diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index bc51722..5e51e77 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.10.5","generation_timestamp":"2024-09-21T09:17:57","documenter_version":"1.7.0"}} \ No newline at end of file +{"documenter":{"julia_version":"1.10.5","generation_timestamp":"2024-09-21T13:51:08","documenter_version":"1.7.0"}} \ No newline at end of file diff --git a/dev/index.html b/dev/index.html index 6a68502..28514fd 100644 --- a/dev/index.html +++ b/dev/index.html @@ -1,2 +1,2 @@ -Home · PereiraMartinez2024.jl
GitHub
+Home · PereiraMartinez2024.jl
GitHub
diff --git a/dev/objects.inv b/dev/objects.inv index 3545eca..10d8bca 100644 Binary files a/dev/objects.inv and b/dev/objects.inv differ diff --git a/dev/pluto_notebooks/proteinA_urea.html b/dev/pluto_notebooks/proteinA_urea.html index 3f52af0..23ade44 100644 --- a/dev/pluto_notebooks/proteinA_urea.html +++ b/dev/pluto_notebooks/proteinA_urea.html @@ -1,231 +1,18 @@ - - - -

BdpA transfer free energy to urea

- - -

Aqueous solutions of urea: experimental data

- - -

In Lin & Timasheff, Biochemistry, 1994, the activity coefficients (\(\gamma_3\)) of urea in aqueous solutions and the corresponding densities are provided. The data provided is that of \(\gamma_3\) as a function of the concentration \(m_3\) (in mol/kg):

- - -

Properties of aqueous solutions of urea

- - -

Experimental data for urea aqueous solutions (at 25\(^\circ\)C):

- - -
m₃γ₃ρ₀
10.0 mol kg^-11.00.997 g mL^-1
20.51 mol kg^-10.961.0084 g mL^-1
31.05 mol kg^-10.9241.016 g mL^-1
42.2 mol kg^-10.8591.0298 g mL^-1
- - -

The data above is used to compute the \((\partial\ln\gamma_3/\partial m_3)\) quantity. For that we need to create first a plot:

- - - - - -

The shape of the plot suggests that a quadratic fit will capture precisely the trend:

- -
urea_fit = fitquadratic(urea_data[!,:m₃],log.(urea_data[!,:γ₃]))
-
------------------- Quadratic Fit -------------
-
-Equation: y = ax^2 + bx + c
-
-With: a = 0.005647775960817326 kg^2 mol^-2
-      b = -0.08138062053973305 kg mol^-1
-      c = -0.00022283020090959187
-
-Pearson correlation coefficient, R = 0.9999787191515568 mol kg^-1
-Average square residue = 1.3341140148130874e-7
-
-Predicted Y: ypred = [-0.00022283020090959187, -0.040257960148764864, ...]
-residues = [-0.00022283020090959187, 0.0005640343714903018, ...]
-
------------------------------------------------
-
- - -

The derivative of the logarithmic of the activity coefficient as a function of the molality is, then, the derivative of the above quadratic fit:

- -
∂lnγ₃∂m₃(m₃) = 2 * urea_fit.a * m₃ + urea_fit.b
-
∂lnγ₃∂m₃ (generic function with 1 method)
- - -

Which can be computed for the concentrations given in Table 2 of the article, and do not agree quantitatively, but follow the same trend:

- - -
4-element Vector{Quantity{Float64, 𝐌 𝐍^-1, Unitful.FreeUnits{(kg, mol^-1), 𝐌 𝐍^-1, nothing}}}:
-  -0.08138062053973305 kg mol^-1
-  -0.07561988905969938 kg mol^-1
-  -0.06952029102201666 kg mol^-1
- -0.056530406312136815 kg mol^-1
- -
urea_data
-
m₃γ₃ρ₀∂lnγ₃∂m₃
10.0 mol kg^-11.00.997 g mL^-1-0.0813806 kg mol^-1
20.51 mol kg^-10.961.0084 g mL^-1-0.0756199 kg mol^-1
31.05 mol kg^-10.9241.016 g mL^-1-0.0695203 kg mol^-1
42.2 mol kg^-10.8591.0298 g mL^-1-0.0565304 kg mol^-1
- - -

We also obtain an expression for the density as a function of the concentration, in mol/L:

- - - - - -

Where the fit is:

- -
density_fit
-
------------------- Quadratic Fit -------------
-
-Equation: y = ax^2 + bx + c
-
-With: a = -0.00314858377052122 g L^2 mL^-1 mol^-2
-      b = 0.021417043069674676 g L mL^-1 mol^-1
-      c = 0.9973483095732278 g mL^-1
-
-Pearson correlation coefficient, R = 0.9988889736362447 g mol mL^-1 L^-1
-Average square residue = 3.1548945450687924e-7 g^2 mL^-2
-
-Predicted Y: ypred = [0.9973483095732278 g mL^-1, 1.0075299894241547 g mL^-1, ...]
-residues = [0.00034830957322784517 g mL^-1, -0.000870010575845237 g mL^-1, ...]
-
------------------------------------------------
-
- - -

BdpA in urea

- - -

With the data above, we can proceed to using equation 5 of Lin and Timasheff:

$$\left(\frac{\partial\mu_2}{\partial m_3}\right)_{T,P,m_2} = --\left(\frac{\partial g_3}{\partial g_2}\right)_{T,P,\mu_3} -\left(\frac{RTM_2}{M_3}\right) -\left(\frac{1}{m_3}+\frac{\partial\ln \gamma_3}{\partial m_3}\right)$$

- - -

where, in the case of urea (species 3) and BpdA (solute, species 2) we have:

- - -
60.06 g mol^-1
- - -
6768.46 g mol^-1
- - -

States considered

Here, fold states are considered, from the set obtained in the simulations: the native state and a denatured state:

These states will be referred to as N and U from here on.

- - -

Converting concentrations:

- - -

For the N and U states, the obtained bulk concentrations of urea are obtained from the simulations:

- - -
NU
10.0938442 mol L^-10.0972688 mol L^-1
20.199496 mol L^-10.200396 mol L^-1
30.29125 mol L^-10.305921 mol L^-1
40.397743 mol L^-10.401483 mol L^-1
50.49524 mol L^-10.501656 mol L^-1
- - -

The densities corresponding to these concentrations, according to the experimental densities, are:

- - -
NU
10.99933 g mL^-10.999402 g mL^-1
21.0015 g mL^-11.00151 g mL^-1
31.00332 g mL^-11.00361 g mL^-1
41.00537 g mL^-11.00544 g mL^-1
51.00718 g mL^-11.0073 g mL^-1
- - -

With which we can compute the molality of urea in each simulated concentration:

- - -
NU
10.0939071 mol kg^-10.097327 mol kg^-1
20.199198 mol kg^-10.200093 mol kg^-1
30.290287 mol kg^-10.304822 mol kg^-1
40.395619 mol kg^-10.399311 mol kg^-1
50.491708 mol kg^-10.49802 mol kg^-1
- - -

Simulated preferential interactions:

- - -

The preferential interaction parameters of BpDA in urea, for the N and U states, are (converted from mol/mol to g/g):

- - -
NU
10.007271680.00416967
20.01117410.0127021
30.01584290.0198925
40.02089960.024073
50.02574810.0303108
- - - - - -

And using the gas constant in kcal/(K mol), and the temperature in K:

- - -
0.0019872036 kcal K^-1 mol^-1
- - -
298 K
- - -

The transfer free energy

- - -

We can define Eq. 5 of the article (written above), as a function of the preferential interactions, molar mass of the cossolvent, and ∂lnγ₃∂m₃:

- -
∂μ₂∂m₃(∂g₃∂g₂, m₃, M2, M3) = -(∂g₃∂g₂) * (R * T * M2 / M3) * (1/m₃ + ∂lnγ₃∂m₃(m₃))
-
∂μ₂∂m₃ (generic function with 1 method)
- - -

Applying this equation to the three possible sets of parameters for the three concentrations studied in the paper, we get:

- - -
NU
1-5.12875 kcal kg mol^-2-2.83677 kcal kg mol^-2
2-3.68459 kcal kg mol^-2-4.16942 kcal kg mol^-2
3-3.55969 kcal kg mol^-2-4.25171 kcal kg mol^-2
4-3.41825 kcal kg mol^-2-3.89981 kcal kg mol^-2
5-3.36433 kcal kg mol^-2-3.90852 kcal kg mol^-2
- - -

Now we can plot those values as function of the concentration of urea:

- - - - - -

The shaded areas are, qualitatively, the transfer free energy of each state up to each concentration.

The plot shows that at very low concentrations, the native state has a lower transfer free energy to urea than the denatured state. Thus, at this low concentration, the native state is stabilized in urea relative to the denatured state.

At higher urea concentrations the integral associated to the denatured state becomes progressively more negative, thus the denatured state is stabilized relative to the native state upon transfer to a urea solution.

Integrating these curves provides a qualitative measure of the free energy of transfer in each case:

- - -
-1.487286307325704 kcal mol^-1
- - -
-1.5714588136472185 kcal mol^-1
- - -

Implying the the denatured state is slightly more favorably transfered to a 0.5 mol/L aqueous urea solution than the native state. The difference probably increases with increasing urea concentration.

- - -

Packages used

- -
using Trapz
- - -
using MolSimToolkit
- - -
using DataFrames
- - -
using Unitful
- - -
using Plots
- - -
using LaTeXStrings
- - -
using EasyFit
- - -
begin
-    using PlutoUI
-    TableOfContents()
-end
- - -
import Pkg
- - -
Pkg.status()
- - - \ No newline at end of file + + + + + + + + +
\ No newline at end of file diff --git a/dev/pluto_notebooks/proteinA_urea/index.html b/dev/pluto_notebooks/proteinA_urea/index.html deleted file mode 100644 index 37362fe..0000000 --- a/dev/pluto_notebooks/proteinA_urea/index.html +++ /dev/null @@ -1,254 +0,0 @@ - -BdpA in urea · PereiraMartinez2024.jl
GitHub
- - - - -

BdpA transfer free energy to urea

- -

Aqueous solutions of urea: experimental data

-
- - -

In Lin & Timasheff, Biochemistry, 1994, the activity coefficients (\(\gamma_3\)) of urea in aqueous solutions and the corresponding densities are provided. The data provided is that of \(\gamma_3\) as a function of the concentration \(m_3\) (in mol/kg):

- - -

Properties of aqueous solutions of urea

- - -

Experimental data for urea aqueous solutions (at 25\(^\circ\)C):

- - -
m₃γ₃ρ₀
10.0 mol kg^-11.00.997 g mL^-1
20.51 mol kg^-10.961.0084 g mL^-1
31.05 mol kg^-10.9241.016 g mL^-1
42.2 mol kg^-10.8591.0298 g mL^-1
- - -

The data above is used to compute the \((\partial\ln\gamma_3/\partial m_3)\) quantity. For that we need to create first a plot:

- - - - - -

The shape of the plot suggests that a quadratic fit will capture precisely the trend:

- -
urea_fit = fitquadratic(urea_data[!,:m₃],log.(urea_data[!,:γ₃]))
-
------------------- Quadratic Fit -------------
-
-Equation: y = ax^2 + bx + c
-
-With: a = 0.0056477759670859945 kg^2 mol^-2
-      b = -0.08138062055484234 kg mol^-1
-      c = -0.0002228301963439134
-
-Pearson correlation coefficient, R = 0.9999787191515569 mol kg^-1
-Average square residue = 1.3341140148130802e-7
-
-Predicted Y: ypred = [-0.0002228301963439134, -0.04025796015027444, ...]
-residues = [-0.0002228301963439134, 0.0005640343699807246, ...]
-
------------------------------------------------
-
- - -

The derivative of the logarithmic of the activity coefficient as a function of the molality is, then, the derivative of the above quadratic fit:

- -
∂lnγ₃∂m₃(m₃) = 2 * urea_fit.a * m₃ + urea_fit.b
-
∂lnγ₃∂m₃ (generic function with 1 method)
- - -

Which can be computed for the concentrations given in Table 2 of the article, and do not agree quantitatively, but follow the same trend:

- - -
4-element Vector{Quantity{Float64, 𝐌 𝐍^-1, Unitful.FreeUnits{(kg, mol^-1), 𝐌 𝐍^-1, nothing}}}:
-  -0.08138062055484234 kg mol^-1
-  -0.07561988906841463 kg mol^-1
-  -0.06952029102396175 kg mol^-1
- -0.056530406299663966 kg mol^-1
- -
urea_data
-
m₃γ₃ρ₀∂lnγ₃∂m₃
10.0 mol kg^-11.00.997 g mL^-1-0.0813806 kg mol^-1
20.51 mol kg^-10.961.0084 g mL^-1-0.0756199 kg mol^-1
31.05 mol kg^-10.9241.016 g mL^-1-0.0695203 kg mol^-1
42.2 mol kg^-10.8591.0298 g mL^-1-0.0565304 kg mol^-1
- - -

We also obtain an expression for the density as a function of the concentration, in mol/L:

- - - - - -

Where the fit is:

- -
density_fit
-
------------------- Quadratic Fit -------------
-
-Equation: y = ax^2 + bx + c
-
-With: a = -0.0031485837712447715 g L^2 mL^-1 mol^-2
-      b = 0.021417043071470018 g L mL^-1 mol^-1
-      c = 0.9973483095726731 g mL^-1
-
-Pearson correlation coefficient, R = 0.9988889736362445 g mol mL^-1 L^-1
-Average square residue = 3.154894545068943e-7 g^2 mL^-2
-
-Predicted Y: ypred = [0.9973483095726731 g mL^-1, 1.007529989424332 g mL^-1, ...]
-residues = [0.0003483095726730667 g mL^-1, -0.0008700105756680454 g mL^-1, ...]
-
------------------------------------------------
-
- -

BdpA in urea

-
- - -

With the data above, we can proceed to using equation 5 of Lin and Timasheff:

$$\left(\frac{\partial\mu_2}{\partial m_3}\right)_{T,P,m_2} = --\left(\frac{\partial g_3}{\partial g_2}\right)_{T,P,\mu_3} -\left(\frac{RTM_2}{M_3}\right) -\left(\frac{1}{m_3}+\frac{\partial\ln \gamma_3}{\partial m_3}\right)$$

- - -

where, in the case of urea (species 3) and BpdA (solute, species 2) we have:

- - -
60.06 g mol^-1
- - -
6768.46 g mol^-1
- - -

States considered

Here, fold states are considered, from the set obtained in the simulations: the native state and a denatured state:

  • Native state: Most representative structure of N8 basin.

  • Denatured state: Most reprentative structure of U6 basin.

These states will be referred to as N and U from here on.

- - -

Converting concentrations:

- - -

For the N and U states, the obtained bulk concentrations of urea are obtained from the simulations:

- - -
NU
10.0938442 mol L^-10.0972688 mol L^-1
20.199496 mol L^-10.200396 mol L^-1
30.29125 mol L^-10.305921 mol L^-1
40.397743 mol L^-10.401483 mol L^-1
50.49524 mol L^-10.501656 mol L^-1
- - -

The densities corresponding to these concentrations, according to the experimental densities, are:

- - -
NU
10.99933 g mL^-10.999402 g mL^-1
21.0015 g mL^-11.00151 g mL^-1
31.00332 g mL^-11.00361 g mL^-1
41.00537 g mL^-11.00544 g mL^-1
51.00718 g mL^-11.0073 g mL^-1
- - -

With which we can compute the molality of urea in each simulated concentration:

- - -
NU
10.0939071 mol kg^-10.097327 mol kg^-1
20.199198 mol kg^-10.200093 mol kg^-1
30.290287 mol kg^-10.304822 mol kg^-1
40.395619 mol kg^-10.399311 mol kg^-1
50.491708 mol kg^-10.49802 mol kg^-1
- - -

Simulated preferential interactions:

- - -

The preferential interaction parameters of BpDA in urea, for the N and U states, are (converted from mol/mol to g/g):

- - -
NU
10.007271680.00416967
20.01117410.0127021
30.01584290.0198925
40.02089960.024073
50.02574810.0303108
- - - - - -

And using the gas constant in kcal/(K mol), and the temperature in K:

- - -
0.0019872036 kcal K^-1 mol^-1
- - -
298 K
- - -

The transfer free energy

- - -

We can define Eq. 5 of the article (written above), as a function of the preferential interactions, molar mass of the cossolvent, and ∂lnγ₃∂m₃:

- -
∂μ₂∂m₃(∂g₃∂g₂, m₃, M2, M3) = -(∂g₃∂g₂) * (R * T * M2 / M3) * (1/m₃ + ∂lnγ₃∂m₃(m₃))
-
∂μ₂∂m₃ (generic function with 1 method)
- - -

Applying this equation to the three possible sets of parameters for the three concentrations studied in the paper, we get:

- - -
NU
1-5.12875 kcal kg mol^-2-2.83677 kcal kg mol^-2
2-3.68459 kcal kg mol^-2-4.16942 kcal kg mol^-2
3-3.55969 kcal kg mol^-2-4.25171 kcal kg mol^-2
4-3.41825 kcal kg mol^-2-3.89981 kcal kg mol^-2
5-3.36433 kcal kg mol^-2-3.90852 kcal kg mol^-2
- - -

Now we can plot those values as function of the concentration of urea:

- - - - - -

The shaded areas are, qualitatively, the transfer free energy of each state up to each concentration.

The plot shows that at very low concentrations, the native state has a lower transfer free energy to urea than the denatured state. Thus, at this low concentration, the native state is stabilized in urea relative to the denatured state.

At higher urea concentrations the integral associated to the denatured state becomes progressively more negative, thus the denatured state is stabilized relative to the native state upon transfer to a urea solution.

Integrating these curves provides a qualitative measure of the free energy of transfer in each case:

- - -
-1.4872863073205036 kcal mol^-1
- - -
-1.5714588136413514 kcal mol^-1
- - -

Implying the the denatured state is slightly more favorably transfered to a 0.5 mol/L aqueous urea solution than the native state. The difference probably increases with increasing urea concentration.

- - -

Packages used

- -
using Trapz
- - -
using MolSimToolkit
- - -
using DataFrames
- - -
using Unitful
- - -
using Plots
- - -
using LaTeXStrings
- - -
using EasyFit
- - -
begin
-    using PlutoUI
-    TableOfContents()
-end
- - -
import Pkg
- - -
Pkg.status()
- - -
diff --git a/dev/pluto_notebooks/rnaset1_urea.html b/dev/pluto_notebooks/rnaset1_urea.html index 3d37769..c6f972b 100644 --- a/dev/pluto_notebooks/rnaset1_urea.html +++ b/dev/pluto_notebooks/rnaset1_urea.html @@ -1,235 +1,18 @@ - - - -

To open and edit this notebook, click here

- - -

RNaseT1 transfer free energy to urea

- - -

Aqueous solutions of urea: experimental data

- - -

In Lin & Timasheff, Biochemistry, 1994, the activity coefficients (\(\gamma_3\)) of urea in aqueous solutions and the corresponding densities are provided. The data provided is that of \(\gamma_3\) as a function of the concentration \(m_3\) (in mol/kg):

- - -

Properties of aqueous solutions of urea

- - -

Experimental data for urea aqueous solutions (25\(^\circ\)C):

- - -
m₃γ₃ρ₀
10.0 mol kg^-11.00.997 g mL^-1
20.51 mol kg^-10.961.0084 g mL^-1
31.05 mol kg^-10.9241.016 g mL^-1
42.2 mol kg^-10.8591.0298 g mL^-1
- - -

The data above is used to compute the \((\partial\ln\gamma_3/\partial m_3)\) quantity. For that we need to create first a plot:

- - - - - -

The shape of the plot suggests that a quadratic fit will capture precisely the trend:

- -
urea_fit = fitquadratic(urea_data[!,:m₃],log.(urea_data[!,:γ₃]))
-
------------------- Quadratic Fit -------------
-
-Equation: y = ax^2 + bx + c
-
-With: a = 0.0056477759585468485 kg^2 mol^-2
-      b = -0.08138062053426073 kg mol^-1
-      c = -0.00022283020256293705
-
-Pearson correlation coefficient, R = 0.9999787191515569 mol kg^-1
-Average square residue = 1.3341140148130736e-7
-
-Predicted Y: ypred = [-0.00022283020256293705, -0.040257960148217885, ...]
-residues = [-0.00022283020256293705, 0.0005640343720372809, ...]
-
------------------------------------------------
-
- - -

The derivative of the logarithmic of the activity coefficient as a function of the molality is, then, the derivative of the above quadratic fit:

- -
∂lnγ₃∂m₃(m₃) = 2 * urea_fit.a * m₃ + urea_fit.b
-
∂lnγ₃∂m₃ (generic function with 1 method)
- - -

Which can be computed for the concentrations given in Table 2 of the article, and do not agree quantitatively, but follow the same trend:

- -
urea_data[:, :∂lnγ₃∂m₃] = ∂lnγ₃∂m₃.(urea_data[!,:m₃])
-
4-element Vector{Quantity{Float64, 𝐌 𝐍^-1, Unitful.FreeUnits{(kg, mol^-1), 𝐌 𝐍^-1, nothing}}}:
- -0.08138062053426073 kg mol^-1
- -0.07561988905654295 kg mol^-1
- -0.06952029102131235 kg mol^-1
-  -0.0565304063166546 kg mol^-1
- -
urea_data
-
m₃γ₃ρ₀∂lnγ₃∂m₃
10.0 mol kg^-11.00.997 g mL^-1-0.0813806 kg mol^-1
20.51 mol kg^-10.961.0084 g mL^-1-0.0756199 kg mol^-1
31.05 mol kg^-10.9241.016 g mL^-1-0.0695203 kg mol^-1
42.2 mol kg^-10.8591.0298 g mL^-1-0.0565304 kg mol^-1
- - -

We also obtain an expression for the density as a function of the concentration, in mol/L:

- - - - - -

Where the fit is:

- -
density_fit
-
------------------- Quadratic Fit -------------
-
-Equation: y = ax^2 + bx + c
-
-With: a = -0.003148583771372212 g L^2 mL^-1 mol^-2
-      b = 0.021417043071786435 g L mL^-1 mol^-1
-      c = 0.9973483095725753 g mL^-1
-
-Pearson correlation coefficient, R = 0.9988889736362447 g mol mL^-1 L^-1
-Average square residue = 3.1548945450691344e-7 g^2 mL^-2
-
-Predicted Y: ypred = [0.9973483095725753 g mL^-1, 1.007529989424363 g mL^-1, ...]
-residues = [0.0003483095725752561 g mL^-1, -0.0008700105756369592 g mL^-1, ...]
-
------------------------------------------------
-
- - -

RNaseT1 in urea

- - -

With the data above, we can proceed to using equation 5 of Lin and Timasheff:

$$\left(\frac{\partial\mu_2}{\partial m_3}\right)_{T,P,m_2} = --\left(\frac{\partial g_3}{\partial g_2}\right)_{T,P,\mu_3} -\left(\frac{RTM_2}{M_3}\right) -\left(\frac{1}{m_3}+\frac{\partial\ln \gamma_3}{\partial m_3}\right)$$

- - -

where, in the case of urea (species 3) and BpdA (solute, species 2) we have:

- - -
60.06 g mol^-1
- - -
11000 g mol^-1
- - -

The RNase-T1 and RCM-T1 constructos of the article, modeling the native and denatured states of RNase-T1, will be named from now on N and U states, respectively.

- - -

Converting concentrations:

- - -

For the N and U states, the obtained bulk concentrations of urea are obtained from the simulations:

- - -
NU
10.5 mol L^-10.5 mol L^-1
21.0 mol L^-11.0 mol L^-1
32.0 mol L^-12.0 mol L^-1
- - -

The densities corresponding to these concentrations, according to the experimental densities, are:

- - -
NU
11.00727 g mL^-11.00727 g mL^-1
21.01562 g mL^-11.01562 g mL^-1
31.02759 g mL^-11.02759 g mL^-1
- - -

With which we can compute the molality of urea in each simulated concentration:

- - -
NU
10.496391 mol kg^-10.496391 mol kg^-1
20.984623 mol kg^-10.984623 mol kg^-1
31.94631 mol kg^-11.94631 mol kg^-1
- - -

Preferential interactions:

- - -

The preferential interaction parameters of BpDA in urea, for the N8 and U6 states, are:

- - -
NU
10.02440.0247
20.03450.0542
30.06470.1445
- - - - - -

And using the gas constant in kcal/(K mol), and the temperature in K:

- - -
0.0019872036 kcal K^-1 mol^-1
- - -
298 K
- - -

The transfer free energy

- - -

We can define Eq. 5 of the article (written above), as a function of the preferential interactions, molar mass of the cossolvent, and ∂lnγ₃∂m₃:

- -
∂μ₂∂m₃(∂g₃∂g₂, m₃, M2, M3) = 
-    -(∂g₃∂g₂) * (R * T * M2 / M3) * (1/m₃ + ∂lnγ₃∂m₃(m₃))
-
∂μ₂∂m₃ (generic function with 1 method)
- - -

Applying this equation to the three possible sets of parameters for the three concentrations studied in the paper, we get:

- - -
NU
1-5.13075 kcal kg mol^-2-5.19384 kcal kg mol^-2
2-3.53738 kcal kg mol^-2-5.55727 kcal kg mol^-2
3-3.18865 kcal kg mol^-2-7.12148 kcal kg mol^-2
- - -

Now we can plot those values as function of the concentration of urea:

- - - - - -

The shaded areas are, qualitatively, the transfer free energy of each state up to each concentration.

The plot shows that at very low concentrations, the native state has a lower transfer free energy to urea than the denatured state. Thus, at this low concentration, the native state is stabilized in urea relative to the denatured state.

At higher urea concentrations the integral associated to the denatured state becomes progressively more negative, thus the denatured state is stabilized relative to the native state upon transfer to a urea solution.

Integrating these curves provides a qualitative measure of the free energy of transfer in each case:

- - -
-5.350177937050242 kcal mol^-1
- - -
-8.720979559297577 kcal mol^-1
- - -

Implying the the denatured state is more favorably transfered to a 0.5 mol/L aqueous urea solution than the native state.

The data above differs from that reported in the paper because they extrapolate the curves to low concentrations, adding an additional constant factor to both energy transfers. Nevertheless, the trends are similar, and indicate that at concentrations higher than \(~\)0.4 mol/L urea distabilizes the protein, being the effect subtler and perhaps opposite at lower concentrations.

- - -

Packages used

- -
using Trapz
- - -
using MolSimToolkit
- - -
using DataFrames
- - -
using Unitful
- - -
using Plots
- - -
using LaTeXStrings
- - -
using EasyFit
- - -
begin
-    using PlutoUI
-    TableOfContents()
-end
- - -
import Pkg
- - -
Pkg.status()
- - - \ No newline at end of file + + + + + + + + +
\ No newline at end of file diff --git a/dev/pluto_notebooks/rnaset1_urea.jl b/dev/pluto_notebooks/rnaset1_urea.jl index a23bd9e..4ed3524 100644 --- a/dev/pluto_notebooks/rnaset1_urea.jl +++ b/dev/pluto_notebooks/rnaset1_urea.jl @@ -33,12 +33,6 @@ begin using PlutoUI TableOfContents() end - -# ╔═╡ fa586575-7aac-4f52-b87a-db84c242d114 -md""" -To open and edit this notebook, [click here](../proteinA_urea.html) -""" - # ╔═╡ 6edcdced-5fb5-41f4-a741-27c98dcbd1b4 md"# RNaseT1 transfer free energy to urea" diff --git a/dev/pluto_notebooks/rnaset1_urea/index.html b/dev/pluto_notebooks/rnaset1_urea/index.html deleted file mode 100644 index 9305a16..0000000 --- a/dev/pluto_notebooks/rnaset1_urea/index.html +++ /dev/null @@ -1,258 +0,0 @@ - -RNase T1 in urea · PereiraMartinez2024.jl
GitHub
- - - - -

To open and edit this notebook, click here

- - -

RNaseT1 transfer free energy to urea

- -

Aqueous solutions of urea: experimental data

-
- - -

In Lin & Timasheff, Biochemistry, 1994, the activity coefficients (\(\gamma_3\)) of urea in aqueous solutions and the corresponding densities are provided. The data provided is that of \(\gamma_3\) as a function of the concentration \(m_3\) (in mol/kg):

- - -

Properties of aqueous solutions of urea

- - -

Experimental data for urea aqueous solutions (25\(^\circ\)C):

- - -
m₃γ₃ρ₀
10.0 mol kg^-11.00.997 g mL^-1
20.51 mol kg^-10.961.0084 g mL^-1
31.05 mol kg^-10.9241.016 g mL^-1
42.2 mol kg^-10.8591.0298 g mL^-1
- - -

The data above is used to compute the \((\partial\ln\gamma_3/\partial m_3)\) quantity. For that we need to create first a plot:

- - - - - -

The shape of the plot suggests that a quadratic fit will capture precisely the trend:

- -
urea_fit = fitquadratic(urea_data[!,:m₃],log.(urea_data[!,:γ₃]))
-
------------------- Quadratic Fit -------------
-
-Equation: y = ax^2 + bx + c
-
-With: a = 0.0056477759456873096 kg^2 mol^-2
-      b = -0.08138062050326553 kg mol^-1
-      c = -0.0002228302119289697
-
-Pearson correlation coefficient, R = 0.9999787191515568 mol kg^-1
-Average square residue = 1.3341140148130882e-7
-
-Predicted Y: ypred = [-0.0002228302119289697, -0.04025796014512112, ...]
-residues = [-0.0002228302119289697, 0.0005640343751340468, ...]
-
------------------------------------------------
-
- - -

The derivative of the logarithmic of the activity coefficient as a function of the molality is, then, the derivative of the above quadratic fit:

- -
∂lnγ₃∂m₃(m₃) = 2 * urea_fit.a * m₃ + urea_fit.b
-
∂lnγ₃∂m₃ (generic function with 1 method)
- - -

Which can be computed for the concentrations given in Table 2 of the article, and do not agree quantitatively, but follow the same trend:

- -
urea_data[:, :∂lnγ₃∂m₃] = ∂lnγ₃∂m₃.(urea_data[!,:m₃])
-
4-element Vector{Quantity{Float64, 𝐌 𝐍^-1, Unitful.FreeUnits{(kg, mol^-1), 𝐌 𝐍^-1, nothing}}}:
-  -0.08138062050326553 kg mol^-1
-  -0.07561988903866447 kg mol^-1
-  -0.06952029101732218 kg mol^-1
- -0.056530406342241366 kg mol^-1
- -
urea_data
-
m₃γ₃ρ₀∂lnγ₃∂m₃
10.0 mol kg^-11.00.997 g mL^-1-0.0813806 kg mol^-1
20.51 mol kg^-10.961.0084 g mL^-1-0.0756199 kg mol^-1
31.05 mol kg^-10.9241.016 g mL^-1-0.0695203 kg mol^-1
42.2 mol kg^-10.8591.0298 g mL^-1-0.0565304 kg mol^-1
- - -

We also obtain an expression for the density as a function of the concentration, in mol/L:

- - - - - -

Where the fit is:

- -
density_fit
-
------------------- Quadratic Fit -------------
-
-Equation: y = ax^2 + bx + c
-
-With: a = -0.003148583773846829 g L^2 mL^-1 mol^-2
-      b = 0.021417043077932497 g L mL^-1 mol^-1
-      c = 0.9973483095706743 g mL^-1
-
-Pearson correlation coefficient, R = 0.9988889736362448 g mol mL^-1 L^-1
-Average square residue = 3.1548945450684954e-7 g^2 mL^-2
-
-Predicted Y: ypred = [0.9973483095706743 g mL^-1, 1.0075299894249685 g mL^-1, ...]
-residues = [0.0003483095706743322 g mL^-1, -0.0008700105750314435 g mL^-1, ...]
-
------------------------------------------------
-
- -

RNaseT1 in urea

-
- - -

With the data above, we can proceed to using equation 5 of Lin and Timasheff:

$$\left(\frac{\partial\mu_2}{\partial m_3}\right)_{T,P,m_2} = --\left(\frac{\partial g_3}{\partial g_2}\right)_{T,P,\mu_3} -\left(\frac{RTM_2}{M_3}\right) -\left(\frac{1}{m_3}+\frac{\partial\ln \gamma_3}{\partial m_3}\right)$$

- - -

where, in the case of urea (species 3) and BpdA (solute, species 2) we have:

- - -
60.06 g mol^-1
- - -
11000 g mol^-1
- - -

The RNase-T1 and RCM-T1 constructos of the article, modeling the native and denatured states of RNase-T1, will be named from now on N and U states, respectively.

- - -

Converting concentrations:

- - -

For the N and U states, the obtained bulk concentrations of urea are obtained from the simulations:

- - -
NU
10.5 mol L^-10.5 mol L^-1
21.0 mol L^-11.0 mol L^-1
32.0 mol L^-12.0 mol L^-1
- - -

The densities corresponding to these concentrations, according to the experimental densities, are:

- - -
NU
11.00727 g mL^-11.00727 g mL^-1
21.01562 g mL^-11.01562 g mL^-1
31.02759 g mL^-11.02759 g mL^-1
- - -

With which we can compute the molality of urea in each simulated concentration:

- - -
NU
10.496391 mol kg^-10.496391 mol kg^-1
20.984623 mol kg^-10.984623 mol kg^-1
31.94631 mol kg^-11.94631 mol kg^-1
- - -

Preferential interactions:

- - -

The preferential interaction parameters of BpDA in urea, for the N8 and U6 states, are:

- - -
NU
10.02440.0247
20.03450.0542
30.06470.1445
- - - - - -

And using the gas constant in kcal/(K mol), and the temperature in K:

- - -
0.0019872036 kcal K^-1 mol^-1
- - -
298 K
- - -

The transfer free energy

- - -

We can define Eq. 5 of the article (written above), as a function of the preferential interactions, molar mass of the cossolvent, and ∂lnγ₃∂m₃:

- -
∂μ₂∂m₃(∂g₃∂g₂, m₃, M2, M3) = 
-    -(∂g₃∂g₂) * (R * T * M2 / M3) * (1/m₃ + ∂lnγ₃∂m₃(m₃))
-
∂μ₂∂m₃ (generic function with 1 method)
- - -

Applying this equation to the three possible sets of parameters for the three concentrations studied in the paper, we get:

- - -
NU
1-5.13075 kcal kg mol^-2-5.19384 kcal kg mol^-2
2-3.53738 kcal kg mol^-2-5.55727 kcal kg mol^-2
3-3.18865 kcal kg mol^-2-7.12148 kcal kg mol^-2
- - -

Now we can plot those values as function of the concentration of urea:

- - - - - -

The shaded areas are, qualitatively, the transfer free energy of each state up to each concentration.

The plot shows that at very low concentrations, the native state has a lower transfer free energy to urea than the denatured state. Thus, at this low concentration, the native state is stabilized in urea relative to the denatured state.

At higher urea concentrations the integral associated to the denatured state becomes progressively more negative, thus the denatured state is stabilized relative to the native state upon transfer to a urea solution.

Integrating these curves provides a qualitative measure of the free energy of transfer in each case:

- - -
-5.3501779370157205 kcal mol^-1
- - -
-8.720979559197184 kcal mol^-1
- - -

Implying the the denatured state is more favorably transfered to a 0.5 mol/L aqueous urea solution than the native state.

The data above differs from that reported in the paper because they extrapolate the curves to low concentrations, adding an additional constant factor to both energy transfers. Nevertheless, the trends are similar, and indicate that at concentrations higher than \(~\)0.4 mol/L urea distabilizes the protein, being the effect subtler and perhaps opposite at lower concentrations.

- - -

Packages used

- -
using Trapz
- - -
using MolSimToolkit
- - -
using DataFrames
- - -
using Unitful
- - -
using Plots
- - -
using LaTeXStrings
- - -
using EasyFit
- - -
begin
-    using PlutoUI
-    TableOfContents()
-end
- - -
import Pkg
- - -
Pkg.status()
- - -
diff --git a/dev/proteinA_urea/index.html b/dev/proteinA_urea/index.html new file mode 100644 index 0000000..8ebb583 --- /dev/null +++ b/dev/proteinA_urea/index.html @@ -0,0 +1,2 @@ + +BdpA in urea · PereiraMartinez2024.jl
GitHub
diff --git a/dev/rnaset1_urea/index.html b/dev/rnaset1_urea/index.html new file mode 100644 index 0000000..3551b81 --- /dev/null +++ b/dev/rnaset1_urea/index.html @@ -0,0 +1,2 @@ + +RNase T1 in urea · PereiraMartinez2024.jl
GitHub
diff --git a/dev/search_index.js b/dev/search_index.js index c0aa63b..b19e658 100644 --- a/dev/search_index.js +++ b/dev/search_index.js @@ -1,3 +1,3 @@ var documenterSearchIndex = {"docs": -[{"location":"pluto_notebooks/proteinA_urea/","page":"BdpA in urea","title":"BdpA in urea","text":"\n\n\n\n\n

BdpA transfer free energy to urea

\n\n","category":"page"},{"location":"pluto_notebooks/proteinA_urea/#Aqueous-solutions-of-urea:-experimental-data","page":"BdpA in urea","title":"Aqueous solutions of urea: experimental data","text":"","category":"section"},{"location":"pluto_notebooks/proteinA_urea/","page":"BdpA in urea","title":"BdpA in urea","text":"
\n
\n\n\n

In Lin & Timasheff, Biochemistry, 1994, the activity coefficients (\\(\\gamma_3\\)) of urea in aqueous solutions and the corresponding densities are provided. The data provided is that of \\(\\gamma_3\\) as a function of the concentration \\(m_3\\) (in mol/kg):

\n\n\n

Properties of aqueous solutions of urea

\n\n\n

Experimental data for urea aqueous solutions (at 25\\(^\\circ\\)C):

\n\n\n
m₃γ₃ρ₀
10.0 mol kg^-11.00.997 g mL^-1
20.51 mol kg^-10.961.0084 g mL^-1
31.05 mol kg^-10.9241.016 g mL^-1
42.2 mol kg^-10.8591.0298 g mL^-1
\n\n\n

The data above is used to compute the \\((\\partial\\ln\\gamma_3/\\partial m_3)\\) quantity. For that we need to create first a plot:

\n\n\n\n\n\n

The shape of the plot suggests that a quadratic fit will capture precisely the trend:

\n\n
urea_fit = fitquadratic(urea_data[!,:m₃],log.(urea_data[!,:γ₃]))
\n
------------------- Quadratic Fit -------------\n\nEquation: y = ax^2 + bx + c\n\nWith: a = 0.0056477759670859945 kg^2 mol^-2\n      b = -0.08138062055484234 kg mol^-1\n      c = -0.0002228301963439134\n\nPearson correlation coefficient, R = 0.9999787191515569 mol kg^-1\nAverage square residue = 1.3341140148130802e-7\n\nPredicted Y: ypred = [-0.0002228301963439134, -0.04025796015027444, ...]\nresidues = [-0.0002228301963439134, 0.0005640343699807246, ...]\n\n-----------------------------------------------\n
\n\n\n

The derivative of the logarithmic of the activity coefficient as a function of the molality is, then, the derivative of the above quadratic fit:

\n\n
∂lnγ₃∂m₃(m₃) = 2 * urea_fit.a * m₃ + urea_fit.b
\n
∂lnγ₃∂m₃ (generic function with 1 method)
\n\n\n

Which can be computed for the concentrations given in Table 2 of the article, and do not agree quantitatively, but follow the same trend:

\n\n\n
4-element Vector{Quantity{Float64, 𝐌 𝐍^-1, Unitful.FreeUnits{(kg, mol^-1), 𝐌 𝐍^-1, nothing}}}:\n  -0.08138062055484234 kg mol^-1\n  -0.07561988906841463 kg mol^-1\n  -0.06952029102396175 kg mol^-1\n -0.056530406299663966 kg mol^-1
\n\n
urea_data
\n
m₃γ₃ρ₀∂lnγ₃∂m₃
10.0 mol kg^-11.00.997 g mL^-1-0.0813806 kg mol^-1
20.51 mol kg^-10.961.0084 g mL^-1-0.0756199 kg mol^-1
31.05 mol kg^-10.9241.016 g mL^-1-0.0695203 kg mol^-1
42.2 mol kg^-10.8591.0298 g mL^-1-0.0565304 kg mol^-1
\n\n\n

We also obtain an expression for the density as a function of the concentration, in mol/L:

\n\n\n\n\n\n

Where the fit is:

\n\n
density_fit
\n
------------------- Quadratic Fit -------------\n\nEquation: y = ax^2 + bx + c\n\nWith: a = -0.0031485837712447715 g L^2 mL^-1 mol^-2\n      b = 0.021417043071470018 g L mL^-1 mol^-1\n      c = 0.9973483095726731 g mL^-1\n\nPearson correlation coefficient, R = 0.9988889736362445 g mol mL^-1 L^-1\nAverage square residue = 3.154894545068943e-7 g^2 mL^-2\n\nPredicted Y: ypred = [0.9973483095726731 g mL^-1, 1.007529989424332 g mL^-1, ...]\nresidues = [0.0003483095726730667 g mL^-1, -0.0008700105756680454 g mL^-1, ...]\n\n-----------------------------------------------\n
\n\n","category":"page"},{"location":"pluto_notebooks/proteinA_urea/#BdpA-in-urea","page":"BdpA in urea","title":"BdpA in urea","text":"","category":"section"},{"location":"pluto_notebooks/proteinA_urea/","page":"BdpA in urea","title":"BdpA in urea","text":"
\n
\n\n\n

With the data above, we can proceed to using equation 5 of Lin and Timasheff:

$$\\left(\\frac{\\partial\\mu_2}{\\partial m_3}\\right)_{T,P,m_2} = \n-\\left(\\frac{\\partial g_3}{\\partial g_2}\\right)_{T,P,\\mu_3}\n\\left(\\frac{RTM_2}{M_3}\\right)\n\\left(\\frac{1}{m_3}+\\frac{\\partial\\ln \\gamma_3}{\\partial m_3}\\right)$$

\n\n\n

where, in the case of urea (species 3) and BpdA (solute, species 2) we have:

\n\n\n
60.06 g mol^-1
\n\n\n
6768.46 g mol^-1
\n\n\n

States considered

Here, fold states are considered, from the set obtained in the simulations: the native state and a denatured state:

These states will be referred to as N and U from here on.

\n\n\n

Converting concentrations:

\n\n\n

For the N and U states, the obtained bulk concentrations of urea are obtained from the simulations:

\n\n\n
NU
10.0938442 mol L^-10.0972688 mol L^-1
20.199496 mol L^-10.200396 mol L^-1
30.29125 mol L^-10.305921 mol L^-1
40.397743 mol L^-10.401483 mol L^-1
50.49524 mol L^-10.501656 mol L^-1
\n\n\n

The densities corresponding to these concentrations, according to the experimental densities, are:

\n\n\n
NU
10.99933 g mL^-10.999402 g mL^-1
21.0015 g mL^-11.00151 g mL^-1
31.00332 g mL^-11.00361 g mL^-1
41.00537 g mL^-11.00544 g mL^-1
51.00718 g mL^-11.0073 g mL^-1
\n\n\n

With which we can compute the molality of urea in each simulated concentration:

\n\n\n
NU
10.0939071 mol kg^-10.097327 mol kg^-1
20.199198 mol kg^-10.200093 mol kg^-1
30.290287 mol kg^-10.304822 mol kg^-1
40.395619 mol kg^-10.399311 mol kg^-1
50.491708 mol kg^-10.49802 mol kg^-1
\n\n\n

Simulated preferential interactions:

\n\n\n

The preferential interaction parameters of BpDA in urea, for the N and U states, are (converted from mol/mol to g/g):

\n\n\n
NU
10.007271680.00416967
20.01117410.0127021
30.01584290.0198925
40.02089960.024073
50.02574810.0303108
\n\n\n\n\n\n

And using the gas constant in kcal/(K mol), and the temperature in K:

\n\n\n
0.0019872036 kcal K^-1 mol^-1
\n\n\n
298 K
\n\n\n

The transfer free energy

\n\n\n

We can define Eq. 5 of the article (written above), as a function of the preferential interactions, molar mass of the cossolvent, and ∂lnγ₃∂m₃:

\n\n
∂μ₂∂m₃(∂g₃∂g₂, m₃, M2, M3) = -(∂g₃∂g₂) * (R * T * M2 / M3) * (1/m₃ + ∂lnγ₃∂m₃(m₃))
\n
∂μ₂∂m₃ (generic function with 1 method)
\n\n\n

Applying this equation to the three possible sets of parameters for the three concentrations studied in the paper, we get:

\n\n\n
NU
1-5.12875 kcal kg mol^-2-2.83677 kcal kg mol^-2
2-3.68459 kcal kg mol^-2-4.16942 kcal kg mol^-2
3-3.55969 kcal kg mol^-2-4.25171 kcal kg mol^-2
4-3.41825 kcal kg mol^-2-3.89981 kcal kg mol^-2
5-3.36433 kcal kg mol^-2-3.90852 kcal kg mol^-2
\n\n\n

Now we can plot those values as function of the concentration of urea:

\n\n\n\n\n\n

The shaded areas are, qualitatively, the transfer free energy of each state up to each concentration.

The plot shows that at very low concentrations, the native state has a lower transfer free energy to urea than the denatured state. Thus, at this low concentration, the native state is stabilized in urea relative to the denatured state.

At higher urea concentrations the integral associated to the denatured state becomes progressively more negative, thus the denatured state is stabilized relative to the native state upon transfer to a urea solution.

Integrating these curves provides a qualitative measure of the free energy of transfer in each case:

\n\n\n
-1.4872863073205036 kcal mol^-1
\n\n\n
-1.5714588136413514 kcal mol^-1
\n\n\n

Implying the the denatured state is slightly more favorably transfered to a 0.5 mol/L aqueous urea solution than the native state. The difference probably increases with increasing urea concentration.

\n\n\n

Packages used

\n\n
using Trapz
\n\n\n
using MolSimToolkit
\n\n\n
using DataFrames
\n\n\n
using Unitful
\n\n\n
using Plots
\n\n\n
using LaTeXStrings
\n\n\n
using EasyFit
\n\n\n
begin\n    using PlutoUI\n    TableOfContents()\nend
\n\n\n
import Pkg
\n\n\n
Pkg.status()
\n\n\n","category":"page"},{"location":"pluto_notebooks/proteinA_urea/","page":"BdpA in urea","title":"BdpA in urea","text":"EditURL = \"https://github.com/m3g/PereiraMartinez2024.jl/blob/main/docs/src/pluto_notebooks/proteinA_urea.jl\"","category":"page"},{"location":"#Supplementary-information","page":"Home","title":"Supplementary information","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"A. F. Pereira, L. Martínez, \"Cossolvent effects on protein folding landscapes\". ","category":"page"},{"location":"pluto_notebooks/rnaset1_urea/","page":"RNase T1 in urea","title":"RNase T1 in urea","text":"\n\n\n\n\n

To open and edit this notebook, click here

\n\n\n

RNaseT1 transfer free energy to urea

\n\n","category":"page"},{"location":"pluto_notebooks/rnaset1_urea/#Aqueous-solutions-of-urea:-experimental-data","page":"RNase T1 in urea","title":"Aqueous solutions of urea: experimental data","text":"","category":"section"},{"location":"pluto_notebooks/rnaset1_urea/","page":"RNase T1 in urea","title":"RNase T1 in urea","text":"
\n
\n\n\n

In Lin & Timasheff, Biochemistry, 1994, the activity coefficients (\\(\\gamma_3\\)) of urea in aqueous solutions and the corresponding densities are provided. The data provided is that of \\(\\gamma_3\\) as a function of the concentration \\(m_3\\) (in mol/kg):

\n\n\n

Properties of aqueous solutions of urea

\n\n\n

Experimental data for urea aqueous solutions (25\\(^\\circ\\)C):

\n\n\n
m₃γ₃ρ₀
10.0 mol kg^-11.00.997 g mL^-1
20.51 mol kg^-10.961.0084 g mL^-1
31.05 mol kg^-10.9241.016 g mL^-1
42.2 mol kg^-10.8591.0298 g mL^-1
\n\n\n

The data above is used to compute the \\((\\partial\\ln\\gamma_3/\\partial m_3)\\) quantity. For that we need to create first a plot:

\n\n\n\n\n\n

The shape of the plot suggests that a quadratic fit will capture precisely the trend:

\n\n
urea_fit = fitquadratic(urea_data[!,:m₃],log.(urea_data[!,:γ₃]))
\n
------------------- Quadratic Fit -------------\n\nEquation: y = ax^2 + bx + c\n\nWith: a = 0.0056477759456873096 kg^2 mol^-2\n      b = -0.08138062050326553 kg mol^-1\n      c = -0.0002228302119289697\n\nPearson correlation coefficient, R = 0.9999787191515568 mol kg^-1\nAverage square residue = 1.3341140148130882e-7\n\nPredicted Y: ypred = [-0.0002228302119289697, -0.04025796014512112, ...]\nresidues = [-0.0002228302119289697, 0.0005640343751340468, ...]\n\n-----------------------------------------------\n
\n\n\n

The derivative of the logarithmic of the activity coefficient as a function of the molality is, then, the derivative of the above quadratic fit:

\n\n
∂lnγ₃∂m₃(m₃) = 2 * urea_fit.a * m₃ + urea_fit.b
\n
∂lnγ₃∂m₃ (generic function with 1 method)
\n\n\n

Which can be computed for the concentrations given in Table 2 of the article, and do not agree quantitatively, but follow the same trend:

\n\n
urea_data[:, :∂lnγ₃∂m₃] = ∂lnγ₃∂m₃.(urea_data[!,:m₃])
\n
4-element Vector{Quantity{Float64, 𝐌 𝐍^-1, Unitful.FreeUnits{(kg, mol^-1), 𝐌 𝐍^-1, nothing}}}:\n  -0.08138062050326553 kg mol^-1\n  -0.07561988903866447 kg mol^-1\n  -0.06952029101732218 kg mol^-1\n -0.056530406342241366 kg mol^-1
\n\n
urea_data
\n
m₃γ₃ρ₀∂lnγ₃∂m₃
10.0 mol kg^-11.00.997 g mL^-1-0.0813806 kg mol^-1
20.51 mol kg^-10.961.0084 g mL^-1-0.0756199 kg mol^-1
31.05 mol kg^-10.9241.016 g mL^-1-0.0695203 kg mol^-1
42.2 mol kg^-10.8591.0298 g mL^-1-0.0565304 kg mol^-1
\n\n\n

We also obtain an expression for the density as a function of the concentration, in mol/L:

\n\n\n\n\n\n

Where the fit is:

\n\n
density_fit
\n
------------------- Quadratic Fit -------------\n\nEquation: y = ax^2 + bx + c\n\nWith: a = -0.003148583773846829 g L^2 mL^-1 mol^-2\n      b = 0.021417043077932497 g L mL^-1 mol^-1\n      c = 0.9973483095706743 g mL^-1\n\nPearson correlation coefficient, R = 0.9988889736362448 g mol mL^-1 L^-1\nAverage square residue = 3.1548945450684954e-7 g^2 mL^-2\n\nPredicted Y: ypred = [0.9973483095706743 g mL^-1, 1.0075299894249685 g mL^-1, ...]\nresidues = [0.0003483095706743322 g mL^-1, -0.0008700105750314435 g mL^-1, ...]\n\n-----------------------------------------------\n
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\n
\n\n\n

With the data above, we can proceed to using equation 5 of Lin and Timasheff:

$$\\left(\\frac{\\partial\\mu_2}{\\partial m_3}\\right)_{T,P,m_2} = \n-\\left(\\frac{\\partial g_3}{\\partial g_2}\\right)_{T,P,\\mu_3}\n\\left(\\frac{RTM_2}{M_3}\\right)\n\\left(\\frac{1}{m_3}+\\frac{\\partial\\ln \\gamma_3}{\\partial m_3}\\right)$$

\n\n\n

where, in the case of urea (species 3) and BpdA (solute, species 2) we have:

\n\n\n
60.06 g mol^-1
\n\n\n
11000 g mol^-1
\n\n\n

The RNase-T1 and RCM-T1 constructos of the article, modeling the native and denatured states of RNase-T1, will be named from now on N and U states, respectively.

\n\n\n

Converting concentrations:

\n\n\n

For the N and U states, the obtained bulk concentrations of urea are obtained from the simulations:

\n\n\n
NU
10.5 mol L^-10.5 mol L^-1
21.0 mol L^-11.0 mol L^-1
32.0 mol L^-12.0 mol L^-1
\n\n\n

The densities corresponding to these concentrations, according to the experimental densities, are:

\n\n\n
NU
11.00727 g mL^-11.00727 g mL^-1
21.01562 g mL^-11.01562 g mL^-1
31.02759 g mL^-11.02759 g mL^-1
\n\n\n

With which we can compute the molality of urea in each simulated concentration:

\n\n\n
NU
10.496391 mol kg^-10.496391 mol kg^-1
20.984623 mol kg^-10.984623 mol kg^-1
31.94631 mol kg^-11.94631 mol kg^-1
\n\n\n

Preferential interactions:

\n\n\n

The preferential interaction parameters of BpDA in urea, for the N8 and U6 states, are:

\n\n\n
NU
10.02440.0247
20.03450.0542
30.06470.1445
\n\n\n\n\n\n

And using the gas constant in kcal/(K mol), and the temperature in K:

\n\n\n
0.0019872036 kcal K^-1 mol^-1
\n\n\n
298 K
\n\n\n

The transfer free energy

\n\n\n

We can define Eq. 5 of the article (written above), as a function of the preferential interactions, molar mass of the cossolvent, and ∂lnγ₃∂m₃:

\n\n
∂μ₂∂m₃(∂g₃∂g₂, m₃, M2, M3) = \n    -(∂g₃∂g₂) * (R * T * M2 / M3) * (1/m₃ + ∂lnγ₃∂m₃(m₃))
\n
∂μ₂∂m₃ (generic function with 1 method)
\n\n\n

Applying this equation to the three possible sets of parameters for the three concentrations studied in the paper, we get:

\n\n\n
NU
1-5.13075 kcal kg mol^-2-5.19384 kcal kg mol^-2
2-3.53738 kcal kg mol^-2-5.55727 kcal kg mol^-2
3-3.18865 kcal kg mol^-2-7.12148 kcal kg mol^-2
\n\n\n

Now we can plot those values as function of the concentration of urea:

\n\n\n\n\n\n

The shaded areas are, qualitatively, the transfer free energy of each state up to each concentration.

The plot shows that at very low concentrations, the native state has a lower transfer free energy to urea than the denatured state. Thus, at this low concentration, the native state is stabilized in urea relative to the denatured state.

At higher urea concentrations the integral associated to the denatured state becomes progressively more negative, thus the denatured state is stabilized relative to the native state upon transfer to a urea solution.

Integrating these curves provides a qualitative measure of the free energy of transfer in each case:

\n\n\n
-5.3501779370157205 kcal mol^-1
\n\n\n
-8.720979559197184 kcal mol^-1
\n\n\n

Implying the the denatured state is more favorably transfered to a 0.5 mol/L aqueous urea solution than the native state.

The data above differs from that reported in the paper because they extrapolate the curves to low concentrations, adding an additional constant factor to both energy transfers. Nevertheless, the trends are similar, and indicate that at concentrations higher than \\(~\\)0.4 mol/L urea distabilizes the protein, being the effect subtler and perhaps opposite at lower concentrations.

\n\n\n

Packages used

\n\n
using Trapz
\n\n\n
using MolSimToolkit
\n\n\n
using DataFrames
\n\n\n
using Unitful
\n\n\n
using Plots
\n\n\n
using LaTeXStrings
\n\n\n
using EasyFit
\n\n\n
begin\n    using PlutoUI\n    TableOfContents()\nend
\n\n\n
import Pkg
\n\n\n
Pkg.status()
\n\n\n","category":"page"},{"location":"pluto_notebooks/rnaset1_urea/","page":"RNase T1 in urea","title":"RNase T1 in urea","text":"EditURL = \"https://github.com/m3g/PereiraMartinez2024.jl/blob/main/docs/src/pluto_notebooks/rnaset1_urea.jl\"","category":"page"}] +[{"location":"proteinA_urea/","page":"BdpA in urea","title":"BdpA in urea","text":"","category":"page"},{"location":"#Supplementary-information","page":"Home","title":"Supplementary information","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"A. F. Pereira, L. Martínez, \"Cossolvent effects on protein folding landscapes\". ","category":"page"},{"location":"rnaset1_urea/","page":"RNase T1 in urea","title":"RNase T1 in urea","text":"","category":"page"}] }