Pheo –/QA
5e + 11
kb4,3
4
P680/ChlD+/A/Pheo –/QA
3
P680/ChlD*/A/Pheo/QA
5e + 11
Primary charge
separation
Charge recombination
kb4,5
4
P680/ChlD+/A/Pheo –/QA
5
P680+/ChlD/A/Pheo –/QA
2.5e + 11
ChlD+ reduction
by P680
kb5,4
5
P680+/ChlD/A/Pheo –/QA
4
P680/ChlD+/A/Pheo –/QA
6.7e + 10
Reverse e flow
from ChlD to P680+
kb5,6
5
P680+/ChlD/A/Pheo –/QA
6
P680+/ChlD/A/Pheo/QA
–
4.8e + 9
QA reduction
by Pheo–
kb6,5
6
P680+/ChlD/A/Pheo/QA
5
P680+/ChlD/A/Pheo –/QA
2.4e + 9
Reverse e flow
from QA– to Pheo
–
Mathematically rigorous model reduction
In engineering science, especially in control theory, many
standard techniques of model reduction have been
developed that are more systematic than the insight-based
approach outlined above. The large number of available
model reduction methods reflects the fact that no single
method is universally superior. Their appropriateness
depends on the type of complexity in the original model,
and on the purpose of the reduction.
Some model reduction methods, such as multiple steady-states or sustained oscillations (Danø et al. 2006) that
aim at understanding of complex dynamic behaviors have
not yet been tested in photosynthesis research. Some other
model reduction techniques have already been employed in
a semi-empiric way of insight-based approaches without
referring to their mathematically rigorous form (see the
above ‘PSII model reduction’ discussion). Most of empiric
photosynthetic models have been reduced based on the
observation that there are states of the model that can not
123
230
Photosynth Res (2007) 93:223–234
Table 5 State matrix elements corresponding to Tab. 4
State matrix
element Aij
j=1
2
3
4
5
6
i=1
0
kb2,1(heat) + kb2,1(fluor.)
kb3,1(heat) + kb3,1(fluor.)
0
0
0
2
0
–kb2,3 – kb2,1(heat) – kb2,1(fluor.)
kb3,2
0
0
0
3
0
Kb2,,3
–kb3,2 – kb3,1(heat)
– kb3,1(fluor.) – kb3,4
kb4,3
0
0
4
0
0
kb3,4
–kb4,3 – kb4,5
kb5,4
0
5
0
0
0
kb4,5
–kb5,4 – kb5,6
kb6,5
6
0
0
0
0
kb5,6
–kb6,5
be reached by certain stimuli, or states that do not contribute to the output/outputs of interest. Thorough mathematical basis in systems theory corresponds to the notions
of controllability and observability (Rugh 1996)4. A
mathematical model reduction method that is based on
controllability and observability is denoted balanced truncation, and it is frequently used in technical applications
(Ljung 1981). However, for biological systems this method
is often less appropriate since the states of the reduced
model do correspond to linear combinations of the states of
the original models and thus may not be easy to interpret
biochemically. A possible use of this method in comprehensive photosynthesis modeling is to reduce only the parts
of the system for which no mechanistic understanding is
required. An example is given in Liebermeister et al.
(2005). Methods that lead to the exclusion of reactions are
sensitivity analysis and term-based methods. Such methods
remove the reactions or terms that have the least impact on
the output variables of the model (Okino and Mavrovouniotis 1998).
As we pointed out earlier, features that are possible to
utilize in the model reduction of photosynthesis are the
widely different timescales at which different reactions
occur. In reaction systems some fast reactions can be
considered instantaneous in comparison to the slower
reactions. This leads to the so-called quasi steady-state
approximation (QSSA). Two methods based on the QSSA
are the computational singular perturbation and the algebraic approximation of the inertial manifold (Okino and
Mavrovouniotis 1998). Unfortunately, also with these
methods the states of the reduced model are usually linear
combinations of the states of the original model complicating detailed molecular interpretation of the new state
variables. This limitation may not be so strict in photosynthesis compared to other biochemical networks because
4
Roughly, controllability reflects existence of input stimuli that could
bring the system in a finite time from an initial state to a particular
system state. If such stimuli are not available, the particular state can
be eliminated from the reduced model. Observability reflects influence that a particular state exerts on any of the modeled outputs.
123
photosynthetic reactions are naturally lumped in complexes
that can be often characterized by a small number of state
variables that approximately characterize the entire
complex.
An example of mathematically rigorous lumping
Lumping is also a model reduction method that is based on
the presence of different timescales within the system.
Lumping is commonly used in biochemical modeling
based on insight into the system, as discussed above using
the PSII as an example system. Nevertheless, lumping may
also be based on a systematic mathematical analysis of the
system (Maertens et al. 2005). The resulting reduced
model contains states and reactions that are lumps of the
states and reactions in the original model.
Figure 1 shows results of such a rigorous lumping procedure for the model of PSII by Holzwarth et al. (2006b).
The lumping procedure starts by defining the relevant
dynamic window, e.g., by high and low frequency limits.
For the sake of simplicity, we shall define the model
reduction only by the high frequency limit. This corresponds, for example, to light detector that cannot capture
fluorescence emission earlier than certain time delay (s)
following the flash. The model reduction algorithm yields
new state variables; some of them being lumped state
variables of the non-reduced model.
Lumping obtained for a slow detector (s = 1 ns) is
shown by the red rectangle in the upper panel of Fig. 1.
Fluorescence kinetics predicted by such a reduced model is
shown in the lower panel of Fig. 1 by the red kinetic trace.
Obviously, the simulations by the original non-reduced
model (black line) and by the reduced model coincide for
the time range 1 nanosecond and longer whereas the
reduced model fails to simulate correctly the fast kinetics.
Modeling of experimental results obtained with light
detector that is of intermediate time response (s = 100 ps)
is represented in Fig. 1 by green color. Match between the
simulations by the non-reduced model (black line) and by
the model limited to 100 ps delay time (green line) is poor
Photosynth Res (2007) 93:223–234
Fig. 1 Model reduction by mathematically rigorous lumping. Four
lumping levels are presented: No lumping (black line) is corresponding to model of Holzwarth et al. (2006b). Slow detector lumping
(s = 1 ns) is represented by red line, intermediate detector lumping
(s = 100 ps) is shown by green line and fast detector lumping
(s = 100 ps) is represented by blue line. The upper panel shows the
structure of the non-reduced and reduced models. The lower panel
shows modeled fluorescence transients
in the fast domain (t < 100 ps) and it is not perfect in the
long time domain (t > 100 ps). This partial failure of the
intermediate reduced model is due to the fact that the
chosen detector response time of 100 ps is not sufficiently
distinct from the turnover times of the system (Table 4).
The model reduction by lumping that corresponds to
the short detector response time (s = 100 ps) is shown in
Fig. 1 by blue color. Clearly, the fluorescence transient
modeled by the reduced model (s = 100 ps, blue line)
closely matches the transient simulated by the original,
non-reduced model (black line). Also, the back-translation
from the lumped state of the reduced model to the
original states of the non-reduced model yields state
dynamics nearly identical to the non-reduced model (not
shown).
Mathematical modeling of
photosynthesis—requirements
The key requirements for comprehensive modeling are
standard formats for the representation of models and
experimental data (Quackenbush 2006). In the area of
231
systems biology, a considerable effort has already been put
into the development of the Systems Biology Markup
Language (SBML, Hucka et al. 2003) that has become a de
facto standard for the representation of biological models.
A further requirement for the successful interdisciplinary
study of large-scale processes, such as photosynthesis, is
the need for collaboration between interdisciplinary
research groups, such as it is done in the Yeast Systems
Biology Network (http://www.ysbn.org). These consortia
recognize that it is beyond capacity of any single research
team to generate comprehensive models of biological
organisms or of complex biological processes. This is true
even for comprehensive modeling of the simplest photosynthetic organisms such as anoxygenic photosynthetic
bacteria, and cyanobacteria. Such a challenging goal can be
reached only by sharing and collaboration of a large multidisciplinary consortium. A further improvement of model
sharing and integration capabilities can be obtained by
common namespace for used biological compartments,
chemical species and reactions, specified within one of the
existing ontologies (e.g., http://www.ebi.ac.uk/ego of the
European Bioinformatics Institute) and within the relevant
databases (e.g., http://www.genome.jp/kegg/ of the Kyoto
Encyclopedia of Genes and Genomes). In the existing
ontologies and databases, the photosynthetic components
are underrepresented which may reflect a general bias
favoring heterotroph and, particularly, mammalian cell
functions. Thus, one of the tasks advocated in this review is
to reach adequate representation of photosynthesis in these
databases.
Photosynthesis as a process that can be modeled in its
full spatio-temporal complexity, using a previously
described global strategy, is currently represented by two
projects aiming at the global Internet-based integration of
photosynthetic models: E-photosynthesis.org (registered in
2004 as http://www.e-photosynthesis.org) organized by
two of the present authors and E-photosynthesis.net
established by S. Long, X.-G. Zhu and R. Syed of
University of Illinois (registered in 2005 as http://www.
e-photosynthesis.net). The E-photosynthesis.net site aims
primarily at predicting ‘‘the dynamics of ATP and
NADPH formation under dynamic light and CO2 conditions’’ and at testing ‘‘hypothesis regarding different
signals related to photosynthesis, such as CO2 uptake
kinetics and fluorescence induction kinetics’’. The
E-photosynthesis.org site is an open modeling platform
that is based on standards shared throughout biology (e.g.,
SBML) and that aims at using, validation, and integration
of reduced photosynthesis models at all levels of systems
complexity. The openness of the platform means that
members of wide community of modelers are invited to
present their models and data simulation in http://www.
e-photosynthesis.org.
123
232
The E-photosynthesis framework
E-photosynthesis.org framework is accessible at the URL:
http://www.e-photosynthesis.org. The project is based on
the Client–Server strategy (Fig. 2). The user’s personal
computer, called the Client computer, can connect via the
Internet to the Server computer that harbors a database of
modeling projects, experiments and simulation software to
perform the model simulation of selected experiments.
To provide a playground for scientists interested in
photosynthesis research, different projects concerning
particular subsystems are implemented in E-photosynthesis.org. Each modeling project has a leader who selects the
states of the photosynthetic apparatus to be involved in the
model. The states are selected from a unique database of
states that is shared in all modeling projects of E-photosynthesis.org. This database may serve for future annotation and ontology of photosynthetic structures and
functions. Then, the model is settled by definition of
chemical reactions. This approach offers a productive way
to remotely develop a particular model with the possibility
of direct feedback from interested users and even competitive project leaders.
Apart from the definition of states and reaction rate
constants, the E-photosynthesis.org modeling framework
also allows the definition of a modulator for each reaction.
These modulators allow the specification of a non-linear
behavior. Modulators will be used to model the interactions
between different subsystems of photosynthesis and to
implement time variation in inputs, rate constants and state
concentration dependencies.
Fig. 2 Client–Server architecture shown in the introduction page of
E-photosynthesis. User (1) connects by web browser of a personal
computer to http://www.e-photosynthesis.org. The Graphical User
Interface of e-photosynthesis (2) allows selection of one of the
supported model projects (3). The model parameters can be
interactively modified. Model projects are supplemented with
experimental data and experiment description (4) that serve to define
external constraints used in model simulation and to verify the model
performance. Model (3) and model constraints reflecting particular
123
Photosynth Res (2007) 93:223–234
An important feature of E-photosynthesis.org is the fact
that models should always be coupled to experimental data.
This means that the project leader should provide the
experimental data that are important for his/her project in
order for users to be able to compare them with results
obtained from model simulations. The data are supplemented by experiment annotation, e.g., reference to relevant papers, taxonomic specification of the organism,
biological treatment, mutations, or chemical intervention
(to be adopted from Zimmermann et al. 2006). The
experiment is further specified for purposes of model
simulation by assumed initial state, irradiance protocol
described by time-irradiance profile, temperature and other
model constraints.
The user can visualize the selected model in the form of
interconnected icons or in the form of standardized model
annotation. The rate constants as well as modulators can be
edited and saved in individual datasets to perform model
simulation with a particular dataset or a default one.
E-photosynthesis.org is a dynamically developing
framework where existing reduced photosynthetic models
can be deposited for sharing, validation, and standardization. The long-perspective goal of the project is to generate
comprehensive model of photosynthesis in the sense
defined above.
A concluding remark
This special issue of Photosynthesis Research is dedicated
to Govindjee who has played an important role in the
experiment (4) are determining set of differential equations (5) to be
solved in the simulation. The set of ordinary differential equations is
solved by ODE solver (6) installed in the server. Results of the
simulation are displayed on-line on the client computer by the
Graphical User Interface of E-photosynthesis. The simulation results
as well as the model described in SBML format can be exported for
further integration (8). Not functional yet is the parameter estimation
that would allow optimization of the model to fit particular data sets
(7, dashed line)
Photosynth Res (2007) 93:223–234
integration of our field for the last half of the century. It is
not co-incidental that his dedication to integrating the
knowledge on photosynthesis led recently to proposing a
new volume of his Advances in Photosynthesis and
Respiration (ISSN: 1572-0233) that will be entitled
‘‘Photosynthesis In Silico: Understanding complexity from
molecules to ecosystems’’. In this review, we have outlined
techniques and approaches that will be applied in reaching
this challenging goal.
Acknowledgments LN was supported by the Czech Academy of
Sciences Grant AV0Z60870520, by the Czech Ministry of Education,
Sports and Youth Grant MSM6007665808, by the Grant Agency of
the Czech Republic GACR 206/05/0894. JČ was supported in part by
the grant 1M0567 of the Czech Ministry of Education. HS was
supported by a grant from the Swedish Foundation for Strategic
Research.
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