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8.2. Fonctions biologiques dans le cadre de Markov quantique

Nous nous tournons vers la dynamique du système ouvert avec l'équation GKSL. Dans notre modélisation, Hamiltonian  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{\mathcal{H}}} et Lindbladian  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{{L}}} représentent une fonction biologique spéciale Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} (voir Khrennikov et al., 2018)[1] pour plus de détails. Son fonctionnement résulte de l'interaction des flux d'informations internes et externes. Dans les sections 10, 11.3, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} est une fonction psychologique; dans le cas le plus simple Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} représente une question posée à Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} (disons est un être humain). Dans la section 7, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} est la régulation génique du métabolisme du glucose/lactose dans la bactérie Escherichia coli. Dans les sections 9, 11.2, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} représente le processus de mutation épigénétique. Symboliquement, la fonction biologique Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} est représentée comme une observable quantique : opérateur Hermitien Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{F}} avec la décomposition spectrale Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{F}=\sum_xx\widehat{E}^F(x)} , où Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} marque les sorties de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} . La théorie de la dynamique des états quantiques de Markov décrit le processus de génération de ces sorties.

Dans le modèle mathématique (Asano et al., 2015b,[2] Asano et al., 2017b,[3] Asano et al., 2017a,[4] Asano et al., 2015a,[5] Asano et al., 2012b,[6] Asano et al., 2011,[7] Asano et al. ., 2012a[8]), les sorties de la fonction biologique Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} sont générées en approchant un état stable de la dynamique GKSL :

  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{t \to \infty}\widehat{\rho}(t)=\widehat{\rho}_{steady}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (25)}

telle qu'elle corresponde à la décomposition spectrale de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{F}} , i.e.,

  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{\rho}_{steady}=\sum_x p_x\widehat{E}^F(x)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (26)}

où

  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_x\geq\sum_xp_x=1} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (26)}

Cela signifie que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{\rho}_{steady}} est diagonal dans une base orthonormée constituée de vecteurs propres de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{F}} . Cet état, ou plus précisément cette décomposition de l'opérateur de densité Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{\rho}_{steady}} , est le mélange statistique classique des états d'information de base déterminant cette fonction biologique. Les probabilités dans la décomposition de l'état (26) sont interprétées statistiquement.

Considérons un grand ensemble de biosystèmes avec l'état Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{\rho}_0} interagissant avec l'environnement Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} . (Nous rappelons que mathématiquement l'interaction est codée dans le Lindbladian Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{{L}}} ) Résultant de cette interaction, la fonction biologique Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} produit la sortie Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} avec probabilité Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_x} . Nous remarquons que dans l'opérateur termes la probabilité est exprimée par Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_x=Tr\widehat{\rho}_{steady}\widehat{E}^F(x)}


Cette interprétation peut être appliquée même à un seul biosystème qui rencontre plusieurs fois le même environnement. Il convient de noter que l'état limite Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{\rho}_{steady}} exprime la stabilité par rapport à l'influence de l'environnement concret Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} . Bien sûr, dans le monde réel, l'état limite ne serait jamais approché. La formule mathématique (25) décrit le processus de stabilisation, d'amortissement des fluctuations. Mais, ils ne disparaîtraient jamais complètement avec le temps.

Nous notons qu'un état stationnaire satisfait l'équation GKSL stationnaire :

  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i[\widehat{H},\widehat{\rho}_{steady}]= \widehat{L}[\widehat{\rho}_{steady}]} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (27)}

Il est également important de souligner que généralement un état stationnaire de l'équation maîtresse quantique n'est pas unique, il dépend de la classe des conditions initiales.

  1. ↑ Khrennikov A., Basieva I., Pothos E.M., Yamato I. Quantum Probability in Decision Making from Quantum Information Representation of Neuronal States, Sci. Rep., 8 (2018), Article 16225
  2. ↑ Asano M., Khrennikov A., Ohya M., Tanaka Y., Yamato I. Quantum Adaptivity in Biology: From Genetics To Cognition Springer, Heidelberg-Berlin-New York(2015)
  3. ↑ Asano M., Basieva I., Khrennikov A., Yamato I. A model of differentiation in quantum bioinformatics. Prog. Biophys. Mol. Biol., 130 (Part A)(2017), pp. 88-98
  4. ↑ Asano M., Basieva I., Khrennikov A., Ohya M., Tanaka Y. A quantum-like model of selection behavior. J. Math. Psychol., 78 (2017), pp. 2-12
  5. ↑ Asano M., Basieva I., Khrennikov A., Ohya M., Tanaka Y., Yamato I. Quantum information biology: from information interpretation of quantum mechanics to applications in molecular biology and cognitive psychology. Found. Phys., 45 (10) (2015), pp. 1362-1378
  6. ↑ Asano M., Basieva I., Khrennikov A., Ohya M., Tanaka Y., Yamato I. Towards modeling of epigenetic evolution with the aid of theory of open quantum systems AIP Conf. Proc., 1508 (2012), p. 75
  7. ↑ Asano M., Ohya M., Tanaka Y., BasievaI., Khrennikov A. Quantum-like model of brain’s functioning: decision making from decoherence. J. Theor. Biol., 281 (1) (2011), pp. 56-64
  8. ↑ Asano M., Basieva I., Khrennikov A., Ohya M., Tanaka Y., I Yamato quantum-like model for the adaptive dynamics of the genetic regulation of e. coli’s metabolism of glucose/lactose Syst. Synth. Biol., 6 (2012), pp. 1-7
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