Difference between revisions of "Team:Heidelberg/Modeling/rtsms"

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Figure 1. IVT model reactions and fits to experimental data. (A) Model reactions describing reversible assembly of templates T and polymerase P to active templates T* that incorporate ATP A into malachite green RNA-aptamers M but also into abortion products, leading to a higher number nA of consumed than ATP molecules nA,M incorporated in malachite green aptamers. (B) Model fits to data at two different polymerase concentrations.
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<strong>Figure 1.</strong> IVT model reactions and fits to experimental data. (A) Model reactions describing reversible assembly of templates T and polymerase P to active templates T* that incorporate ATP A into malachite green RNA-aptamers M but also into abortion products, leading to a higher number $n_A$ of consumed than ATP molecules $n_{A,M}$ incorporated in malachite green aptamers. (B) Model fits to data at two different polymerase concentrations.
 
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Because of two unexpected findings, a basic model could not explain the data. Counter-intuitively, malachite green showed a linear increase while the ATP-spinach intensity was exponentially decreasing. Furthermore, it was surprising that doubling the amount of polymerase increased the production of malachite green by even more than two-fold. Both phenomena could be explained by an optimal model variant (Fig. 1B, 1A), in which the polymerase inaccuracy increased with increasing ratios between ATP and active templates that could not be further reduced without losing fit quality (Fig. 2B, 2C).
 
Because of two unexpected findings, a basic model could not explain the data. Counter-intuitively, malachite green showed a linear increase while the ATP-spinach intensity was exponentially decreasing. Furthermore, it was surprising that doubling the amount of polymerase increased the production of malachite green by even more than two-fold. Both phenomena could be explained by an optimal model variant (Fig. 1B, 1A), in which the polymerase inaccuracy increased with increasing ratios between ATP and active templates that could not be further reduced without losing fit quality (Fig. 2B, 2C).
 
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Figure 2. IVT inaccuracy depends on the ATP to active template ratio. (A) A basic model with constant numbers of nA and synthesis parameters ksyn,M, was extended to variants with nA and ksyn,M depending on the polymerase concentration (Variant 2), A- and T*-dependent nA with exponents k and l (Variant 3) or only an exponent for T* (Variant 4). Fitting improvement is indicated by decreasing Akaike information criterion (AIC) values. (B) Reducing the optimal variant 4 by assuming a steady state for T*, no  degradation of P or no degradation of A disproved model fits. (C) Model variant 4 can explain increasing inefficiency (higher nA) with decreasing A/T* ratios.
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<strong>Figure 2.</strong> IVT inaccuracy depends on the ATP to active template ratio. (A) A basic model with constant numbers of $n_A$ and synthesis parameters $k_{syn,M}$, was extended to variants with $n_A$ and $k_{syn,M}$ depending on the polymerase concentration (variant 2), A- and T*-dependent $n_A$ with exponents $k$ and $l$ (variant 3) or only an exponent for T* (variant 4). Fitting improvement is indicated by decreasing Akaike information criterion (AIC) values. (B) Reducing the optimal variant 4 by assuming a steady state for T*, no  degradation of P or no degradation of A disproved model fits. (C) Model variant 4 can explain increasing inefficiency (higher $n_A$) with decreasing A/T* ratios.
 
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Table 3. Stepwise changes from the basic model to the optimal variant 4 and to simplifications of variant 4 to variants 4a to 4c</p>
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<strong>Table 3.</strong> Stepwise changes from the basic model to the optimal variant 4 and to simplifications of variant 4 to variants 4a to 4c</p>
 
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Table 4. Model equations for the basic model and Variants 1 to 4c</p>
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Table 4. Model equations for the basic model and variants 1 to 4c</p>
 
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Variants 1 to 4, 4c</p>
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Revision as of 06:31, 2 October 2015

Studying determinants of polymerase efficiency based on an aptamer sensor

Our subproject on small molecule sensing facilitates quantitatively studying in vitro transcription (IVT) by ATP-spinach and malachite green RNA-aptamers. In particular, we could study the inaccuracy of polymerases reflected by an excess of consumed ATP molecules over the number of ATP molecules in synthesized malachite green aptamers [1], based on a mathematical model (Fig. 1A).


Figure 1. IVT model reactions and fits to experimental data. (A) Model reactions describing reversible assembly of templates T and polymerase P to active templates T* that incorporate ATP A into malachite green RNA-aptamers M but also into abortion products, leading to a higher number $n_A$ of consumed than ATP molecules $n_{A,M}$ incorporated in malachite green aptamers. (B) Model fits to data at two different polymerase concentrations.  

Because of two unexpected findings, a basic model could not explain the data. Counter-intuitively, malachite green showed a linear increase while the ATP-spinach intensity was exponentially decreasing. Furthermore, it was surprising that doubling the amount of polymerase increased the production of malachite green by even more than two-fold. Both phenomena could be explained by an optimal model variant (Fig. 1B, 1A), in which the polymerase inaccuracy increased with increasing ratios between ATP and active templates that could not be further reduced without losing fit quality (Fig. 2B, 2C).


Figure 2. IVT inaccuracy depends on the ATP to active template ratio. (A) A basic model with constant numbers of $n_A$ and synthesis parameters $k_{syn,M}$, was extended to variants with $n_A$ and $k_{syn,M}$ depending on the polymerase concentration (variant 2), A- and T*-dependent $n_A$ with exponents $k$ and $l$ (variant 3) or only an exponent for T* (variant 4). Fitting improvement is indicated by decreasing Akaike information criterion (AIC) values. (B) Reducing the optimal variant 4 by assuming a steady state for T*, no degradation of P or no degradation of A disproved model fits. (C) Model variant 4 can explain increasing inefficiency (higher $n_A$) with decreasing A/T* ratios.  

Table 3. Stepwise changes from the basic model to the optimal variant 4 and to simplifications of variant 4 to variants 4a to 4c

Model variant

Subsequent modifications relative to basic model or previous variant

Changes in fitting quality

1

Michaelis-Menten instead of linear kinetics for active template

no improvement

2

Individual $k_{syn}$ and $n_A$ values for different polymerase concentrations

improvement

3

$n_A$ depends on function of $T_{act}$ and $A$

$n_A=n_{A,0} A^{k} /T_{act}^{l}$

improvement, $k\approx0$

 

4, best model

Setting $k=0$

improvement

4a

No degradation of P in variant 4

decrease

4b

No degradation of A in variant 4

decrease

4c

Binding of $P$ to $T$ in steady state in variant 4

decrease

 


 

Table 4. Model equations for the basic model and variants 1 to 4c

Model species

variant

Equation

$P$

Basic model

variants 1 to 4, 4c

$\frac{d[P]}{dt}=-k_{on}[T][P]+k_{off}[T_{act}]-k_{deg,P}[P]$

variant 4a

$[P](t)=[P](t_{0})\exp\left(-k_{deg,P}t\right)$

Variant 4b

$\frac{d[P]}{dt}=-k_{on}[T][P]+k_{off}[T_{act}]$

$T$

Basic model

Variants 1 to 4, 4b, 4c

$\frac{d[T]}{dt}=-k_{on}[T][P]+k_{off}[T_{act}]$

Variant 4a

$[T]=[T_{tot}]-[T_{act}]$

$T_{act}$

Basic model

Variants 1 to 4, 4b, 4c

$\frac{d[T_{act}]}{dt}=k_{on}[T][P]-k_{off}[T_{act}]$

Variant 4a

$[T_{act}]=\frac{[T_{tot}][P]}{K_{d,P}}$

$A$

Basic model

Variants 2 to 4, 4a, 4b

$\frac{d[A]}{dt}=-k_{syn}[A][T_{act}]-k_{deg,A}[A]$

Variant 1

$\frac{d[A]}{dt}=-k_{syn}\frac{[A][T_{act}]}{K_{m,T}+[T_{act}]}-k_{deg,A}[A]$

 

Variant 4c

$\frac{d[A]}{dt}=-k_{syn}[A][T_{act}]$

$M$

Basic model,

Variant 2

$\frac{d[M]}{dt}=\frac{k_{syn}}{n_{A}}[A][T_{act}]$

Variants 1

$\frac{d[M]}{dt}=\frac{k_{syn}}{n_{A}}\frac{[A][T_{act}]}{K_{m,T}+[T_{act}]}$

Variant 3

$\frac{d[M]}{dt}=\frac{k_{syn}}{n_{A,0}\frac{[A]^{k}}{[T_{act}]^{l}}}[A][T_{act}]=\frac{k_{syn}}{n_{A,0}}[A]^{1-k}[T_{act}]^{1+j}$

Variants 4, 4a, 4b, 4c

$\frac{d[M]}{dt}=\frac{k_{syn}}{n_{A,0}\frac{[A]}{[T_{act}]^{l}}}[A][T_{act}]=\frac{k_{syn}}{n_{A,0}}[T_{act}]^{1+j}$