In this part, we have two main aims to achieve:
1. Estimating the expression quantity of ferritin and preparing for temperature calculation in the next part.
2. Analyzing the noise in ferritin expression and identifying its sability.
(If you want to learn more details about the ferritin or magnetic-receiver, you can click here)
Deterministic Model
With the purpose of controlling the expression, we used PQE-80L, which has two lac I binding sites, to construct our genetic circuit.
The schematic figure is shown below:
And the schematic diagram can be simplified as follow:
Using Michaelis-Menten kineticsmass-action laws, we constructed the following set of differential equations:
Because the vector we used was PQE-80L, which has two lac I binding sites, we created a parameter “α” to described the probability that RNA polymerase can be bound to the promoter of interest.
Parameter Choice
Modeling induction effect with α
To describe the induction effect of IPTG, we created the parameter “α” with thermodynamic model. We made the assumption that the gene expression level is proportional to the equilibrium probability that RNA polymerase is bound to the promoter of interest.
This statistical mechanics provides a framework for computing these probabilities.Once these states have been enumerated, we weight each of them according to the Boltzmann law: if e is the energy of a state, its statistical weight is exp(-ε/kBT). Finally, to compute the probability of promoter occupancy, we construct the ratio of the sum of the weights for the favorable outcome (i.e. promoter occupied) to the sum over all of the weights[1].
In our modeling, there are three states for the lac I binding sites: none of them is occupied, either of them is occupied and both of them are occupied. And only in the third state can the gene be transcribed.
So the “α” can be described as follow:
In order to simplify the representation, we divided the numerator and denominator by exp(-ε1/kBT), then we can get a simple form:
In the formula, Δε stands for:
and its another form is
and the “a”in the formula means:
So we can get a simple from of “α”:
Results
With “α” calculated, we can simulate the induction effect of IPTG as shown blow.
From the result shown above, we can drew the conclusion that the leak from the gene expression was terrible and this result got great feedback from our wet lab experiments.
Sensitivity analysis
Local sensitivity analysis studies the impact of small fluctuations on the model outputs, thus we did the work in local analysis to observe which parameter takes the greatest importance.
Local sensitivity analysis involves computing the relative change of the steady state with respect to a change in the parameter. So we used Matlab and a finite-difference approximation of the derivative, sensitivities were calculated for 5% changes in the parameters. The sensitivity below was defined as the ratio of the output with changed parameters and the output with unchanged parameters. So the larger the ratio was, the more sensitive the parameter was.
From the table above we could find that the degradation rates showed great influence on the result of the deterministic model.
Stochastic Model And Noise Analysis
As a magnetic-receiver, we need a relative stable steady-state for the coming heating process. So the noise from the ferritin’s expression should be really small.
Consequently, we constructed the stochastic model with the condition that the promoter was fully induced by IPTG. We used the Gillespie SSA to simulation the reaction and calculate is noise though the stationary distribution of ferritin molecules[23].
According to the stationary distribution of ferritin molecules, we calculated the noise value to be 0.0058 which was very small. It means that ferritin’s expression level would be stable for a long time. So in the heating processing, we can could get a relative stable change in the whole-cell temperature distribution.
References
[1]Bintu L, Buchler N E, Garcia H G, et al. Transcriptional regulation by the numbers: models[J]. Current opinion in genetics & development, 2005, 15(2): 116-124.
[2] Rydenfelt M, Cox III R S, Garcia H, et al. Statistical mechanical model of coupled transcription from multiple promoters due to transcription factor titration[J]. Physical Review E, 2014, 89(1): 012702.
[3] Ozbudak E M, Thattai M, Kurtser I, et al. Regulation of noise in the expression of a single gene[J]. Nature genetics, 2002, 31(1): 69-73.
[4] Li G, Rabitz H, Yelvington P E, et al. Global sensitivity analysis for systems with independent and/or correlated inputs[J]. The Journal of Physical Chemistry A, 2010, 114(19): 6022-6032.
[5] Gibson M A, Bruck J. Efficient exact stochastic simulation of chemical systems with many species and many channels[J]. The journal of physical chemistry A, 2000, 104(9): 1876-1889.
[6] Saltelli A, Ratto M, Tarantola S, et al. Sensitivity analysis for chemical models[J]. Chemical reviews, 2005, 105(7): 2811-2828.
[7] Thattai M, Van Oudenaarden A. Intrinsic noise in gene regulatory networks[J]. Proceedings of the National Academy of Sciences, 2001, 98(15): 8614-8619.
[8] Pedraza J M, van Oudenaarden A. Noise propagation in gene networks[J]. Science, 2005, 307(5717): 1965-1969.
[9] Gillespie D T. Exact stochastic simulation of coupled chemical reactions[J]. The journal of physical chemistry, 1977, 81(25): 2340-2361.
[10] Zi Z. Sensitivity analysis approaches applied to systems biology models[J]. Systems Biology, IET, 2011, 5(6): 336-346.
[11] Jones D L, Brewster R C, Phillips R. Promoter architecture dictates cell-to-cell variability in gene expression[J]. Science, 2014, 346(6216): 1533-1536.
[12] Zhang J, Nie Q, He M, et al. An effective method for computing the noise in biochemical networks[J]. The Journal of chemical physics, 2013, 138(8): 084106.
[13] Fortin J P, Gazeau F, Wilhelm C. Intracellular heating of living cells through Néel relaxation of magnetic nanoparticles[J]. European Biophysics Journal, 2008, 37(2): 223-228.
[14] Gilles C, Bonville P, Rakoto H, et al. Magnetic hysteresis and superantiferromagnetism in ferritin nanoparticles[J]. Journal of Magnetism and Magnetic Materials, 2002, 241(2): 430-440.
[15] Derfus A M, von Maltzahn G, Harris T J, et al. Remotely triggered release from magnetic nanoparticles[J]. ADVANCED MATERIALS-DEERFIELD BEACH THEN WEINHEIM-, 2007, 19(22): 3932.
[16] Wong P, Gladney S, Keasling J D. Mathematical model of the lac operon: inducer exclusion, catabolite repression, and diauxic growth on glucose and lactose[J]. Biotechnology progress, 1997, 13(2): 132-143.
[17] Stanley S A, Sauer J, Kane R S, et al. Remote regulation of glucose homeostasis in mice using genetically encoded nanoparticles[J]. Nature medicine, 2015, 21(1): 92-98.
[18] Fortin J P, Wilhelm C, Servais J, et al. Size-sorted anionic iron oxide nanomagnets as colloidal mediators for magnetic hyperthermia[J]. Journal of the American Chemical Society, 2007, 129(9): 2628-2635.
[19] Huang H, Delikanli S, Zeng H, et al. Remote control of ion channels and neurons through magnetic-field heating of nanoparticles[J]. Nature nanotechnology, 2010, 5(8): 602-606.
[20] Stanley S A, Gagner J E, Damanpour S, et al. Radio-wave heating of iron oxide nanoparticles can regulate plasma glucose in mice[J]. Science, 2012, 336(6081): 604-608.
[21] Lian T, Locke B, Kholodenko Y, et al. Energy flow from solute to solvent probed by femtosecond IR spectroscopy: malachite green and heme protein solutions[J]. The Journal of Physical Chemistry, 1994, 98(45): 11648-11656.
[22] Paulsson J. Summing up the noise in gene networks[J]. Nature, 2004, 427(6973): 415-418.
[23] Erban R, Chapman J, Maini P. A practical guide to stochastic simulations of reaction-diffusion processes[J]. arXiv preprint arXiv:0704.1908, 2007.
