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8.2. Funciones biológicas en el marco cuántico de Markov

Pasamos a la dinámica de sistemas abiertos con la ecuación GKSL. En nuestro modelo, 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}}} y 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}}} representan alguna función biológica especial 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}  (ver Khrennikov et al., 2018)[1]para detalles. Su funcionamiento resulta de la interacción de flujos de información internos y externos. En las Secciones 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} es alguna función psicológica; en el caso más 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} representa una pregunta hecha 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} (digamos que es un ser humano). En la Sección 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}   es la regulación génica del metabolismo de la glucosa/lactosa en la bacteria Escherichia coli. En las Secciones 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} representa el proceso de mutación epigenética. Función simbólicamente biológica 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} se representa como un observable cuántico: operador 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 descomposición espectral 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)} ,dónde 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} etiqueta las salidas 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 teoría de la dinámica de estado cuántica de Markov describe el proceso de generación de estos resultados.

En el modelo matemático (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]), los resultados de la función biológica 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} se generan acercándose a un estado estable de la dinámica 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)}

tal que coincide con la descomposición espectral 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}} ,es decir.,

  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)}

dónde

  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}

Esto significa 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}} es diagonal en una base ortonormal que consta de vectores propios 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}} . Este estado, o más precisamente, esta descomposición del operador de densidad 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}} ,es la mezcla estadística clásica de los estados básicos de información que determinan esta función biológica. Las probabilidades en la descomposición de estados (26) se interpretan estadísticamente.

Considere un gran conjunto de biosistemas con el estado 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}  interactuando con el medio 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} . (Recordemos que matemáticamente la interacción está codificada en el 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}}} ) Como resultado de esta interacción, la función biológica 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 salida 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 probabilidad 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} .Observamos que en los términos del operador la probabilidad se expresa como 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)}

Esta interpretación se puede aplicar incluso a un solo biosistema que se encuentra con el mismo entorno muchas veces.

Cabe señalar que el estado límite 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}} expresa la estabilidad con respecto a la influencia del entorno 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} .Por supuesto, en el mundo real nunca se alcanzaría el estado límite. La fórmula matemática (25) describe el proceso de estabilización, amortiguación de fluctuaciones. Pero, nunca desaparecerían por completo con el tiempo.

Observamos que un estado estacionario satisface la ecuación GKSL estacionaria:

  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)}

También es importante señalar que, en general, un estado estacionario de la ecuación maestra cuántica no es único, depende de la clase de condiciones iniciales.

  1. ↑ Khrennikov A., Basieva I., PothosE.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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