Fig 1.
Representation of a guard cell as a multi-membrane complex.
The model involves two membranes and a hand-picked selection of transporters. Only the outer membrane offers mechanical resistance to volume changes, which results in an equal pressure in both compartments. Each compartment contains a certain amount of hydrogen, chloride, potassium and water. All these chemical amounts compose the system state, from which the energy function is evaluated. (Gear image adapted from Issi at openclipart.org).
Fig 2.
Electric representation of a two-membrane complex.
Each capacitor represents a membrane and each compartment corresponds to a node in the circuit. Variations of ion amounts in each compartment result in adding or removing charges in these compartments. The circuit representation suggests that adding charged particles into compartment 2 contributes to charging both capacitors C1 and C2. This circuit analogy is used to evaluate the electrostatic energy term.
Fig 3.
A toy multi-membrane complex with one membrane and three transporters.
The hydrogen pump is an active transporter, while the hydrogen/chloride symporter is passive. Water transport across the membrane is also considered passive. As there are three species and one compartment, the system state is a vector of size 3. We use this toy problem to illustrate the proposed quasi-static setting.
Fig 4.
Slice view of the space of states in the plane (,
) for the toy problem.
The level curves in the background represent the energy function G. Once the hydrogen pump has brought the system from to
, the passive evolution toward an equilibrium is only possible along the affine subspace
, which is directed by
and
(
is out of the plane). The minimum of G in
corresponds to points where the level curves of G are tangent to
. As x increases, the plane
continuously moves, dragging the equilibrium state
, which follows the thick curved trajectory. Thus, the system state always minimizes the same function G, but in a subspace that moves as x increases.
Table 1.
Numerical values of chosen parameters and initial conditions for the simulation.
Fig 5.
Simulation results for the guard cell model.
The simulation output consists of the evolution of the number of moles of each chemical species as the progress of active transporters increases. Number of moles of water, chloride and potassium are plotted in a cumulated way, with the upper part representing the number of moles in the cytoplasm and the lower part representing the number of moles in the vacuole. While pH reflects the hydrogen concentration, pressure and electric potentials are obtained from derivatives of the energy function.
Fig 6.
Simulation results with constant ratios between pump rates.
The rates of the vacuole pump are (left),
(middle), and
(right). The plotted values are the numbers of moles of chloride (top), potassium (middle) and water (bottom), with the amount of water being proportional to the compartment volume. All plots on a same row share the same y-axis. The ratio between osmolyte and water transport at the vacuole membrane and at the plasma membrane is asymptotically similar to the ratio between the hydrogen pump rates, which corresponds to equal concentrations of osmolytes between compartments.
Fig 7.
Simulation result involving a constant-volume constraint.
The rate of the vacuole membrane pump is controlled to keep the cytoplasm volume constant. The plot (b) shows the progress of both pumps along the simulation. This is an example of tuning the model input parameters to qualitatively reproduce experimentally observed phenomena.
Fig 8.
Simulation result for the guard cell model with hydrogen buffer.
The plotted physical quantities are the same as in Fig 5. The cytoplasm pH values follow a more realistic behavior when hydrogen buffering is taken into account.
Fig 9.
Hessian matrices for each term of the energy function, evaluated at the final state of the simulation.
Hessian coefficients (in ) are represented in logarithmic scale, i.e. 27.2 means 1027.2 and a blank case means that the coefficient is zero. The Hessian matrices define the sensitivity of chemical, electrostatic and mechanical forces to perturbations.
Fig 10.
Spectral information about the total energy Hessian matrix.
(a) Eigenvectors of the total energy Hessian at the final state, with corresponding eigenvalues
. (b) Evolution of the Hessian eigenvalues along the simulation steps. Eigenvalues change but keep the same ordering during the process, and the corresponding eigenvectors evolve continuously during the simulation. (c) Angle between each eigenvector
at a given step and its final position. A small angle means that eigenvectors are close to each other. After a short transient regime, eigenvectors remain less than one degree apart from their final position.
Fig 11.
Summary of the transport cascade at the plasma membrane in the guard cell model without buffering, showing the hierachy of forces at work upon a charge imbalance.
Priorities in equilibria is decreasing from 1 to 4. The vacuole is not represented in the figure. In steps 2 and 3, one balance is restored at the expense of another less strictly enforced balance.