Thermodynamic Prediction of Glycine Polymerization as a Function of Temperature and pH Consistent with Experimentally Obtained Results

Abstract

Prediction of the thermodynamic behaviors of biomolecules at high temperature and pressure is fundamental to understanding the role of hydrothermal systems in the origin and evolution of life on the primitive Earth. However, available thermodynamic dataset for amino acids, essential components for life, cannot represent experimentally observed polymerization behaviors of amino acids accurately under hydrothermal conditions. This report presents the thermodynamic data and the revised HKF parameters for the simplest amino acid “Gly” and its polymers (GlyGly, GlyGlyGly and DKP) based on experimental thermodynamic data from the literature. Values for the ionization states of Gly (Gly⁺ and Gly⁻) and Gly peptides (GlyGly⁺, GlyGly⁻, GlyGlyGly⁺, and GlyGlyGly⁻) were also retrieved from reported experimental data by combining group additivity algorithms. The obtained dataset enables prediction of the polymerization behavior of Gly as a function of temperature and pH, consistent with experimentally obtained results in the literature. The revised thermodynamic data for zwitterionic Gly, GlyGly, and DKP were also used to estimate the energetics of amino acid polymerization into proteins. Results show that the Gibbs energy necessary to synthesize a mole of peptide bond is more than 10 kJ mol⁻¹ less than previously estimated over widely various temperatures (e.g., 28.3 kJ mol⁻¹ → 17.1 kJ mol⁻¹ at 25 °C and 1 bar). Protein synthesis under abiotic conditions might therefore be more feasible than earlier studies have shown.

Appendices

Appendix 1

In this appendix, the author summarizes the revised HKF equations of state and the thermodynamic conventions adopted in this study. Further detailed information for the calculation methodology is available in Amend and Helgeson (1997) and Dick et al. (2006).

Standard State Conventions

The standard state convention adopted for H₂O is unit activity of the pure solvent at any temperature and pressure. The convention for other aqueous species corresponds to unit activity of the species in a hypothetical one molal solution referenced to infinite dilution at any temperature and pressure. The conventional standard molal thermodynamic properties of a charged aqueous species are given as the following.

$$\Xi = \Xi^{\text{abs}} - Z\Xi_{{H^{ + } }}^{\text{abs}}$$

(21)

Therein, Ξ and Ξ ^abs, respectively, stand for any conventional and absolute standard molal property of the aqueous species of interest, and $\Xi_{{H^{ + } }}^{\text{abs}}$ denotes the corresponding absolute standard molal property of the hydrogen ion. Also, Z represents the charge of the aqueous species of interest.

The standard molal Gibbs energies (∆G ^°) and enthalpies (∆H ^°) of aqueous species are expressed as apparent standard molal Gibbs energies and enthalpies of formation, which are defined as

$$\Delta H^{^\circ } \equiv \Delta_{f} H^{{^{^\circ } }} + \left( {H_{P,T}^{{^{^\circ } }} - H_{{P_{\text{r}} ,T_{\text{r}} }}^{{^{^\circ } }} } \right)$$

(22)

and

$$\Delta G^{^\circ } \equiv \Delta_{f} G^{{^{^\circ } }} + \left( {G_{P,T}^{{^{^\circ } }} - G_{{P_{\text{r}} ,T_{\text{r}} }}^{{^{^\circ } }} } \right)$$

(23)

where ∆ _f G ^° and ∆ _f H ^° represent the standard molal Gibbs energy and enthalpy of formation of the species from the elements at the reference temperature (T _r = 298.15 K) and pressure (P _r = 1 bar), and where $G_{P,T}^{{^{^\circ } }} - G_{{P_{\text{r}} ,T_{\text{r}} }}^{{^{^\circ } }}$ and $H_{P,T}^{{^{^\circ } }} - H_{{P_{\text{r}} ,T_{\text{r}} }}^{{^{^\circ } }}$, respectively, denote the differences between the standard molal Gibbs energy and enthalpy at the temperature (T) and pressure (P) of interest, and those at T _r and P _r.

The values of ∆ _f G ^° and ∆ _f H ^° and the standard molal entropy at 25 °C and 1 bar ($S_{{P_{\text{r}} ,T_{\text{r}} }}^{{^{^\circ } }}$) are related by the following.

$$\Delta G_{f}^{{^{^\circ } }} = \Delta H_{f}^{{^{^\circ } }} - T_{\text{r}} \left( {S_{{P_{\text{r}} ,T_{\text{r}} }}^{{^{^\circ } }} - S_{{P_{\text{r}} , T_{\text{r}} , {\text{elements}}}}^{{^{^\circ } }} } \right)$$

(24)

In that equation, $S_{{P_{\text{r}} , T_{\text{r}} , {\text{elements}}}}^{{^{^\circ } }}$ represents the total standard molal entropy at 25 °C and 1 bar of the elements making up the species of interest. The values of $S_{{P_{\text{r}} ,T_{\text{r}} }}^{{^{^\circ } }}$ of the elements used in this study were referred from a report by Cox et al. (1989).

Summary of the Revised HKF Equation of State

The revised HKF equations of state are consistent with the separation of variables represented as shown below.

$$\Xi = \Delta \Xi_{n} + \Delta \Xi_{s}$$

(25)

Therein, Ξ stands for any standard molal property of an aqueous species. In addition, ∆Ξ _n and ∆Ξ _s , respectively, refer to the nonsolvation and solvation contributions to Ξ.

The nonsolvation contributions to the standard molal isobaric heat capacity (C ^° _P ), volume (V ^°), and isothermal compressibility (κ ^°_T ) of an aqueous species ($\Delta C_{P,n}^{{^{^\circ } }}$, $\Delta V_{n}^{{^{^\circ } }}$, and $\Delta \kappa_{T,n}^{{^{^\circ } }}$, respectively) are given in the literature (Tanger and Helgeson 1988) as

$$\Delta C_{P,n}^{{^{^\circ } }} = c_{1} + \frac{{c_{2} }}{{\left( {T - \Theta } \right)^{2} }} - \left( {\frac{2T}{{\left( {T - \Theta } \right)^{3} }}} \right)\left( {a_{3} \left( {P - P_{\text{r}} } \right) + a_{4} \ln \left( {\frac{\Psi + P}{{\Psi + P_{\text{r}} }}} \right)} \right)$$

(26)

$$\Delta V_{n}^{o} = a_{1} + \frac{{a_{2} }}{\Psi + P} - \left( {a_{3} + \frac{{a_{4} }}{\Psi + P}} \right)\left( {\frac{1}{T - \Theta }} \right)$$

(27)

and

$$\Delta \kappa_{T,n}^{{^{^\circ } }} = \left( {a_{2} + \frac{{a_{4} }}{T - \varTheta }} \right)\left( {\frac{1}{\varPsi + P}} \right)^{2}$$

(28)

where a ₁, a ₂, a ₃, a ₄, c ₁, and c ₂ represent temperature-independent and pressure-independent parameters of the species of interest, and where Θ and Ψ, respectively, represent solvent parameters equal to 228 K and 2,600 bar.

The solvation contributions to C ^° _P , V ^°, and$\kappa_{T}^{o}$ are expressed as

$$\Delta C_{P,s}^{{^{^\circ } }} = \omega TX + 2TY\left( {\frac{\partial \omega }{\partial T}} \right)_{P} - T\left( {\frac{1}{\epsilon } - 1} \right)\left( {\frac{{\partial^{2} \omega }}{{\partial T^{2} }}} \right)_{P}$$

(29)

$$\Delta V_{s}^{{^{^\circ } }} = \omega Q + \left( {\frac{1}{\epsilon } - 1} \right)\left( {\frac{\partial \omega }{\partial P}} \right)_{T}$$

(30)

and

$$\Delta \kappa_{T,s}^{{^{^\circ } }} = \omega N + 2Q\left( {\frac{\partial \omega }{\partial P}} \right)_{T} - \left( {\frac{1}{\epsilon } - 1} \right)\left( {\frac{{\partial^{2} \omega }}{{\partial P^{2} }}} \right)_{T}$$

(31)

where ω denotes the solvation parameter of the species of interest, $\epsilon$ stands for the dielectric constant of H₂O, and Q, N, Y, and X designate the Born functions defined, respectively, as $- \left( {\frac{{\partial \left( {1/\epsilon } \right)}}{\partial P}} \right)_{T}$, $\left( {\frac{\partial Q}{\partial P}} \right)_{T}$, $- \left( {\frac{{\partial \left( {1/epsilon } \right)}}{\partial T}} \right)_{P}$, and $\left( {\frac{\partial Y}{\partial T}} \right)_{P}$. The values of Q, X, and Y used in this study were taken from Shock et al. (1992b), but those of N correspond to the values reported by Tanger and Helgeson (1988). The partial derivatives of ω with respect to temperature and pressure are taken as

$$\left( {\frac{\partial \omega }{\partial T}} \right)_{P} = \left( {\frac{\partial \omega }{\partial P}} \right)_{T} = \left( {\frac{{\partial^{2} \omega }}{{\partial T^{2} }}} \right)_{P} = \left( {\frac{{\partial^{2} \omega }}{{\partial P^{2} }}} \right)_{T} = 0$$

(32)

for all the neutral and charged groups and species considered below (Amend and Helgeson 1997, 2000; Dick et al. 2006).

At P ≈ P_r, Eq. 26 reduces to the equation shown below.

$$\Delta C_{P,n}^{{^{^\circ } }} = c_{1} { + }\frac{{c_{2} }}{{\left( {T - \varTheta } \right)^{2} }}$$

(33)

Contributions by the pressure-dependent terms are negligible for pressures less than a few hundred bars (Dick et al. 2006). Eq. 27 can be written as

$$\Delta V_{n}^{{^{^\circ } }} = \sigma + \frac{\xi }{T - \varTheta }$$

(34)

by defining the parameters σ and ξ as

$$\sigma \equiv a_{1} + \frac{{a_{2} }}{\varPsi + p}$$

(35)

and the following expression.

$$\xi \equiv a_{3} + \frac{{a_{4} }}{\varPsi + P}$$

(36)

By taking account of Eqs. 25 and 32 together with Eqs. 29 and 33 for C ^° _P or Eqs. 30 and 34 for V ^°, it is possible to write

$$C_{P}^{{^{^\circ } }} = c_{1} + \frac{{c_{2} }}{{\left( {T - \varTheta } \right)^{2} }} + \omega TX$$

(37)

and the following.

$$V^{o} = \sigma + \frac{\xi }{T - \varTheta } - \omega Q$$

(38)

The revised HKF equations of state for ∆G ^°, ∆H ^°, and S ^° can be written as (Tanger and Helgeson 1988)

$$\begin{gathered} \Delta G^{o} =\, \Delta G_{f}^{o} - S_{{P_{r} ,T_{r} }}^{o} \left( {T - T_{r} } \right) - c_{1} \left[ {T\ln \left( {\frac{T}{{T_{r} }}} \right) - T + T_{r} } \right] - c_{2} \left\{ {\left[ {\left( {\frac{1}{T - \varTheta }} \right) - \left( {\frac{1}{{T_{r} - \varTheta }}} \right)} \right]\left( {\frac{\varTheta - T}{\varTheta }} \right)} \right\} \hfill \\ + c_{2} \left\{ {\frac{T}{{\varTheta^{2} }}\ln \left[ {\frac{{T_{r} \left( {T - \varTheta } \right)}}{{T\left( {T_{r} - \varTheta } \right)}}} \right]} \right\} + a_{1} \left( {P - P_{r} } \right) + a_{2} \ln \left( {\frac{\varPsi + P}{{\varPsi + P_{r} }}} \right) + \left( {\frac{1}{T + \varTheta }} \right)\left[ {a_{3} \left( {P - P_{r} } \right) + a_{4} \ln \left( {\frac{\varPsi + P}{{\varPsi + P_{r} }}} \right)} \right] \hfill \\ + \omega \left( {\frac{1}{\epsilon } - 1} \right) - \omega_{{P_{r} ,T_{r} }} \left[ {Y_{{P_{r} ,T_{r} }} \left( {T_{r} - T} \right) + \frac{1}{{\epsilon_{{P_{r} ,T_{r} }} }} - 1} \right] \hfill \\ \end{gathered}$$

(39)

$$\begin{gathered} \Delta H^{o} = \Delta H_{f}^{o} + c_{1} \left( {T - T_{r} } \right) - c_{2} \left[ {\left( {\frac{1}{T - \varTheta }} \right) - \left( {\frac{1}{{T_{r} - \varTheta }}} \right)} \right] + a_{1} \left( {P - P_{r} } \right) + \hfill \\ a_{2} \ln \left( {\frac{\varPsi + P}{{\varPsi + P_{r} }}} \right) + \left[ {\frac{2T - \varTheta }{{\left( {T - \varTheta } \right)^{2} }}} \right]\left[ {a_{3} \left( {P - P_{r} } \right) + a_{4} \ln \left( {\frac{\varPsi + P}{{\varPsi + P_{r} }}} \right)} \right] + \hfill \\ \omega \left( {TY + \frac{1}{\epsilon } - 1} \right) - T\left( {\frac{1}{\epsilon } - 1} \right)\left( {\frac{\partial \omega }{\partial T}} \right)_{P} - \omega_{{P_{r} ,T_{r} }} \left( {T_{r} Y_{{P_{r} ,T_{r} }} + \frac{1}{{\epsilon_{{P_{r} ,T_{r} }} }} - 1} \right) \hfill \\ \end{gathered}$$

(40)

and

$$\begin{gathered} S^{o} =\, S_{{P_{r} ,T_{r} }}^{o} + c_{1} \ln \left( {\frac{T}{{T_{r} }}} \right) - \frac{{c_{2} }}{\varTheta }\left\{ {\left( {\frac{1}{T - \varTheta }} \right) - \left( {\frac{1}{{T_{r} - \varTheta }}} \right) + \frac{1}{\varTheta }\ln \left[ {\frac{{T_{r} \left( {T - \varTheta } \right)}}{{T\left( {T_{r} - \varTheta } \right)}}} \right]} \right\} + \hfill \\ \left( {\frac{1}{T - \varTheta }} \right)^{2} \left[ {a_{3} \left( {P - P_{r} } \right) + a_{4} \ln \left( {\frac{\varPsi + P}{{\varPsi + P_{r} }}} \right)} \right] + \omega Y - \left( {\frac{1}{\epsilon } - 1} \right)\left( {\frac{\partial \omega }{\partial T}} \right)_{P} - \omega_{{P_{r} ,T_{r} }} Y_{{P_{r} ,T_{r} }} \hfill \\ \end{gathered}$$

(41)

respectively.

Appendix 2

See Fig. 10.

Fig. 10

Experimental (symbols) and calculated (curves) standard molal heat capacity (C ^° _P ) of Gly^± as a function of temperature. No error bar is shown in this figure because the reported uncertainties are less than the symbol size