Lecture #3

The Process of Simulating Dynamic Mass Balances

Outline

• Basic simulation procedure• Graphical representation• Post-processing the solution• Multi-scale representation

BASIC PROCEDURE

Dynamic Simulation

1. Formulate the mass balances:

2. Specify the numerical values of the parameters

3. Obtain the numerical solution

4. Analyze the results

dtdx =Sv(x;k)

ki=value; I.C. xio=xi(t=0)

Run a software package

xi

t

viGraphically:

t

Obtaining Numerical Solutions:use anyone you like – we use mathematica

Set up the equationsSet up the equations

Specify parametersSpecify parameters

SolveSolve

Specify graphical outputSpecify graphical output

Plot and exportPlot and export

Typical Mathematica WorkbookStep 1

Step 2

Step 3

Step 4

Results

The Results of Simulation:a table of numbers

1 2 3 4 50

xi

t

Can correlate variables, r2

•off-set in time•on a particular time scale

GRAPHICAL REPRESENTATION

Example Graphical Representation of the Solution, x(t)

Multi-time scale representation:segmenting the x-axis

Dynamic Phase Portrait

Can also plot fluxes vi(t) and pools pi(t) in the same way; this is a very useful representation of the results

Characteristic “Signatures” in the Phase Portrait

POST-PROCESSING THE SOLUTION

Post-processing the Solution1. Computing the fluxes

stst

vi

t

vi

vj

t=ot ∞

v(t)=v(x(t))

dynamic response phase portrait

Post-processing the Solution2. Forming pools; two examples

Compute “pools”: p(t)=Px(t) t

p(t)

Example #2

ATP ADP

AMP ADP

then plot

x1 x2

Keq=1

Example #1

total mass

Post-processing the Solution3. Computing auto correlations

n=3first segment

n=mentire range

n=moff-set in time, =2

Example 1

Example 2

Example 3

MULTI-SCALE REPRESENTATION

Tiling Phase Portraitsx1

x2

x2

x1

x3

x1

k

slope=kr2~1

A series of tiled phase portraits can be prepared,one for each time scale of interest

x1

x2

x3

x1 vs. x2 same as x2 vs. x1

symmetric array

use corresponding locations in array to conveydifferent information•graph•statistics

Representing Multiple Time-Scales:tiling phase portraits on separate time scales

Time: 0 -> 3 sec Time: 3 -> 300 sec

Example System:

Example of Time-Scale Decomposition

time

x10=1

x20=x30=x40=0

x3, x4 donot move

Fast

x4 doesnot move

Intermediate Slow

All Times Fast

Intermediate Slow

Tiled Phase Portraits:overall and on each time scale

expanded scalex3 ~ 0

perfect correlation

Post-processing the Solution1. Forming pools

p1 and p2 are dis-equilibrium variables

p3 is a conservation variable

Tiled Phase Portraits for Pools:L-shaped; dynamically independent

(1/2,1)

(0,0)

(0,0) (0,0)

(1,1) (1,1/2)

no correlations

Summary• Network dynamics are described by dynamic mass balances

dx/dt=Sv(x;k) that are formulated after applying a series of simplifying assumptions

• To simulate the dynamic mass balances we have to specify the numerical values of the kinetic constants (k), the initial conditions (x), and any fixed boundary fluxes.

• The equations with the initial conditions can be integrated numerically using a variety of available software packages.

• The solution is in a file that contains numerical values for the concentration variables at discrete time points. The solution is graphically displayed as concentrations over time, or in a phase portrait.

• The solution can be post-processed following its initial analysis to bring out special dynamic features of the network. We will describe such features in more detail in the following three chapters.