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Stochastic Mirror Symmetry Breaking: Theoretical Models and Simulation of Experiments

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Biochirality

Part of the book series: Topics in Current Chemistry ((TOPCURRCHEM,volume 333))

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Acknowledgements

The authors are grateful to Professor Josep M. Ribó (University of Barcelona) and to Professor Meir Lahav (Weizmann Institute of Science) for collaboration and for many useful discussions over the past few years which have helped to shape and temper our own perspectives on the subject of chiral symmetry breaking and chiral amplification at the molecular level. DH acknowledges the Grant AYA2009-13920-C02-01 from the Ministerio de Ciencia e Innovación (Spain) and forms part of the COST Action CM07030: Systems Chemistry. CB acknowledges a Calvo-Rodés graduate student contract from the Instituto Nacional de Técnica Aeroespacial (INTA).

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Authors and Affiliations

  1. Centro de Astrobiología (CSIC-INTA), Carretera Ajalvir Kilómetro 4, 28850, Torrejón de Ardoz, Madrid, Spain

    Celia Blanco & David Hochberg

Authors
  1. Celia Blanco
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  2. David Hochberg
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Correspondence to David Hochberg .

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Editors and Affiliations

  1. Dpto. Quimica Organica e Inorganica, Facultad de Ciencias-UEX, Badajoz, 06006, Spain

    Pedro Cintas

Appendix

Appendix

The differential rate equations for the chiral copolymerization scheme [61] are collected here. We begin with the rate equations for the two enantiomers:

$$ \begin{array}{lll} \frac{{{\text{d}}c_{{1,0}}^{\rm{A}}}}{{{\text{d}}t}} = - {{k}_{{aa}}}c_{{1,0}}^{\rm{A}}\left( {2c_{{1,0}}^{\rm{A}} + \sum\limits_{{n = 2}}^{{N - 1}} {c_{{n,0}}^{\rm{A}} + \sum\limits_{{n = 2}}^{{N - 2}} {c_{{1,n}}^{\rm{A}} + \sum\limits_{{r = 2}}^{{N - 2}} {\sum\limits_{{s = 1}}^{{N - 1 - r}} {c_{{r,s}}^{\rm{A}}} } } } } \right) \cr - {{k}_{{ba}}}c_{{1,0}}^{\rm{A}}\left( {\sum\limits_{{n = 2}}^{{N - 1}} {c_{{0,n}}^{\rm{B}} + \sum\limits_{{n = 2}}^{{N - 2}} {c_{{n,1}}^{\rm{B}} + \sum\limits_{{s = 2}}^{{N - 2}} {\sum\limits_{{r = 1}}^{{N - 1 - s}} {c_{{r,s}}^{\rm{B}}} } } } } \right) \cr - {{k}_h}c_{{1,0}}^{\rm{A}}c_{{0,1}}^{\rm{B}} - {{k}_{{ha}}}c_{{1,0}}^{\rm{A}}{{c}_{{1,1}}} + k_{{aa}}^{*}\left( {2c_{{2,0}}^{\rm{A}} + \sum\limits_{{n = 3}}^N {c_{{n,0}}^{\rm{A}}} + \sum\limits_{{n = 2}}^{{N - 2}} {c_{{2,n}}^{\rm{A}} + \sum\limits_{{r = 3}}^{{N - 1}} {\sum\limits_{{s = 1}}^{{N - r}} {c_{{r,s}}^{\rm{A}}} } } } \right) \cr + k_{{ba}}^{*}\left( {\sum\limits_{{n = 2}}^{{N - 1}} {c_{{1,n}}^{\rm{A}} + \sum\limits_{{n = 2}}^{{N - 2}} {c_{{2,n}}^{\rm{A}} + \sum\limits_{{r = 3}}^{{N - 1}} {\sum\limits_{{s = 1}}^{{N - r}} {c_{{r,s}}^{\rm{A}}} } } } } \right) + k_h^{*}{{c}_{{1,1}}} + k_{{ha}}^{*}c_{{2,1}}^{\rm{A}},\end{array} $$
$$ \begin{array}{lll} \frac{{{\text{d}}c_{{0,1}}^{\rm{B}}}}{{{\text{d}}t}} = - {{k}_{{bb}}}c_{{0,1}}^{\rm{B}}\left( {2c_{{0,1}}^{\rm{B}} + \sum\limits_{{n = 2}}^{{N - 1}} {c_{{0,n}}^{\rm{B}} + \sum\limits_{{n = 2}}^{{N - 2}} {c_{{n,1}}^{\rm{B}} + \sum\limits_{{s = 2}}^{{N - 2}} {\sum\limits_{{r = 1}}^{{N - 1 - s}} {c_{{r,s}}^{\rm{B}}} } } } } \right) \cr - {{k}_{{ab}}}c_{{0,1}}^{\rm{B}}\left( {\sum\limits_{{n = 2}}^{{N - 1}} {c_{{n,0}}^{\rm{A}} + \sum\limits_{{n = 2}}^{{N - 2}} {c_{{1,n}}^{\rm{A}} + \sum\limits_{{r = 2}}^{{N - 2}} {\sum\limits_{{s = 1}}^{{N - 1 - r}} {c_{{r,s}}^{\rm{A}}} } } } } \right) \cr - {{k}_h}c_{{1,0}}^{\rm{A}}c_{{0,1}}^{\rm{B}} - {{k}_{{hb}}}c_{{0,1}}^{\rm{B}}{{c}_{{1,1}}} + k_{{bb}}^{*}\left( {2c_{{0,2}}^{\rm{B}} + \sum\limits_{{n = 3}}^N {c_{{0,n}}^{\rm{B}}} + \sum\limits_{{n = 2}}^{{N - 2}} {c_{{n,2}}^{\rm{B}} + \sum\limits_{{s = 3}}^{{N - 1}} {\sum\limits_{{r = 1}}^{{N - s}} {c_{{r,s}}^{\rm{B}}} } } } \right) \cr + k_{{ab}}^{*}\left( {\sum\limits_{{n = 2}}^{{N - 1}} {c_{{n,1}}^{\rm{B}} + \sum\limits_{{n = 2}}^{{N - 2}} {c_{{n,2}}^{\rm{B}} + \sum\limits_{{s = 3}}^{{N - 1}} {\sum\limits_{{r = 1}}^{{N - s}} {c_{{r,s}}^{\rm{B}}} } } } } \right) + k_h^{*}{{c}_{{1,1}}} + k_{{hb}}^{*}c_{{1,2}}^{\rm{B}}.\end{array} $$

The equations describing the concentration of the homopolymers, for 2 ≤ n ≤ N − 1:

$$ \begin{array}{lll} \frac{{{\text{d}}c_{{n,0}}^{\rm{A}}}}{{{\text{d}}t}} = {{k}_{{aa}}}c_{{1,0}}^{\rm{A}}\left( {c_{{n - 1,0}}^{\rm{A}} - c_{{n,0}}^{\rm{A}}} \right) - {{k}_{{ab}}}c_{{n,0}}^{\rm{A}}c_{{0,1}}^{\rm{B}} + k_{{aa}}^{*}\left( {c_{{n + 1,0}}^{\rm{A}} - c_{{n,0}}^{\rm{A}}} \right) + k_{{ab}}^{*}c_{{n,1}}^{\rm{B}}, \hfill \\\frac{{{\text{d}}c_{{0,n}}^{\rm{B}}}}{{{\text{d}}t}} = {{k}_{{bb}}}c_{{0,1}}^{\rm{B}}\left( {c_{{0,n - 1}}^{\rm{B}} - c_{{0,n}}^{\rm{B}}} \right) - {{k}_{{ba}}}c_{{0,n}}^{\rm{B}}c_{{1,0}}^{\rm{A}} + k_{{bb}}^{*}\left( {c_{{0,n + 1}}^{\rm{B}} - c_{{0,n}}^{\rm{B}}} \right) + k_{{ba}}^{*}c_{{1,n}}^{\rm{A}}. \end{array} $$

It is necessary to treat the kinetic equations of the maximum length N homopolymers individually. Since these do not elongate further, they cannot directly react, and cannot be the product of an inverse reaction involving a longer chain:

$$ \begin{array}{lll} \frac{{{\text{d}}c_{{N,0}}^{\rm{A}}}}{{{\text{d}}t}} = {{k}_{{aa}}}c_{{1,0}}^{\rm{A}}c_{{N - 1,0}}^{\rm{A}} - k_{{aa}}^{*}c_{{N,0}}^{\rm{A}}, \hfill \\\frac{{{\text{d}}c_{{0,N}}^{\rm{B}}}}{{{\text{d}}t}} = {{k}_{{bb}}}c_{{0,1}}^{\rm{B}}c_{{0,N - 1}}^{\rm{B}} - k_{{bb}}^{*}c_{{0,N}}^{\rm{B}}. \end{array} $$

The differential equations describing the concentration of each type of heteropolymer (included the heterodimer), for 2 ≤ n ≤ N − 2:

$$ \begin{array}{lll} \frac{{{\text{d}}{{c}_{{1,1}}}}}{{{\text{d}}t}} = {{k}_h}c_{{1,0}}^{\rm{A}}c_{{0,1}}^{\rm{B}} - {{k}_{{ha}}}{{c}_{{1,1}}}c_{{1,0}}^{\rm{A}} - {{k}_{{hb}}}{{c}_{{1,1}}}c_{{0,1}}^{\rm{B}} - k_h^{*}{{c}_{{1,1}}} + k_{{ha}}^{*}c_{{2,1}}^{\rm{A}} + k_{{hb}}^{*}c_{{1,2}}^{\rm{B}}, \hfill \cr \frac{{{\text{d}}c_{{1,n}}^{\rm{A}}}}{{{\text{d}}t}} = - {{k}_{{aa}}}c_{{1,0}}^{\rm{A}}c_{{1,n}}^{\rm{A}} - {{k}_{{ab}}}c_{{0,1}}^{\rm{B}}c_{{1,n}}^{\rm{A}} + {{k}_{{ba}}}c_{{0,n}}^{\rm{B}}c_{{1,0}}^{\rm{A}} + k_{{aa}}^{*}c_{{2,n}}^{\rm{A}} + k_{{ab}}^{*}c_{{1,n + 1}}^{\rm{B}} - k_{{ba}}^{*}c_{{1,n}}^{\rm{A}}, \hfill \cr \frac{{{\text{d}}c_{{n,1}}^{\rm{B}}}}{{{\text{d}}t}} = - {{k}_{{bb}}}c_{{0,1}}^{\rm{B}}c_{{n,1}}^{\rm{B}} - {{k}_{{ba}}}c_{{1,0}}^{\rm{A}}c_{{n,1}}^{\rm{B}} + {{k}_{{ab}}}c_{{n,0}}^{\rm{A}}c_{{0,1}}^{\rm{B}} + k_{{bb}}^{*}c_{{n,2}}^{\rm{B}} + k_{{ba}}^{*}c_{{n + 1,1}}^{\rm{A}} - k_{{ab}}^{*}c_{{n,1}}^{\rm{B}}. \end{array} $$

It is useful to treat individually the maximum length polymers N:

$$ \begin{array}{lll} \frac{{{\text{d}}c_{{1,N - 1}}^{\rm{A}}}}{{dt}} = {{k}_{{ba}}}c_{{0,N - 1}}^{\rm{B}}c_{{1,0}}^{\rm{A}} - k_{{ba}}^{*}c_{{1,N - 1}}^{\rm{A}}, \hfill \\\frac{{{\text{d}}c_{{N - 1,1}}^{\rm{B}}}}{{{\text{d}}t}} = {{k}_{{ab}}}c_{{N - 1,0}}^{\rm{A}}c_{{0,1}}^{\rm{B}} - k_{{ab}}^{*}c_{{N - 1,1}}^{\rm{B}}. \end{array} $$

Each kind of trimer \( c_{{2,1}}^{\rm{A}} \) and \( c_{{1,2}}^{\rm{B}} \) must have its own differential equation in terms of k ha , k hb :

$$ \begin{array}{lll} \frac{{{\text{d}}c_{{2,1}}^{\rm{A}}}}{{{\text{d}}t}} = - {{k}_{{aa}}}c_{{1,0}}^{\rm{A}}c_{{2,1}}^{\rm{A}} - {{k}_{{ab}}}c_{{0,1}}^{\rm{B}}c_{{2,1}}^{\rm{A}} + {{k}_{{ha}}}{{c}_{{1,1}}}c_{{1,0}}^{\rm{A}} + k_{{aa}}^{*}c_{{3,1}}^{\rm{A}} + k_{{ab}}^{*}c_{{2,2}}^{\rm{B}} - k_{{ha}}^{*}c_{{2,1}}^{\rm{A}}, \hfill \\\frac{{{\text{d}}c_{{1,2}}^{\rm{B}}}}{{{\text{d}}t}} = - {{k}_{{bb}}}c_{{0,1}}^{\rm{B}}c_{{1,2}}^{\rm{B}} - {{k}_{{ba}}}c_{{1,0}}^{\rm{A}}c_{{1,2}}^{\rm{B}} + {{k}_{{hb}}}{{c}_{{1,1}}}c_{{0,1}}^{\rm{B}} + k_{{bb}}^{*}c_{{1,3}}^{\rm{B}} + k_{{ba}}^{*}c_{{2,2}}^{\rm{A}} - k_{{hb}}^{*}c_{{1,2}}^{\rm{B}}. \end{array} $$

For 2 ≤ n ≤ N − 3:

$$ \frac{{{\text{d}}c_{{2,n}}^{\rm{A}}}}{{{\text{d}}t}} = {{k}_{{aa}}}c_{{1,0}}^{\rm{A}}\left( {c_{{1,n}}^{\rm{A}} - c_{{2,n}}^{\rm{A}}} \right) - {{k}_{{ab}}}c_{{0,1}}^{\rm{B}}c_{{2,n}}^{\rm{A}} + {{k}_{{ba}}}c_{{1,n}}^{\rm{B}}c_{{1,0}}^{\rm{A}} + k_{{aa}}^{*}\left( {c_{{3,n}}^{\rm{A}} - c_{{2,n}}^{\rm{A}}} \right) + k_{{ab}}^{*}c_{{2,n + 1}}^{\rm{B}} - k_{{ba}}^{*}c_{{2,n}}^{\rm{A}}, $$
$$ \frac{{{\text{d}}c_{{n,2}}^{\rm{B}}}}{{{\text{d}}t}} = {{k}_{{bb}}}c_{{0,1}}^{\rm{B}}\left( {c_{{n,1}}^{\rm{B}} - c_{{n,2}}^{\rm{B}}} \right) - {{k}_{{ba}}}c_{{1,0}}^{\rm{A}}c_{{n,2}}^{\rm{B}} + {{k}_{{ab}}}c_{{n,1}}^{\rm{A}}c_{{0,1}}^{\rm{B}} + k_{{bb}}^{*}\left( {c_{{n,3}}^{\rm{B}} - c_{{n,2}}^{\rm{B}}} \right) + k_{{ba}}^{*}c_{{n + 1,2}}^{\rm{A}} - k_{{ab}}^{*}c_{{n,2}}^{\rm{B}}. $$

As before, the equations corresponding to the maximum length N homopolymers are

$$ \begin{array}{lll} \hfill \frac{{{\text{d}}c_{{2,N - 2}}^{\rm{A}}}}{{{\text{d}}t}} = {{k}_{{aa}}}c_{{1,0}}^{\rm{A}}c_{{1,N - 2}}^{\rm{A}} + {{k}_{{ba}}}c_{{1,N - 2}}^{\rm{B}}c_{{1,0}}^{\rm{A}} - k_{{aa}}^{*}c_{{2,N - 2}}^{\rm{A}} - k_{{ba}}^{*}c_{{2,N - 2}}^{\rm{A}}, \\\hfill \frac{{{\text{d}}c_{{N - 2,2}}^{\rm{B}}}}{{{\text{d}}t}} = {{k}_{{bb}}}c_{{0,1}}^{\rm{B}}c_{{N - 2,1}}^{\rm{B}} + {{k}_{{ab}}}c_{{N - 2,1}}^{\rm{A}}c_{{0,1}}^{\rm{B}} - k_{{bb}}^{*}c_{{N - 2,2}}^{\rm{B}} - k_{{ab}}^{*}c_{{N - 2,2}}^{\rm{B}}.\end{array} $$

For 3 ≤ r ≤ N − 2 and 1 ≤ s ≤ N − 1 − r:

$$ \begin{array}{lll} \frac{{{\text{d}}c_{{r,s}}^{\rm{A}}}}{{{\text{d}}t}} = {{k}_{{aa}}}c_{{1,0}}^{\rm{A}}\left( {c_{{r - 1,s}}^{\rm{A}} - c_{{r,s}}^{\rm{A}}} \right) - {{k}_{{ab}}}c_{{0,1}}^{\rm{B}}c_{{r,s}}^{\rm{A}} + {{k}_{{ba}}}c_{{r - 1,s}}^{\rm{B}}c_{{1,0}}^{\rm{A}}, \\\quad +\; k_{{aa}}^{*}\left( {c_{{r + 1,s}}^{\rm{A}} - c_{{r,s}}^{\rm{A}}} \right) + k_{{ab}}^{*}c_{{r,s + 1}}^{\rm{B}} - k_{{ba}}^{*}c_{{r,s}}^{\rm{A}}.\end{array} $$

For 3 ≤ s ≤ N − 2 and 1 ≤ r ≤ N − 1 − s:

$$ \begin{array}{lll} \frac{{{\text{d}}c_{{r,s}}^{\rm{B}}}}{{{\text{d}}t}} = {{k}_{{bb}}}c_{{0,1}}^{\rm{B}}\left( {c_{{r,s - 1}}^{\rm{B}} - c_{{r,s}}^{\rm{B}}} \right) - {{k}_{{ba}}}c_{{1,0}}^{\rm{A}}c_{{r,s}}^{\rm{B}} + {{k}_{{ab}}}c_{{r,s - 1}}^{\rm{A}}c_{{0,1}}^{\rm{B}} \\\quad +\; k_{{bb}}^{*}\left( {c_{{r,s + 1}}^{\rm{B}} - c_{{r,s}}^{\rm{B}}} \right) + k_{{ba}}^{*}c_{{r + 1,s}}^{\rm{A}} - k_{{ab}}^{*}c_{{r,s}}^{\rm{B}}.\end{array} $$

For 3 ≤ n ≤ N − 1:

$$ \begin{array}{lll} \hfill \frac{{{\text{d}}c_{{n,N - n}}^{\rm{A}}}}{{{\text{d}}t}} = {{k}_{{aa}}}c_{{1,0}}^{\rm{A}}c_{{n - 1,N - n}}^{\rm{A}} + {{k}_{{ba}}}c_{{n - 1,N - n}}^{\rm{B}}c_{{1,0}}^{\rm{A}} - k_{{aa}}^{*}c_{{n,N - n}}^{\rm{A}} - k_{{ba}}^{*}c_{{n,N - n,}}^{\rm{A}} \\\hfill \frac{{{\text{d}}c_{{N - n,n}}^{\rm{B}}}}{{{\text{d}}t}} = {{k}_{{bb}}}c_{{0,1}}^{\rm{B}}c_{{N - n,n - 1}}^{\rm{B}} + {{k}_{{ab}}}c_{{N - n,n - 1}}^{\rm{A}}c_{{0,1}}^{\rm{B}} - k_{{bb}}^{*}c_{{N - n,n}}^{\rm{B}} - k_{{ab}}^{*}c_{{N - n,n}}^{\rm{B}}.\end{array} $$

The complete reaction scheme must satisfy mass conservation in a closed system, implying that the mass variation rate must be strictly zero:

$$ \begin{array}{lll} 0 = 2{{{\dot{c}}}_{{1,1}}} + 3\left( {\dot{c}_{{2,1}}^{\rm{A}} + \dot{c}_{{1,2}}^{\rm{B}}} \right) + \sum\limits_{{n = 1}}^N {n\left( {\dot{c}_{{n,0}}^{\rm{A}} + \dot{c}_{{0,n}}^{\rm{B}}} \right) + } \sum\limits_{{n = 2}}^{{N - 1}} {(n + 1)} \left( {\dot{c}_{{1,n}}^{\rm{A}} + \dot{c}_{{n,1}}^{\rm{B}}} \right) \cr + \sum\limits_{{n = 2}}^{{N - 2}} {(n + 2)} \left( {\dot{c}_{{2,n}}^{\rm{A}} + \dot{c}_{{n,2}}^{\rm{B}}} \right) + \sum\limits_{{r = 3}}^{{N - 1}} {\sum\limits_{{s = 1}}^{{N - 1}} {(r + s)} \left( {\dot{c}_{{r,s}}^{\rm{A}} + \dot{c}_{{r,s}}^{\rm{B}}} \right)},\end{array} $$

where the overdot stands for the time-derivative.

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Blanco, C., Hochberg, D. (2012). Stochastic Mirror Symmetry Breaking: Theoretical Models and Simulation of Experiments. In: Cintas, P. (eds) Biochirality. Topics in Current Chemistry, vol 333. Springer, Berlin, Heidelberg. https://doi.org/10.1007/128_2012_362

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