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8.2. Funzioni biologiche nel framework quantistico di Markov

Passiamo alla dinamica dei sistemi aperti con l'equazione GKSL. Nella nostra modellazione, Hamiltoniano  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}}} e Lindbladiano 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}}}  rappresentano una funzione biologica speciale 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} (vedi Khrennikov et al., 2018)[1] per i dettagli. Il suo funzionamento risulta dall'interazione di flussi informativi interni ed esterni. Nelle Sezioni 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} è una funzione psicologica; nel caso più semplice 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} rappresenta una domanda posta a 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}  (diciamo che 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} è un essere umano). Nella Sezione 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} è la regolazione genica del metabolismo del glucosio/lattosio nel batterio Escherichia coli. Nelle Sezioni 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} rappresenta il processo di mutazione epigenetica. Simbolicamente la funzione biologica 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} è rappresentata come osservabile quantistica: l'operatore Hermitiano  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}} con la decomposizione spettrale 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)} , dove 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} etichetta gli output di 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 teoria della dinamica di stato di Markov quantistica descrive il processo di generazione di questi output.

Nel modello matematico (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]), gli output della funzione biologica 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} sono generati avvicinandosi a uno stato stazionario della dinamica 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)}

tale che corrisponda alla decomposizione spettrale di 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}} , cioè,

  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)}

dove

  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)}

Ciò significa che 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}} è diagonale in una base ortonormale costituita da autovettori di 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}} . Questo stato, o più precisamente, questa scomposizione dell'operatore di densità 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}} , è la classica miscela statistica degli stati informativi di base che determinano questa funzione biologica. Le probabilità nella decomposizione dello stato (26) sono interpretate statisticamente.

Consideriamo un grande insieme di biosistemi con lo stato 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} che interagiscono con l'ambiente 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} . (ricordiamo che matematicamente l'interazione è codificata nel 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}}} ) Risultando da questa interazione, la funzione biologica 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} produce output 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} con probabilità 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} . Osserviamo che in termini operatore la probabilità è espressa come 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)}

Questa interpretazione può essere applicata anche a un singolo biosistema che incontra più volte lo stesso ambiente.

Va notato che lo stato 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}} esprime la stabilità rispetto all'influenza dell'ambiente concreto 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} . Naturalmente, nel mondo reale lo stato limite non verrebbe mai avvicinato. La formula matematica (25) descrive il processo di stabilizzazione, smorzamento delle fluttuazioni che, però, non sarebbero mai scomparsi completamente con il tempo.

Notiamo che uno stato stazionario soddisfa l'equazione GKSL stazionaria:

  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)}

È anche importante sottolineare che generalmente uno stato stazionario dell'equazione master quantistica non è unico, dipende dalla classe delle condizioni iniziali.

  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., Basieva I., 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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