Team:Jilin China/Model/Curve Fitting
Model
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Overview and Motivation
From our experiment, we got the raw data of thermosensors at different temperatures. However, using the original data merely, we could hardly measure the pure impact of temperature on our RNA-based thermosensors and described its switch behavior. Therefore, we defined normalized fluorescence, which means a normalized activity of thermosensors, and introduced Statistical Postulate to describe the probability distribution of thermosensors’ states Furthermore, we built a model to describe the thermodynamics of them and obtained a continuous temperature-dependent curve.
The goal of this model was to answer the following questions:
How to eliminate the temperature effect of bacterial expression system?
How to describe the switch behavior of massive RNA molecules statistically?
How to fit a continuous temperature-dependent expression curve?
How to extract features of RNA theromosensors from the curve?Data Normalization
Temperature dependence of global factors such as the survival rate of RNA, fluorescence parameters of sfGFP or enzyme activity may contribute to the individual measurement. They should, however, affect all thermosensors in a similar function. Therefore, we focus on the relative difference in the thermosensor activities.[1]
To measure the relative difference of fluorescence expression, we used BBa_R0040, a device without sfGFP coding sequence, as our negative control (Neg.). And our positive control (Pos.) whose sequence predicted by the software will not form a stem-loop structure, always express sfGFP and its activity won't change sharply.
$$Normalization={Fluorescence/Abs600_{(Device)}-Fluorescence/Abs600_{(Neg.)}\over{Fluorescence/Abs600_{(Pos.)-Fluorescence/Abs600_{(Neg.)}}}}$$The value of Normalized Fluorescence reflects the ratio of pure RNA expression between thermosensors and PC group, giving us a relatively impartial value to measure the relative difference in the thermosensors activities.
Two-state Distribution follows from the Statistical Postulate
From our design, we knew that each RNA thermosensor has two temperature-dependent states (Fod/Unfolded). When there are massive thermosensors in E.coli, individual actors are behaving randomly and yet a clear pattern emerges statistically. We replaced the idea of a definite state by the idea of a definite probability distribution of states.[2]
At low temperatures all samples displayed as folding state with low expression, whereas at high temperatures the expression increase, which indicates the increasing statistical rate of unfolded thermosensors. As the increasing value of expression of each thermosensor is constant, the normalized fluorescence of our thermosensor system is a linear function of the fraction of unfolded thermosensors, which also represent the probability distribution that a system will be in a certain state.
Thermodynamics
We defined \(f_T\) as the function of the fraction of unfolded molecules(f) vs temperature(T) and defined melting temperature \(T_m\) as the temperature for which \(f{(T_m)}=0.5\). We can use the equation appropriate for monomolecular to calculate equilibrium constant.[3]
$$K_{eq}={f\over{1-f}}$$The Van ’t Hoff equation is the equation revealing the relationship between the change in the equilibrium constant(\(K_{eq}\)) of a chemical reaction and the change in temperature(T). The Van 't Hoff equation of between temperatures T1 and T2 is [4]
$${ln{K_1\over{K_2}}}=-{\triangle{H^\Theta}\over{R}}{({1\over{T_2}}-{1\over{T_1}})}$$Where R is the ideal gas constant and 公式is standard enthalpy change.
Logistic Regression
Logistic function is a common "S" shape (sigmoid curve), S-curve function that can be used to describe the probability distribution of a two-state system with equation[5]
$$f(x)={L\over{1+e^{k(x-x_0)}}}$$
Figure 1. The features we extracted
Basing on the melting temperature T_m, we transform our equations ①② into the form of logistic function. $$f(T)={1\over{1+e^{[-k({1\over T} -{1\over T_m})]}}}$$ As the normalized fluorescence F(T) of our system is a linear function of fraction of unfolded molecules f(T)
$$F(T)=a\times{f(T)}+b$$The final function of normalized fluorescence vs temperature is
$$f(T)={a\over{1+e^{[-k({1\over T} -{1\over T_m})]}}+b}$$Where a, b, c are the parameters of the function.
Annotation of Parameters
From our function, we could analyze that
$$f(T)=0$$ $$F(T)=a\times0+b=b$$
when all thermosensors are folded,The value of its expression is the constant value b.
$$f(T)=1$$ $$F(T)=a\times1+b=a+b$$
When all thermosensors are unfolded,The value of its expression is the constant value a+b.
The value of k can be analogized to the Hill coefficient, for the first item of the Taylor expansion on ln(1+x) is 1/x. Our function can be changed into the form of Hill equation when doing an approximate treatment.
Curve Fitting
We employed least square method to regress a temperature-dependent expression curve for each thermosensor. R-squared is used to measure fitting goodness and to measure the desirableity of thermosensors. An example of the curve is shown below.
The result of desirableity classification is used in our machine learning model RNA Thermosensors Intelligent Screening System, which provided us an intelligent mean to screen sequences of desirable thermosensors. (超链接或see more)(figure2)
Feature Extraction
After communicating with our HP group, we found several features of our thermosensors are vital in practice. We extracted the values of features from our expression curve and built a database to describe behaviors of each thermosensor.(超链接到result)
The features include:
Melting Temperature \(T_m\): the temperature at which a 50% switch in expression occurs, with the value of \(T_m\)[6]
Sensitivity: the expression sensitivity to temperature, expression as the value of derivative of \(f(T)\) at \(T_m\)
Relative Intensity (M)：the value of predicted normalized fluorescence when all RNA molecules are unfolded, with the value of a+b
Threshold (m): the value of predicted expression when all RNA are folded, with the value of b
figure3Reference
[1] Sen S, Apurva D, Satija R, et al. Design of a Toolbox of RNA Thermometers[J]. Acs Synthetic Biology, 2017, 6(8).
[2] Nelson P C. Biological physics:energy, information, life[M]. W.H. Freeman and Co, 2004.
[3] Mergny J L, Lacroix L. Analysis of thermal melting curves.[J]. Oligonucleotides, 2003, 13(6):515.
[4] Ives, D. J. G. (1971). Chemical Thermodynamics. University Chemistry. Macdonald Technical and Scientific. ISBN 0-356-03736-3.
[5]Vogels M, Zoeckler R, Stasiw D M, et al. P. F. Verhulst's “notice sur la loi que la populations suit dans son accroissement” from correspondence mathematique et physique. Ghent, vol. X, 1838[J]. Journal of Biological Physics, 1975, 3(4):183-192.
[6]Sadler F W, Dodevski I, Sarkar C A. RNA Thermometers for the PURExpress System[J]. Acs Synthetic Biology, 2017, 7(1):292-296.Sub-title 1
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