Team:Nanjing-China/Model/h2
The Signalling Cascade
Protein hoxB and hoxC form a Heterodimeric core(RH) they acts as an hydrogen sensor in concert with the histidine protein kinase HoxJ . This RH-HoxJ (hoxBC-hoxJ) complex detects the presence of hydrogen in the environment and transmits the signal to a response regulator HoxA. The response regulator, HoxA, is active in its non-phosphorylated form and inactivated by HoxJ-mediated phosphorylation. In the hydrogen atmosphere, HoxA is dephosphorylated and interacts with a promotor hoxP, and it is capable of activating downstream gfp gene transcription.
Thus, our key assumptions for interactions among proteins are:
1.Ignoring the procession of hoxB hoxC and hoxJ combination.
2. the Law of Mass Action kinetics is suitable for Hydrogen detection.
3. the Law of Michaelis–Menten kinetics is suitable for hoxBC-hoxJ catalyst.
4.Because HoxA has been phosphorylated as soon as it’s translated, we assume there is only hoxA-P before hydrogen induces dephosphorization.
Signalling Kinetics
In cell signalling, molecule Hydrogen binds to their receptor sites hoxBC-hoxJ which triggers a conformational change in the receptor. The donor bind to their receptors with varying rates and affinities depending on the molecules involved.
We model this rate using the Law of Mass Action kinetics. This principle governs that the rate of a reaction (d/dt) is proportional to the concentrations of the substrates [2].
For a particular reaction, for example:
the rate can be written mathematically as:
But most biological reactions are reversible, so
Applying these principles to the interaction between hoxBC-hoxJ and Hydrogen yields:
Signalling Cascade Model Parameters |
Value |
Units |
| kf | Hydrogen binding constant |
8(est) |
| kr | Hydrogen dissociation |
0.8(est) |
In cell signaling, the enzyme hoxBC-hoxJ, combined with Hydrogen, catalyzes the reaction of hoxA –P dephosphorization, We model this rate using the Law of Michaelis–Menten kinetics, this model involves an enzyme, E, binding to a substrate, S, to form a complex, ES, which in turn releases a product, P, regenerating the original enzyme. This may be represented schematically as
where (forward rate), (reverse rate), and (catalytic rate) denote the rate constants, the double arrows between S (substrate) and ES (enzyme-substrate complex) represent the fact that enzyme-substrate binding is a reversible process, and the single forward arrow represents the formation of P (product).
Under certain assumption– such as the enzyme concentration being much less than the substrate concentration – the rate of product formation is given by
Applying these principles to the interaction between hoxBC-hoxJ and hoxA-P yields:
Signalling Cascade Model Parameters |
Value |
Units |
kcat |
the turnover number |
950（est） |
km |
The substrate concentration at which the reaction rate is half of |
0.015（est） |
Transcription
Transcription is a complex procession. However, it is well established that initiation of transcription is the major factor in determining overall rate of mRNA production. Transcription initiation is a process influenced by a number of key steps including: promoter escape, binding of the RNA Polymerase onto the promoter, isomerization, and formation of the closed RNA Pol complex from the open complex, and your Escherichia coli strain. In this model, we will assume that binding of the RNA polymerase is the rate determining step in transcription initiation.
Thus, our key assumptions for transcription are:
1. For the constitutively expressed genes (hoxA, hoxB, hoxC, hoxJ), we can assume that they are at maximal transcription.
2. hoxP transcription rate is determined primarily by the amount of substrate (hoxA) bound to the DNA and the strength of the promoter hoxP. Maximum transcription rate occurs when all binding sites are filled.
3. Promoter strength (in Polymerases per second) is a constant, and represents the rate of transcription at maximal expression.
As for the reporter genes, the model is complicated by the binding of signalling molecules, Which is contrast to the constitutively expressed genes, we model this rate using Hill equation[3].
- - Fraction of the receptor protein concentration that is bound to ligand.
- [L]- Free (unbound) ligand concentration
- - Apparent dissociation constant derived from the law of mass action (the equilibrium constant for dissociation), which is equal to the ratio of the dissociation rate of the ligand-receptor complex to its association rate ().[4]
- - The ligand concentration producing half occupation (ligand concentration occupying half of the binding sites). Because is defined so that , this is also known as the microscopic dissociation constant. In recent literature, this constant is sometimes referred to as .
- n>1, Positively cooperative binding: Once one ligand molecule is bound to the enzyme, its affinity for other ligand molecules increases.
- n<1, Negatively cooperative binding: Once one ligand molecule is bound to the enzyme, its affinity for other ligand molecules decreases.
- n=1, Noncooperative (completely independent) binding: The affinity of the enzyme for a ligand molecule is not dependent on whether or not other ligand molecules are already bound.
So
Model Parameters |
Value |
Units |
T5 Promoter strength |
0.012 |
mRNA molecules/sec |
| Vmax | 0.23 |
mRNA molecules/sec |
Kd |
The substrate concentration at which the reaction rate is half of |
0.012（est） |
Translation
After transcription, it must be translated into a functional chromo protein. Translation involves 3 main stages: initiation, elongation of the polypeptide, and termination. The key molecular machinery driving this process are ribosomes[5], who's binding and movements determine the overall rate of transcription.
Translation rate can be affected by 3 major processes:
a) the binding of the ribosome to the RBS,
b) formation of initiation complexes,
c) speed of the ribosome across the mRNA transcript.
we will assume that the movement of the ribosome (in amino acids formed per second) is the rate determining step in the translation of the proteins, and the average translation rate of 17.1 amino acids/sec.[6]
Product |
Length (amino acids) |
Translation rate (1/s) |
hoxBC-hoxJ |
3628 |
0.0047 |
HoxA |
482 |
0.0353 |
Gfp |
230 |
0.0739 |
Degradation and Dilution
The concentrations of each species in the model is affected by it’s degradation (inherent to the molecule) and dilution (due to cell growth). While these processes are completely unrelated, they have the same net effect: Decreasing the concentration of the species[7]. Therefore, we represent the two rates as a combined degradation/dilution parameter in our model. The promotor of hox gene is constitutive, so we assume only the degradation/dilution of Gfp mRNA and Gfp.Name |
Value |
Units |
Gfp mRNA degradation/dilution |
0.0022 |
molecules/sec |
Gfp protein degradation/dilution |
0.0013（est） |
molecules/sec |
Cellular Equations
Using the parameters found above, we can construct a series of ordinary differential equations:
The system described by these equations can be represented diagramatically by SimBiology:
Result
The simulation was run with an initial ethylene concentration of 20 mM, over a time period of 10 hr, The graphs below show the time course concentrations of each species, with species of similar concentrations being plotted on the separate axes.
Reference:
-
[1] Tanja Burgdorf, Oliver Lenz, Thorsten Buhrke: [NiFe]-Hydrogenases of Ralstonia eutropha H16:
Modular Enzymes for Oxygen-Tolerant Biological Hydrogen Oxidation. J Mol Microbiol Biotechnol 2005;10:181–196; DOI: 10.1159/000091564
[2] Péter Érdi; János Tóth: Mathematical Models of Chemical Reactions: Theory and Applications of Deterministic and Stochastic Models.1989 Manchester University Press. p. 3. ISBN 978-0-7190-2208-1.
[3] Coval, ML: Analysis of Hill interaction coefficients and the invalidity of the Kwon and Brown equation.1970, J. Biol. Chem. 245 (23): 6335–6. PMID 5484812
[4] Nelson, David L.; Cox, Michael M. (2013). Lehninger principles of biochemistry (6th ed.). New York: W.H. Freeman. pp. 158–162. ISBN 978-1429234146.
[5] S Proshkin, AR Rahmouni, A Mironov, E Nudler: Cooperation Between Translating Ribosomes and RNA Polymerase in Transcription Elongation. Science, 2010, 328 (5977) :504
[6] R Young, H Bremer: Polypeptide-chain-elongation rate in Escherichia coli B/r as a function of growth rate. Biochemical Journal, 1976 , 160 (2) :185
[7] Selinger, D. W. :Global RNA Half-Life Analysis In Escherichia Coli Reveals Positional Patterns Of Transcript Degradation. Genome Research 13.2 (2003): 216-223. Web. 18 Oct. 2016.
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