Model

What is modeling?

Modeling with systems biology is based on a system of reactions between different states, where states may be for example steps in a larger chemical reaction (see example below). By knowing the parameters for each reaction between the states, differential equations (with changes over time) for the different states can be created. These differential equations can be displayed as individual graphs over time.

When you have one or more models for a system, laboratory data can be used to reject incorrect models. The final model can then be optimized and more (previosly unseen) data can be used to confirm or deny overfitting of the model to the previous data.

Example:

The following reactions occur with the following reaction rates, assuming mass action kinetics:

\[(1): A \xrightarrow{k_1} B, r_1 = k_1*[A]\] \[(2): B \xrightarrow{k_2} C, r_2 = k_2*[B]\] \[(3): A + C \xrightarrow{k_3} D, r_3 = k_3*[A]*[C]\] \[(4): D \xrightarrow{k_4} , r_4 = k_4*[D]\]

From these four reactions, the following differential equations are made, assuming mass action kinetics:

\[\frac{d}{dt}[A] = -r_1 - r_3 = -k_1*[A] - k_3*[A]*[C] \] \[\frac{d}{dt}[B] = r_1 - r_2 = k_1*[A] - k_2*[B] \] \[\frac{d}{dt}[C] = r_2 - r_3 = k_2*[B] - k_3*[A]*[C] \] \[\frac{d}{dt}[D] = r_3 - r_4 = k_3*[A]*[C] - k_4*[D] \] Depending in the rate constants (\(k_1, k_2, ...\)) and the initial concentrations of the compounds (\([A]_0, [B]_0, ...\)), the spectra of the differential equations will look different based on the mentioned parameters (rate constants and initial concentrations).

The model we use and how

The MATLAB model we created is based on the Wolfram Mathematica model FoldEco for protein folding in E. coli [1] by Powers, E. T., Powers, D. L., & Gierasch, L. M. [2].

FoldEco - using Wolfram Mathematica

FoldEco is a computational model [1] of the E. coli protein homeostasis (also known as proteostasis). With FoldEco, it is possible to follow a single cytosolic protein during its production, as it can enter different stages. The stages in FoldEco are [2]:

Protein Synthesis and Folding.
The KJE System.
The GroELS System.
The Disaggregation System.
Protein Degradation.

Protein Synthesis and Folding contains the synthesis of the protein with the ribosome and trigger factor (TF). After the unfolded protein has been synthesized, it can be folded natively, misfold (and may aggregate), enter the KJE System or enter the GroELS System. [2]

The KJE System unfolds misfolded or unfolded proteins with the proteins DnaK, DnaJ and GrpE. If a protein has been unfolded and is bound to ATP-bound DnaK, it may re-enter the KJE System. [2]

In the GroELS System, unfolded or misfolded proteins can be folded into its unfolded, misfolded or native state by the proteins GroEL and GroES, by providing an isolated environment for the protein folding. [2]

The Disaggregation System contains the proteins DnaK, DnaJ and GrpE which unfolds monomers from small aggregates (four or less monomers in aggregate). DnaK and DnaJ are also involved in the aggregates binding to the protein ClpB, where ClpB removes one aggregate monomer at a time as a unfolded monomer. [2]

Protein Degradation can occur in two ways in FoldEco, either by the protease Lon which degrades unfolded or misfolded proteins, or by ClpAP-type proteases (Dn) which degrades tagged native proteins. [2]

FoldEco contains different approximations. These include: assuming free mixing of the components in the cytoplasm, in vitro measurements are used for in vivo, and that the concentration of ATP is constant. All assumptions and simplifications can be found in the supplemental information to the original article. [2]

FoldEco has been used to compare real performed experiments. The authors noticed that the simulation matched the experimental results well, although not completely. The authors says that models can be used to explain experimental results, meaning that something is missing in the model if it doesn’t explain the results. [2]

Our translated model - using MATLAB

The differential equations based on the FoldEco model are visualized in MATLAB using IQM Tools [3], to obtain graphs of the changes for the different states. By having data from the laboratory, we can either confirm or deny the model. If the model is confirmed, we change different parameters to try to obtain the desired task - as much folded protein as possible.

The model we created based on the FoldEco model has not been tested! Our translated model is available from here and is given as is.

Sources

1. Link to the FoldEco website. URL: http://foldeco.scripps.edu:8080/webMathematica/foldecoindex.html
2. Powers, E. T., Powers, D. L., & Gierasch, L. M. (2012). FoldEco: A Model for Proteostasis in E. coli. Cell Reports, 1(3), 265–276. http://doi.org/10.1016/j.celrep.2012.02.011
3. Link to IQM Tools. URL: http://www.intiquan.com/iqm-tools/#toggle-id-3