Modelling of Population Dynamics

Introduction

A graphical user interface is included to model single and multipulse population dynamics for an ensemble or randomly oriented particles that interacts with a polarised laser beam. The orientation of a molecule determines its transition dipole moment tdm. The orientation of that tdm with respect to the polarisation of the laser beam determines the probability it is photo-excited (or photolysed). Assuming no rotational diffusion, the excited molecule can then return to the ground state (having the same orientation distribution function as it had before being photo-excited). This ground state recovery is time-dependent, and is generally dependent on its molecular properties and enviroment. Ideally, the ground state recovery is measured, but this is not always feasible. Alternatively, a model that describes the photophysics upon excitation can be used to estimate the time-dependent concentration for each specific (meta-) stable spectroscopic state. Knowledge about the time-dependent ground state recovery fraction (i.e. the photophysical model) is required to model the interaction between that fraction and a second resonant laser pulse. The presented toolbox does this for you. Molecular properties and its time course (determined by the model) are needed to estimate the fraction molecules that is photolysed by the first laser pulse, and that of a second laser pulse that comes after a certain time delay. That second pulse can interact with the fraction molecules that has not been excited by the first pulse, and that fraction that has recovered after a time delay.

The global analyis package can supply you the time-dependent ground state recovered fraction. A global analysis requires a model describing the photophysical patways and connectivities between (meta-stable) states that occur your system after photo-activation. If is known which state corresponds to the ground state, the concentration profiles generated can give you an estimate on the ground state recovered fraction after a specific time delay. Alternatively, a model can be constructed in the Simbiology toolbox.

Single pulse experiment demo

Here a demonstration is given on how to use this toolbox by using real- world values of phytochrome. The Symbolic Math toolbox is required to run this package.

Start the gui by excuting (in the command window):

The main gui appears.

Enter the requested parameters for the single pulse laser experiment (i.e. pump-probe). Choose for the modelling of a Gaussian or flat top (multi-mode) laser beam profile by ticking the appropriate box. Phytochrome is used here as an example. A box is also shown for pH, but this is not used for any calculation. The main reason it is there is to make you aware of the fact that the absorption cross-section can significantly change with acidity.

The 'Calculate single pulse fraction' button starts the calculation for a one-pulse experiment (i.e. a pump-probe), and shows the influence of the power density on the orientation distribution. In other words, the probablity a molecule is excited dependes on x, the relative orientation with respect to the polarisation of the laser pulse (left panel). The integral under these curces represents the ensemble averaged photolysed fraction (middle panel). The right panel shows the beam size dependence on the same integrated photolysed fraction. Note that the first time you execute this script the Symbolic Math toolbox needs to be loaded. Therefore, the second time you execute the script it will complete faster.

In addition, the calculated ensemble-averaged photolysed fraction is given in the gui. The difference between the resulting two values is that one is calculated analytically, the other one is approximated by a Taylor-series. Depending on the number of orders included the accuracy improves, but the calculation time obviously increases. Set the orders to include in the Taylor series:

For instance, for 12 orders:

And for 25 orders the Taylor series approximates the theoretical value:

The number of relative orientations x can be set by adjusting its corresponding spacing parameter. Since x ranges from -1 to 1, a spacing of 0.1 leads to 21 values. You can also set the number of 'power factors', a multiplier of the used laser power density, to see the influence of the excitation power on the orientation distribution of the photolysed fraction (left panel in the results figure). The number of power densities (the x-axis values in the middle panel) used to calculate the ensemble-averaged photolysed fraction can also be set. Similarly, the number of x-coordinates (which correspond to multipliers of the beam diameter set in the gui) for the right panel can be set.

Multi-pulse calculations

If a multi-pulse experiment is modelled, proceed with defining the properties concerning the second pulse. For instance:

And give the delay(s) between the first and second pulse, and their corresponding ground state recovered fractions need to be defined. These can come from measurements, or from a model (see above). The global analyis package can be used for this, or the Simbiology toolbox. See the introduction above for more information. Fill the requested boxes. The delay times are only used for plotting (they represent the x-coordinates for the bottom-right panel in the results figure below).

After having pressed the two-pulse fraction calculation button, the photolysed fractions are calculated for pulse 1 (top row figures) and for the two-pulse experiment (bottom row). Depending on the ground state recovery fraction, which is user-defined (0, 0.14 0.65 and 0.9 in this case), the photolysed fraction of the second pulse changes. The ground state recovery fraction needs to be modelled by the user, which can be done by using the Globe_Toolbox. Use the concentration profile plots to identify the recovered fraction for your system. The recovery of the orientation distribution of the initially photoexcited population is shown in the bottom-left panel (1-β represents the ground state recovered population). The interaction of the second pulse with the recovering population is shown in the bottom-middle panel. The ensemble-averaged photolysed fraction of the first and second pulses are shown in the last figure as function of the delay time between those pulses.

The calculated fractions of the last figure are also output to the gui for your convenience. The numerical solution of the averaged photolysed fraction of the second pulse that comes after the given time delay approximates the analytical solution if a Taylor series cut-off of above 40 is used (see above to see how to set the Taylor series order).

Further reading

For more details the reader is referred to: Modelling multi-pulse population dynamics from ultrafast spectrscopy, LJGW van Wilderen, CN Lincoln and JJ van Thor.

See Also

- global analyis package