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Cells_in_medium

Cells in medium

Synchronization of cells through transport of metabolites to the medium

This follows the work of Wolf and Heinrich (1), which takes a simple model of glycolytic oscillations from Sel'kov (2) as the base unit (a cell with oscillating glycolysis) and creates a new model with suspensions consisting of an arbitrary number N of interacting units. The individual cells interact via the flux of metabolites, which are produced in all cells and may permeate through the cell membranes (1). Two specific cases are examined: in Model I the coupling is via the product of the autocatalytic reaction (species Y, case 1 and 2), and in Model II the coupling is via the substrate of the autocatalytic reaction (species X, case 3).

We create three examples following (1) using either Model I or II. Each example is codified in a shell script that calls sbmodelr, using a base COPASI file encoding the basic model for a single unit. We use a different base file for each case, however they only differ in the value of the constant k and the settings for the time course (this allows one to run the resulting file without first having to adjust parameters).

Invoking sbmodelr with appropriate options, creates a new model consisting of several cells that export one species (Y for Model I, and X for Model II) to a medium with the appropriate volume (5-fold larger than the total volume of all cells, according to ref. 1). We also set slightly different initial conditions for each unit (variance of ~10%).

For each example there is a shell script that shows the complete command invoking sbmodelr, the base file, the resulting complex model file (case[123].cps), and an image file with results. The images were obtained by specifying an appropriate plot in the resulting files using the COPASI GUI, as sbmodlr does not copy plot settings to the new models.

Case 1

This uses Model I, with 2 cells producing synchronous oscillations (file ex1case1.sh). We set the two cells to have different initial conditions (options --pn), set the transport rate constant for transport to be 3.2, as in Fig. 3 of ref. 1), and the medium volume to be 10 (to keep the ratio of volumes to 0.2 as in ref. 1).

command line options comment
sbmodelr run sbmodelr
--output case1.cps name the output file
--add-medium include a medium unit
--medium-volume 10 set medium volume (5*N)
--transport Y transport the species Y (Model I)
--transport-k 3.2 value of rate constant for transport
--pn X 0.1 uni initial values of X inside +/-10% interval around value of X in base
--pn Y 0.1 uni initial values of Y inside +/-10% interval around value of Y in base
Selkov-Wolf-Heinrich.cps COPASI file with the base unit
2 create 2 units

Running the command explained above (e.g. by running file ex1case1.sh) results in a new model file case1.cps. Loading that file into COPASI, creating a plot for Y_1, Y_2 and Y_medium, and adjusting its axes, reproduces Fig. 3 of reference 1.

Reproduction of Fig.3 of reference 1, the two intracellular Y oscillate in phase and Y in the medium oscillates with lower amplitude and out of phase

Case 2

This uses Model I, with 5 cells producing regular asynchronous oscillations (file ex1case2.sh). We set the five cells to have different initial conditions (options --pn), set the transport rate constant for transport to be 1, as in Fig. 3 of ref. 1), and the medium volume to be 25 (to keep the ratio of intracellular over extracellular volume to 0.2 as in ref. 1). We use a different base file, Selkov-Wolf-Heinrich_k2.5.cps, which only differs from the one used in case 1 by setting the constant k=2.5 (instead of 3.84).

command line options comment
sbmodelr run sbmodelr
--output case2.cps name the output file
--add-medium include a medium unit
--medium-volume 25 set medium volume (5*N)
--transport Y transport the species Y (Model I)
--transport-k 1 value of rate constant for transport
--pn X 0.1 uni initial values of X inside +/-10% interval around value of X in base
--pn Y 0.1 uni initial values of Y inside +/-10% interval around value of Y in base
Selkov-Wolf-Heinrich_k2.5.cps COPASI file with the base unit
5 create 5 units

Running the command explained above (e.g. by running file ex1case2.sh) results in a new model file case2.cps. Loading that file into COPASI, creating a plot for Y_1, Y_2, Y_3, Y_4, Y_5 and Y_medium, and adjusting its axes, reproduces Fig. 5 of reference 1.

Reproduction of Fig.5 of reference 1, the five intracellular Y oscillate out phase shifted by 1/5 the value of the period; Y in the medium oscillates with very low amplitude, nearly constant

Case 3

This uses Model II, with 3 cells producing a stable steady state but with oscillations before convergence (file ex1case3.sh). We set the three cells to have different initial conditions (options --pn), set the transport rate constant for transport to be k=4.16, as in, and the medium volume to be 15 (to keep the ratio of volumes to 0.2 as in ref. 1).

command line options comment
sbmodelr run sbmodelr
--output case3.cps name the output file
--add-medium include a medium unit
--medium-volume 15 set medium volume (5*N)
--transport X transport the species X (Model II)
--transport-k 4.16 value of rate constant for transport
--pn X 0.1 uni initial values of X inside +/-10% interval around value of X in base
--pn Y 0.1 uni initial values of Y inside +/-10% interval around value of Y in base
Selkov-Wolf-Heinrich_k1.5.cps COPASI file with the base unit
3 create 3 units

Running the command explained above (e.g. by running file ex1case3.sh) results in a new model file case3.cps. Loading that file into COPASI, creating a plot for X_1, Y_1, X_2, Y_2, X_3, Y_3, and Y_medium displays the complex approach to the steady state with oscillations. Note that not all units converge to the same steady state. This corresponds to a small region in the bifurcation diagram of Fig. 12 of reference 1 (between lines kappa_c and kappa_a).

Complex approach to a stable steady state

References

  1. Wolf J, Heinrich R (1997) Dynamics of two-component biochemical systems in interacting cells; synchronization and desynchronization of oscillations and multiple steady states. BioSystems 43:1–24
  2. Sel’kov EE (1968) Self-oscillations in glycolysis. 1. A simple kinetic model. European Journal of Biochemistry 4:79–86