PLoS Computational Biology, 2005; 1(2): (más artículos en esta revista)

Los interruptores moleculares en el Synapse Emerge Receptor de Tráfico y Kinase

Biblioteca Pública de la Ciencia
Hayer Arnold [1], Upinder Bhalla S [1]
[1] Centro Nacional de Ciencias Biológicas, Bangalore, India
[2] École Supérieure de Biotechnologie de Strasbourg, Estrasburgo, Francia
Resumen

Los cambios en la fuerza sináptica conexión entre neuronas se cree que desempeñan un papel en la formación de la memoria. Un importante mecanismo para la modificación sináptica fuerza es a través de movimiento de los receptores de neurotransmisores y proteínas reguladoras y de la sinapsis. Varias actividades de control de eventos bioquímicos desencadenados estos movimientos. En este sentido, el uso de modelos de computadora para explorar la forma en que estos putativo de memoria relacionados con los cambios pueden ser estabilizado mucho después de que el disparador inicial, y más allá de la vida útil de las moléculas sináptica. Basamos nuestros modelos de datos y bioquímicos publicado los experimentos realizados en la actividad dependiente de un movimiento de los receptores de glutamato, AMPAR, y una cinasa dependiente de calcio, CaMKII. Nos parece que estas dos moléculas de participar en distintas bistable interruptores. Estos interruptores son simuladas efectiva durante largos períodos a pesar de la rotación molecular y bioquímica de las fluctuaciones derivadas del pequeño número de moléculas en la sinapsis. El AMPAR cambiar surge de un novedoso proceso de la libre contratación en la que la presencia de receptores suficiente sesgos ciclo de la circulación del receptor para insertar aún más receptores en la sinapsis. El CaMKII cambiar surge de autophosphorylation de la quinasa. Los conmutadores pueden funcionar en una forma firmemente y, o relativamente independiente. Este último caso da lugar a múltiples estados estables de la sinapsis. Proponemos que la libre contratación similares ciclos puede ser importante para mantener los niveles de muchas moléculas que se someten a circulación regulada, y que pueden conducir a la posible combinatoria estable como los sistemas de los estados de la sinapsis.

Introducción

Almacenamiento a largo plazo de la información neuronal se cree que se producen por alteraciones en la eficacia sináptica. Muchos mecanismos se han definido para los cambios en la fuerza sináptica, incluyendo la modulación de la liberación de neurotransmisores, los receptores de los cambios en la conductividad, los cambios en el número de receptores o activa las sinapsis, modificaciones estructurales y de la sinapsis. Entre estos, la inserción de receptores de glutamato de la alfa-amino-3-hidroxi-5-metil-4-isoxazol propionato (AMPA) subtipo en la membrana postsináptica y la modulación de la conductancia de la fosforilación de receptores son hechos clave en la modulación de la eficacia sináptica. Una cuestión fundamental todos los desafíos de estos mecanismos: ¿cómo pueden durar toda la vida?

Sináptica recuerdos pueden caries en por lo menos tres formas: el volumen de negocios, el intercambio difusivos, y stochasticity. Volumen de negocios de las principales moléculas postsináptica varía entre períodos de pocos minutos hasta varios días y puede ser mejorado aún más por la actividad sináptica [1]. Una solución a la pérdida de memoria debido a la rotación molecular es el concepto de auto-sostenible interruptores moleculares [2]. Estos suelen implicar algún tipo de información molecular, dando lugar a sistemas químicos, que pueden instalarse establemente en uno de estos dos estados. Esos dos Estados, o bistable, sistemas pueden almacenar la información de una manera binaria. Siempre hay un suministro constante de sustitución de las moléculas, el volumen de negocios molecular puede ser tolerado, ya que recién sintetizadas, ingenuo moléculas arrastrado a convertirse en el estado actual del sistema. Algunas propuestas actuales para tales bistable sináptica switches incluyen el calcio calmodulina quinasa tipo II (CaMKII) hipótesis [3], una mitogen-activated proteína quinasa (MAPK) bucle de [4, 5], y, recientemente, el objetivo de mamíferos rapamycin ( MTOR), la síntesis de proteínas bucle [6]. De ellos, la CaMKII modelo se ha planteado en la mayoría de los detalles con la más completa correlatos estructurales. Según este modelo, CaMKII en la sinapsis puede someterse autophosphorylation, lo que lleva a la activación de la quinasa. La quinasa activada moléculas de catalizar la fosforilación de CaMKII moléculas aún más, resultando en un ciclo de auto-sostenible. El bucle de MAPK modelo también implica una auto-sostenible ciclo, pero en este caso varias moléculas intermedias participar en el bucle. La síntesis de proteínas bucle modelo se basa en la observación de los mecanismos de traducción de proteínas asociadas a las sinapsis. ARN mensajero para varias proteínas, entre ellas la ribosomas sí mismos, también está presente. Así, la síntesis de proteínas de alta local crea los mecanismos para mantener altos niveles de síntesis. Esta proteína sintética bucle se rige por mTOR.

El segundo mecanismo de descomposición de la memoria sináptica es difusiva intercambio de proteínas sinápticas, lo que de lavado de determinados Estados de la sinapsis. Extrapolaciones de libre difusión constantes difusivos sugieren que el intercambio entre la columna vertebral y la dendrita sináptica es probable que sea rápido, de menos de 10 s incluso en el caso de las proteínas [7]. La densidad postsináptica (PSD) es un elaborado complejo citoesqueleto y la señalización que proporciona proteínas anclas para sináptica cerca de la región de la liberación presináptica de neurotransmisores. Este anclaje se soluciona el problema de la libre difusión de lavado de activos y moléculas, sino que introduce el problema de la regulación de la inserción de las moléculas en los lugares correctos. Hay pruebas considerables para dirigir el tráfico de moléculas a partir de la PSD y. Una de ellas es el ciclo de la trata de inserción y eliminación de los receptores de glutamato de la AMPA subtipo en la membrana sináptica [8]. Un ejemplo importante y fisiológica de los receptores de inserción es la conversión de sinapsis silenciosa, que carecen de los receptores AMPA (AMPARs), en las sinapsis activa con una dotación completa de los receptores (revisado en [8]]. La entrega de AMPARs a la membrana sináptica implica dos corrientes: una vía constitutiva de los receptores de glutamato heteromers 2 y 3 (GluR23), y una actividad que dependen de la participación vía GluR12 [9]. Sobre la base de las pruebas actuales, que dependen de la actividad de GluR12 inserción en la membrana sináptica es estimulada por fosforilación en Ser845 [10, 11]. También hay pruebas de tales fosforilación implicados en la plasticidad sináptica [12, 13]. CaMKII también translocates al PSD a calmodulina (CaM), de carácter vinculante y de estimulación [14]. Así pues, además de su conocida participación en la plasticidad sináptica, AMPARs y CaMKII tienen mecanismos de la actividad dependiente de la División del Sector Privado para la contratación de un modo que los actos contrarios a los procesos de lavado [15]. Esta combinación de atributos hace que estas moléculas interesantes candidatos para el análisis de los mecanismos moleculares de memoria. Sin embargo, a largo plazo, incluso de anclaje eventos son reversibles y otros procesos deben ser considerados para la estabilidad.

Un tercer obstáculo importante a la formación de la memoria es estable bioquímicos stochasticity. Esto provoca incertidumbre (ruido) en el resultado de las reacciones bioquímicas de un pequeño número de moléculas. Esas fluctuaciones son graves en la sinapsis, en los que muchas importantes moléculas de señalización están presentes en números bajos, es decir, menos de 100 moléculas. En un típico sináptica volumen de 0,1 fl [16] se estima que hay cinco libres iones Ca 2 +. Bajo condiciones estocástico, existe una probabilidad finita de estado espontánea gira en bistable interruptores moleculares [7, 17]. La vida útil de los estados estable depende tanto de las tasas de reacción y sobre el número de moléculas. Por ejemplo, la propuesta de cambiar MAPK no son muy adecuados para los volúmenes sináptica, y despliega espontáneamente en el estado escala de tiempo de minutos [7]. Sin embargo, estas estimaciones de tiempo son altamente dependientes de las hipótesis acerca de la difusión, anclaje, y los niveles de ruido en las vías de otras sináptica.

Poner estos temas en conjunto, un mecanismo plausible sináptica de memoria puede tener un aspecto como un interruptor molecular bistable que es resistente al volumen de negocios, incorpora el tráfico de moléculas y de la División del Sector Privado, y es poco probable que espontáneamente flip estado incluso cuando sináptica pequeña molécula números se tienen en cuenta . En este estudio un nuevo informe de glutamato (AMPA) del receptor de cambiar de base que surge de un examen de su tráfico y responde a estos criterios. También examinará una posible CaMKII cambiar en el contexto de estos criterios. Por último, integrar estos conmutadores para estudiar la forma en múltiples estados puedan surgir sináptica [18].

Resultados

Nuestro estudio se desarrolló en tres fases: la construcción y el modelo de exploración, entonces el examen de la reglamentación y la bistability, y finalmente el examen de las interacciones entre las dos formas de bistability.

Modelo de la Construcción

Las moléculas clave en la simulación se CaMKII, AMPAR, y sus principales reguladores: la proteína quinasa A (PKA), la proteína fosfatasa 2B (PP2B, también conocida como la calcineurina), y la proteína fosfatasa 1 (PP1). Como se discutió en los métodos, los parámetros de estas vías se derivan de los modelos publicados anteriormente disponible en la base de datos DOQCS [19] y se perfeccionaron utilizando parámetros conocidos sináptica. Descanso Ca 2 + concentraciones fueron 80 nM en todos los modelos, y en estos niveles es muy escasa o PP2B CaM activación. Hemos ampliado para incluir modelos de la translocación pasos utilizando simples medidas vinculantes con proteínas de anclaje sináptica. Estas medidas vinculantes se supone que incluyen difusivos movimiento de las moléculas entre el citosol y la División del Sector Privado. Como se muestra en la Figura 1, la activación de CaMKII o su Thr286-fosforilados estado a través de CaM vinculante causas que se mueva hacia el PSD. Asimismo, la fosforilación de AMPAR sobre Ser845 causas que pasar de las piscinas interiores a la membrana sináptica.

Nuestro uso de simulaciones seis modelos.

Modelo de exploración

En este punto hemos tenido un modelo (modelo 0) que se adapten a bioquímicos y biológicos de células de las observaciones de AMPAR y CaMKII tráfico. Este modelo es descriptivo en el sentido de que fue eficaz en la reproducción de los datos que se han utilizado a tal fin, en primer lugar. A continuación pregunta si el modelo era lo suficientemente completa para que coincida con efectos en el sistema más complejo que no se habían usado para fijar los parámetros para el modelo. Esto es a menudo una valiosa forma de evaluar si un modelo es probable que haya una representación razonable de un sistema complejo.

En primer lugar, hemos examinado AMPAR comportamiento en diferentes concentraciones de Ca 2 + constante (Figura 3 A). El descanso nivel de Ca 2 + fue de 80 nM. A partir de las simulaciones, encontramos que hubo una pequeña disminución del número de receptores AMPA en la sinapsis cuando Ca 2 + superado los 300 nM. Esto se debió a la activación de la calcineurina, que dephosphorylates receptor en el sitio de la PKA, y acelera su regreso a la citosol.

También se computa el mismo calcio dependiente de la curva de CaMKII (Figura 3 B). Aquí frente al citoplasma de la actividad de la quinasa levanta, seguido por translocación a la División del Sector Privado. El primer chapuzón PP2B debido a la actividad no estuvo presente en nuestro modelo de CaMKII. Esto se debe a que hemos tenido muy poco autophosphorylated CaMKII, en la División del Sector Privado en el estado basal, de modo que había poco sustrato presentes en PP2B activados para actuar sobre PP1.

AMPAR conductancia es una función tanto del número de receptores en la superficie de membrana, y de su estado de fosforilación (véase Materiales y Métodos]. En la Figura 3 C hemos calculado conductancia. A las bajas de Ca 2 +, es un seguimiento de cerca el número de la membrana de la AMPARs inserta. En cerca de 1 μ M Ca 2 + la conductancia de los receptores de la rosa, porque CaMKII fue activado y los receptores fosforilados en Ser831. El efecto neto de estos acontecimientos fue que compiten AMPAR conductancia por declinar por debajo de línea de base, y luego pasó por encima de la base de referencia. Esto es coherente con la teórica Bienenstock-Cooper-Munro (BCM) curva [22], los experimentos con estímulos eléctricos [23], y Ca 2 + plasticidad inducida [24]. Así, el modelo 0 es compatible con un número de observaciones experimentales para las que no se ha sintonizado. Sin embargo, el modelo 0 no tiene la capacidad para mantener estos cambios en la producción cuando los insumos regresaron a los niveles de reposo.

Luego utiliza el modelo 0 a analizar AMPAR respuestas a los cambios sostenidos en cuatro parámetros que pueden actuar como sitios de regulación de tráfico AMPAR. Los parámetros fueron las actividades de CaMKII, PKA, y PP1, y las tasas de reciclado y de los receptores a la superficie de la membrana. Cada uno de estos parámetros es sináptica implicados en el cambio, y es una posible señal de control aguas arriba de la conductancia AMPAR. Se sabe que los estímulos adecuados pueden conducir a cambios en la conductancia sináptica en un radio de aproximadamente 50% a 200% de los niveles basales de la transmisión sináptica [25]. Le preguntamos cuál de estos parámetros de control podría AMPAR conductancia más de esta gama.

En nuestro experimento simulado, que escala cada uno de los cuatro parámetros de 0,1 a diez veces su valor basal, una a la vez. En el caso de CaMKII, PKA, y PP1, esta ampliación fue realizada por buffering numéricamente el nivel de la forma activa de la enzima para el valor deseado. En el caso de las tasas de reciclado, exocitosis y endocitosis escala de tasas que se describen a continuación. En cada caso, las concentraciones de los restantes parámetros moleculares (CaMKII, PKA, y PP1) se permitió que se asienten en el estado de las nuevas concentraciones. En términos biológicos, esto correspondería a la aplicación de los inhibidores de los activadores o parámetro seleccionado. Registramos lo que el estado de equilibrio número de AMPARs sináptica (Figura 4]. Como en la figura 3, también AMPAR la conductancia calculada como un porcentaje de la máxima conductancia. El nivel máximo es la conductancia conductancia si todos los receptores en la membrana en el doblemente Ser831-fosforilados estado. En cada uno de estos cálculos se mantiene el número total de internos de los receptores de la membrana sináptica, más de 80 moléculas, y la concentración de Ca 2 + está en su nivel de reposo de 80 nM.

Las parcelas en la Figura 4 muestran los resultados para todo el rango de la escala activa de los insumos, de una proporción de 0,1 a diez veces la concentración basal de la entrada. CaMKII y PP1 (Figura 4 Ay 4 C) y no tenía poco efecto en la membrana sináptica de localización de los receptores. En lugar de ello actuó de manera complementaria en la modificación sináptica conductancia a través de la fosforilación y dephosphorylation de GluR1 sobre Ser831.

PKA (Figura 4 B) tuvo el mayor efecto total sobre la conductancia sináptica, con un rango de casi cero a un conductancia de casi el 70% de máxima. En baja actividad PKA había poca AMPARs de inserción en la membrana sináptica, por lo que la conductancia fue pequeño. En PKA de alta actividad, la mayoría de los receptores se insertaron. Además, la entrada de PKA activa indirectamente activado CaMKII, llevando a la fosforilación de receptores de Ser831, lo que resulta en un nuevo aumento de la conductancia. Esta activación se produce indirecta a través de dos pasos sucesivos inhibitoria. Primera PKA inhibe PP1 porque fosforilados inhibidor 1 del PP1 y se une a bloques PP1. En segundo lugar, la propia PP1 inhibe CaMKII por dephosphorylation de la quinasa (ver Figura 1 E).

Receptor de reciclaje ha sido sugerida como un mecanismo para alterar la conductancia sináptica [9, 10]. En las simulaciones que se escala la endocitosis AMPAR tasa especificada por el reciclado de, y al mismo tiempo la tasa de exocitosis por su inverso. Por lo tanto, un ratio de 0,1 endocitosis tendría una tasa de 0,1 veces basal, y una tasa de exocitosis de diez veces basal. Curiosamente, no es fácil atraer a un mayor número de receptores en la membrana sináptica (Figura 4 D). Esto fue en parte porque el modelo ya tenía la mayoría de sus receptores en la membrana. En altos valores de la tasa de reciclaje del receptor, la endocitosis tasa fue mayor que la tasa de exocitosis, por lo que la cantidad de sináptica de los receptores de membrana es muy agotado.

Por lo tanto, como una primera predicción, nuestras simulaciones señaló PKA o bien una combinación de CaMKII, PP1, y el reciclado de ser un suficiente control a largo plazo de dar cuenta de señal bidireccional AMPAR cambios, incluso en un régimen donde no se cuenta con receptor de cambio. Estos efectos fueron de control, no es de extrañar, ya que estas interacciones con AMPAR reciclaje fueron específicamente incluidos en nuestro modelo. Sin embargo, el modelo sí ilustran la cantidad de AMPAR inserción o supresión que se espera de las diferentes manipulaciones. Una combinación similar de regulación de los insumos ha sido implicado en el aprendizaje (por ejemplo, [26]]. Sin embargo, las simulaciones en este momento no abordó la cuestión de cómo esas largo plazo señales de control podría mantenerse.

AMPAR Bistability

Las exploraciones del modelo de las respuestas ha sugerido que el modelo era razonablemente coherente con una serie de hallazgos experimentales, incluidos varios que no se había ajustado para. Sin embargo, estas primeras pruebas utilizadas modelo 0, que no consideró el volumen de negocios molecular. ¿Cómo podría la inclusión de la síntesis y degradación del receptor de alterar el tráfico AMPAR?

En las simulaciones preliminares (no se muestra) hemos añadido o eliminado AMPAR moléculas de modelo 0 en el momento de iniciar el modelo, y preguntó cómo redistribuir los receptores cuando el modelo se agoten para el estado de equilibrio. El AMPAR moléculas se añadieron a la doblemente Ser845-fosforilados piscina interior de los receptores, pero el sitio de adición no afecta a la final el estado de la distribución. Inesperadamente, encontramos que la adición de los receptores de modelo a 0 en realidad, se redujo el número de receptores nativos. Nativo de los receptores se definen como unphosphorylated receptores en la endocytosed piscina. Esto sugiere que los receptores se redistribuye en el modelo de la sinapsis de una manera que podría dar lugar a dos estados estables. Se procedió a probar esta usando una serie de simulaciones sobre el modelo completo incluyendo AMPAR síntesis y degradación, es decir, el modelo 1. Detalles del modelo 1 en el Protocolo son parámetros S1.

En el modelo 1, se supone que recién sintetizadas AMPARs están presentes en la dendrita (denominado grueso AMPAR), y que el intercambio con el nativo de receptores en la columna vertebral (Figuras 1 A, 1 B, y 5 A). También se supone que existe una lenta degradación de la doblemente Ser845-receptores fosforilados piscina. Modelo 1 tiene 164 proteínas de anclaje situado en el PSD. Hemos realizado varias pruebas sobre el modelo 1 para examinar si en realidad exhiben dos estados estables con diferentes números de los receptores inserta en la membrana sináptica. Examinamos llegada de los receptores en la columna vertebral en diferentes condiciones. A continuación realiza el estado de los análisis, incluyendo un análisis de sensibilidad de parámetros, para mostrar bistability. Hemos simulado el estado de cambios en el modelo de respuesta a los estímulos y stochasticity. Estos resultados se describen a continuación y en la Figura 5, y acumulativamente caracterizar el bistable propiedades del modelo.

En primer lugar, pregunta si existen dos estados en los que había una afluencia de los receptores en la columna vertebral. Esta afluencia sería una condición necesaria para compensar la degradación, y la presencia de dos de esos estados sería una indicación de bistability. Hemos calculado el flujo entre los receptores de la dendrita y el nativo receptores en la columna vertebral, como una función del número total de AMPARs en la sinapsis (Figura 5 B). Hemos definido el número total de sináptica AMPARs como la suma de AMPARs sináptica en el interior de la piscina y en la membrana sináptica (Figura 5 A). Con el fin de calcular el flujo, se realizaron las siguientes manipulación: AMPAR moléculas se añadieron a la doblemente Ser845-fosforilados piscina interior de los receptores. Receptor intercambio con grueso AMPAR se inhabilitó para permitir que el sistema para resolver el estado de equilibrio para 5000 s sin pérdida de los receptores. Entonces los receptores de intercambio fue permitido volver a la simulación y se ha ejecutado por un nuevo 1000 s de resolver. El flujo entre la mayor parte de los receptores y los nativos AMPAR receptores se calculó en este momento crítico para la obtención de la Figura 5 B. Este cálculo preliminar confirmó nuestras observaciones. Hemos encontrado que hay dos distintos y muy distantes regiones de los receptores de afluencia, uno en que hay menos de 20 sináptica AMPARs total, y en el que hubo 180 o más.

Esto sugiere que, en efecto, dos regímenes estables, donde los receptores de la dendrita afluencia podría equilibrar la degradación del receptor. En el régimen de bajos números de sináptica AMPARs, era un simple equilibrio entre la mayor parte de receptores AMPA piscina y el receptor nativo piscina. En el régimen de alto el número, los receptores en la columna vertebral se redistribuían a la membrana de modo que el receptor nativo piscina fue agotado, de nuevo líder de los receptores de afluencia. La presencia de dos regímenes de los receptores de afluencia, según el número de AMPARs sináptica, puede ser una predicción comprobables experimentalmente.

Para confirmar que esta formado una bistable sistema, que computa los estados estables del sistema bajo diferentes condiciones de regulación (Figura 5 Cy 5 D). Se utilizó Ca 2 + y de la adenosina monofosfato cíclico (AMPc) como regulador insumos. Se obtuvieron los estados estable ejecutando modelo 1 al estado de equilibrio (120000 s), ya sea a partir de un bajo estado (bajo número de sináptica AMPARs) o un alto estado (un número elevado de sináptica AMPARs). En los casos en que sólo hay un estado estable, los convergentes de dos carreras con el mismo valor constante. En los casos en que existen distintos estados estables, la dos carreras reiterada a sus respectivos alto o bajo estados estables. También se encontró la inestabilidad de punto fijo (el umbral para el cambio de estado) por medio de una bisección iterativo método descrito en Materiales y Métodos.

Estas simulaciones nos dan las curvas de dosis-dependencia bistable que ilustran la naturaleza del sistema. El Ca 2 + dependencia de la dosis interruptor es interesante y poco habitual (Figura 5 C). En el bajo régimen de Ca 2 +, el sistema es bistable. El sistema se instala en tanto un estado alto o bajo dependiendo de las condiciones iniciales. En el 0,12 a 0,6 μ M gama, el sistema entra en un solo estado de baja actividad debido a la acción de la calcineurina (PP2B). Calcineurina se activa en estas concentraciones de Ca 2 +, y es capaz de dephosphorylate AMPAR rápidamente. El unphosphorylated receptor entra en la piscina del receptor nativo, y luego sale a la dendrita. En Ca 2 + concentraciones de más de 0,8 μ M, el sistema entra en un solo estado de alta actividad debido a la activación de CaM. La actividad de CaM conduce a un aumento de la fosforilación AMPAR tanto a través de PKA y CaMKII. CaM activado adenylyl-ciclasa (véase la figura 1] produce cAMP, el aumento de la actividad PKA. CaM también directamente activa CaMKII. Estos eventos son similares a las descritas para la Figura 3.

De esta manera, se aplica Ca 2 + que pueden acceder el estado del interruptor en cualquier dirección, en función de Ca 2 + amplitudes. El interruptor es inusual debido a que ambos estados se llegó por un aumento en las concentraciones de regulación de Ca 2 +. Como se analiza a continuación de AMPc, que es mucho más común de un estado a ser activado por baja regulador de las concentraciones, y la otra por el alto estado regulador de las concentraciones. Esta regulación bidireccional por el aumento de Ca 2 + se observa también en los resultados de la muestra en la Figura 3 A, pero no es tan sorprendente. Es posible que haya una muy estrecha bistable región en torno a 0,7 μ M Ca 2 + en el modelo, como lo sugiere la pequeña separación entre las curvas de baja y alta, pero esto no estaba dentro de la resolución numérica de nuestros cálculos. No esperamos que esa multa separación sería biológicamente relevantes. También estima que los umbrales de la bistable interruptor (abrir círculos en la Figura 5 C). Estos no son muy dependientes de las concentraciones de Ca 2 +. En términos biológicos, la sinapsis podría ser cambiado a la baja por el estado de recaudación de Ca 2 + al bajo régimen (0,12 a 0,6 μ M), que permite el flujo de los receptores, que se inicie, y en rápida disminución de Ca 2 + a la bistable régimen. Para alcanzar el alto estado, la entrada de Ca 2 + debe ser más de 0,8 μ M en el tiempo suficiente para el paso de resolver y, a continuación, Ca 2 + que rápidamente caen por debajo de 0,1 μ M en la región bistable. Al hablar después, los detalles biológicos de la dinámica de Ca 2 + están fuera del alcance de nuestros actuales modelos de estado.

El cAMP dependencia de la dosis fue cambiar más convencionales. Tomó la forma de una simple curva de histéresis, donde el bajo estado se debió a una disminución de AMPc y el alto estado de un aumento (Figura 5 D). La región de bistable el interruptor está en un rango intermedio de cAMP. En este caso, la sinapsis cambia a la baja cAMP estado si se reduce por debajo de 20 nM, y al alto estado cAMP si se plantea a más de 35 nM. Para completar el análisis, encontramos la inestabilidad de los puntos fijos de la bistable interruptor (abrir círculos en la Figura 5 D). Estos puntos pueden ser interpretados como umbrales para el cambio de un estado a otro. Como era de esperar para un sistema de bistable, estos puntos fijos inestables curva hacia atrás en dirección de los límites del ciclo de histéresis en una curva en forma de S-(Figura 5 D).

¿Cómo "robusto" bistability es el modelo de 1? Una medida de esto es preguntar si los efectos persisten bistable importante cuando los parámetros del modelo son variadas. Estamos sistemáticamente variados importantes parámetros del modelo y buscó bistability. Para probar para bistability, empezamos el modelo fuera, ya sea en el estado superior o inferior, y luego corrió hacia fuera para el estado de equilibrio con los parámetros alterados. Si el modelo de conmutación de estado ya no era bistable. Se encontró que el modelo 1 mantenerse bistable su comportamiento en una amplia gama de la mayoría de los parámetros, ilustra en la Figura 5 E. La clave "sensibles" parámetros son exactamente los indicados en la figura 4 como clave de los reguladores de estado sináptica conductancia: CaMKII, PKA, el reciclado, y Ca 2 +. La mayoría de los parámetros fueron capaces de un factor de escala, al menos, dos arriba o hacia abajo sin perder bistability.

Como simple firma de bistability contábamos con un tiempo de simulación de curso en el que el modelo explícitamente conmutación constante entre dos estados (Figura 5 F). En este sentido, el modelo de primera reiterada a la baja del estado donde hay pocos AMPARs en la membrana sináptica. Tras un pulso de cAMP (0,2 μ M, 1000 s), el modelo pasa a la gran estado, con muchos AMPARs inserta en la membrana sináptica. El sistema se enciende de nuevo a la baja por estado numéricamente cAMP reducir a cero (6000 s). Si bien este número de la conmutación de la membrana de la insertará AMPARs es posiblemente una predicción comprobables, que sólo indica la presencia de bistability y no arrojar mucha luz sobre el mecanismo, que se analiza a continuación.

El tiempo de conmutación es lento, del orden de una hora. Esto es compatible con el receptor de las tasas de tráfico en el modelo, que se deriva de el estado de mediciones. Más rápido puede ser transitoria las tasas aplicables durante la estimulación sináptica, como consideramos en la discusión.

Otra manifestación de la robustez es la capacidad de mantener el modelo de estado, a pesar de stochasticity. Stochasticity en la sinapsis se origina a partir de la aparición de reacciones probabilística entre un pequeño número de moléculas y da lugar a la aparente bioquímicos ruido. Este noisiness impone severas limitaciones a la fiabilidad de cualquier propuesta de mecanismo de señalización sináptica [7]. En particular, bistable sistemas bioquímicos están sujetos a espontáneo estado de gira porque bioquímicos ruido [7, 17]. Pusimos a prueba las respuestas de la simulación estocástica el modelo 1 utilizando el método estocástico Gillespie exacta [27]. Todo el modelo fue simulado stochastically, incluidas todas las moléculas en la dendrita, jefe de la columna vertebral, y la División del Sector Privado (detalles en los materiales y métodos). Empezamos el modelo de estado en la baja, donde unos AMPARs se inserta en la membrana sináptica (Figura 5 G). En la mitad de las carreras hemos aplicado una cAMP estímulo para cambiar el modelo de Estado a la alta en alrededor de 1 h (negro línea en la Figura 5 G). A continuación, el modelo de simulación de un período de 120000 s (> 33 h) para comprobar su resistencia a la espontánea o bien cambiar de estado. En el ejemplo en la Figura 5 G, el bajo estado (línea gris) no cambió, pero el alto estado (negro línea) espontáneamente apagado, en torno al 16 h.

Sobre la base de nuestro análisis se muestra en la Figura 5 B, de la estabilidad de cada estado debe ser una función de la concentración de la mayor parte AMPAR. Si la concentración de la mayor parte AMPAR es alto, entonces aumenta la afluencia de receptores. En estas condiciones, una parte relativamente pequeña fluctuación debe empujar el pasado el sistema de régimen de flujo de salida en la parte superior del régimen de flujo (Figura 5 B). Por el contrario, a un bajo grueso AMPAR que debe ser fácil de la cubierta de alta a la baja estado. Hemos repetido nuestra simulaciones estocásticas para una serie de concentraciones a granel AMPAR mientras que otra retención de los parámetros del modelo 1. En cada concentración de la mayor parte AMPAR hemos repetido las simulaciones por lo menos 24 veces para crear un perfil de las horas de conmutación (Figura 5 H). En nuestra gama de referencia de 11,11 nM de grueso AMPAR, frente a la situación se mantuvo estable durante más de 360 h de media, y la situación se mantuvo estable en alrededor de 42 h. Como discutimos más adelante, otros procesos sináptica puedan hacerse cargo de la tarea de mantener la información de estado, dentro de este marco temporal.

En la mayor parte de baja AMPAR el alto estado era muy corta vida, pero la baja no ha estado en todos los flip durante toda la duración de nuestras simulaciones (indicado por los grandes símbolos en la Figura 5 H). Por el contrario, en la mayor parte AMPAR alto, el bajo estado era inestable, pero el alto estado duró muy largo tiempo.

A continuación, examinó la manera en que el modelo de vida de los estados podría escala con el número de proteínas de anclaje en la División del Sector Privado. Mientras que el número de proteínas de anclaje establece el número máximo de receptores que se puede insertar en la membrana, este parámetro es importante para la solidez y la estabilidad del modelo. Hemos simulado el tiempo la estabilidad de la transición para una serie de números de la proteína de anclaje (Figura 5]. El número predeterminado de las proteínas de anclaje en el modelo es 164. En los números más bajos de proteína de anclaje cambiar la vida fue más bien corta, del orden de unas pocas horas. A mayores ancla cambiar los números de la vida aumentó rápidamente para ambos estados. La vida útil de la alta estado siguió aumentando y superó el 2 mo cuando había más de 320 proteínas de anclaje (grandes símbolos en la Figura 5 I indican que el modelo de estado no cambió durante toda la duración de nuestras simulaciones). El tiempo de baja estado fue mayor (aproximadamente 2 meses) a 240 de anclaje proteínas, y luego se redujo a alrededor de 200 h cuando más de anclaje proteínas estaban presentes. Esto puede deberse a que la presencia de proteínas de anclaje adicionales parciales de la circulación hacia el receptor de la columna vertebral. Biológicamente, un aumento de la proteína de anclaje número puede correlacionar con el tamaño de la cabeza de la columna vertebral. Por lo tanto, nuestro modelo predice que espinas más grandes deberían ser más estable, una observación que tiene algunas apoyo experimental [28, 29].

En esta fase del estudio se han analizado ampliamente las propiedades de la bistability del modelo 1 con respecto al número de AMPARs en la sinapsis. Nos ha considerado la posibilidad de la dependencia de estado y de los reguladores de flujo. Nos ha demostrado que la bistability fue robusto y, en particular, ha demostrado cuánto tiempo puede mantener el modelo de estado stochasticity información cuando se tuvo en cuenta. Sobre la base de estos cálculos, sugerimos que el modelo podría ser un candidato para retener información sináptica estado durante horas a meses.

¿Cómo funciona el bistability surgir? Modelo 1 es muy complejo e involucra a 16 estados de fosforilación AMPAR cada uno en la piscina y en el interior de la membrana sináptica. Para facilitar el trabajo, hicimos una esquelética modelo con la misma topología general. The skeletal model retained only two phosphorylation/dephosphorylation steps involving AMPAR ( Figure 6 A). This is called model 2. We used the same parameters as for the full model, with the exception of lower concentrations of PP1 (0.333 μM). The concentration of PP1 was reduced as its other substrate, CaMKII, was not present in model 2. The K m for PKA and PP1 was halved as compared to model 1, as each receptor phosphorylation site in model 2 corresponds to two receptor phosphorylation sites in model 1. For model simplicity we had the degradation steps feeding into the bulk AMPAR pool, which was numerically buffered to a steady value. The full parameters for this model are presented in Protocol S1 . We used this simpler model to analyse the mechanistic basis of this form of bistability.

We repeated the analysis of receptor flux as a function of the number of total synaptic receptors. Model 2 also had two regions of AMPAR influx separated by a region of efflux from the synapse ( Figure 6 B). This indicated that it shared the same mechanism for bistability as model 1. We then performed a simulation of the time course of stimulus-triggered transitions between the stable states of the model. We took advantage of the smaller number of molecular species in model 2 to monitor all the molecular concentrations during PKA-triggered transitions between the low and high states. In this simulation the steady-state amount of PKA was one molecule. The switch was turned on using a stimulus of 40 active PKA molecules for 2400 s, and turned off using zero active PKA molecules for 5000 s. The responses of several molecules are illustrated in Figure 6 C– 6 F. The low stable state (0 to 3 h) is characterised by low numbers of all forms of the receptor, and consequently little saturation of PP1. During the switch to the high state at 3 h, the concentration of unphosphorylated internal receptor drops, leading to a large influx of receptors. These are rapidly phosphorylated by the high numbers of PKA, and the number of IR** (see Figure 6 legend for explanation of abbreviations) rises sharply.

As the IR pool exchanges rapidly with bulk AMPAR, its concentration rapidly returned to basal levels after the PKA stimulus ended. Once in the IR** state, the receptor translocated to the synaptic membrane, into the MR** pool. Due to the large numbers of MR**, the PP1 became saturated ( Figure 6 E and 6 F). Thus, the combination of PKA phosphorylation of MR, and translocation from the IR** pool, formed MR** at a greater rate than the PP1 could dephosphorylate it.

At 7 h we switched the system back to the low state by removing all PKA. This allowed PP1 to complete dephosphorylation of the phosphorylated receptor pools. There was a large transient rise in MR due to the dephosphorylation of MR** and MR*, and the slow traffic back to the IR pool. Finally, when we restored active PKA to its resting level the system settled back into the lower stable state.

The translocation step appears to be important—we were unable to obtain bistability without it—but it was not possible to completely explore parameter space so it was not clear whether this is an absolute requirement. We were also able to obtain bistability with a single phosphorylation step, provided that the translocation step was second order in the receptor (data not shown). In all these processes, it was assumed that the bulk AMPAR was constant, that is, that the balance of synthesis and degradation was sufficient to rapidly add and remove receptors from the spine. In the biological system the situation is more complex and synthesis itself may be activity dependent [ 3032 ].

In summary, the simple model retains the fundamental features of the AMPAR translocation-based bistability and facilitates an analysis of its mechanism. The low state of this form of bistability occurs when few synaptic AMPAR molecules are present, so that PKA can act on only a few substrate molecules, and PP1 is not saturated. Therefore, few internal AMPARs are in the phosphorylated state and only a few AMPARs are translocated to the surface. The upper state of activity is characterised by high numbers of AMPARs in the phosphorylated states both internally and in the synaptic membrane. This state persists because of the higher basal activity of PKA as compared to PP1. This leads to PP1 saturation. The final step in maintaining the upper state is the translocation of phosphorylated receptor to the membrane. This removes receptors from the internal pools and keeps the native receptor (internal receptor) at sufficiently low levels that receptor influx is favoured. A similar PP1 saturation effect is seen in the complete model (model 1) when it is in the high state (data not shown). This is a possible testable prediction of the model.

CaMKII Bistability

Having characterised AMPAR translocation bistability, we wished to examine the well-studied CaMKII autophosphorylation system in the context of translocation. Although model 1 included CaMKII translocation, the CaMKII portion of the model was not bistable. A key assumption for CaMKII bistability is that upon binding to NMDAR, the CaMKII becomes susceptible to autophosphorylation even in the absence of bound CaM [ 33 ]. In our model, as in others (eg, [ 34 ]), this assumption is also linked to PP1 saturation as we now describe. In order to analyse the CaMKII responses in a simpler context, we derived a reduced model from the basic model discussed above. This reduced model contained only CaMKII in the cytosol and PSD, and its immediate regulators, PP1, CaM, and PP2B. PKA was included only as the active enzyme, without regulatory steps. We modified this reduced model of CaMKII by including the autophosphorylation in the absence of CaM. These changes are illustrated by the dashed line in Figure 1 C and the bold lines in Figure 7 A. This model is called model 3.

Despite the simplicity of our CaMKII model as compared to previous work [ 34 , 35 ], we were able to obtain bistability. To do so we used somewhat different phosphorylation and dephosphorylation rates in the PSD as compared to the model 1 (see Protocol S1 ). Since CaMKII phosphorylates itself, we had to adapt our previous analysis, which relies on separate input and output molecules [ 4 ]. We did this by numerically bifurcating the autonomously active CaMKII–PSD (Aut-CaMKII; Figure 7 A) into two simulated molecular pools: Aut-CaMKII enzyme and Aut-CaMKII readout. We set the number of Aut-CaMKII enzyme pools to specified values, and monitored the number of molecules in the Aut-CaMKII readout pool ( Figure 7 B). This manipulation was facilitated because the level of Aut-CaMKII is computed as the sum of the autonomously active states of Thr286-phosphorylated and Thr286/Thr305-phosphorylated CaMKII, indicated in gray in Figure 7 A. So our enzyme assignment bypassed this summation, and directly set the number of Aut-CaMKII molecules. Our readout number was the sum of Thr286-phosphorylated and Thr286/Thr305-phosphorylated CaMKII.

The results of these calculations for a range of Aut-CaMKII enzyme values are shown in Figure 7 B. The intersection points of this curve with the 45° line are stable points of the system, because at these points the enzyme and readout activities of Aut-CaMKII are identical. In other words, at these points the autonomous CaMKII would exactly sustain its own activity. The upper and lower intersection points define the stable numbers of Aut-CaMKII, and the intermediate point is a transition point. This behaviour can be seen by considering a small increase in the number of Aut-CaMKII enzyme above one of the stable points. The resulting Aut-CaMKII readout (read off from the y -axis) would be smaller than the new input number. This would tend to restore the CaMKII activity toward the stable point. A similar argument applies to small negative deflections from the stable points. Around the transition point the situation is reversed: any small deflection will be amplified until the system converges to either the upper or lower stable point.

We confirmed the presence of bistability by simulating a time series in which the system was turned on with a calcium pulse of 2.7 μM for 500 s and later turned off by raising the k cat of the PSD-localised PP1 by 5-fold for 500 s ( Figure 7 C). Two distinct stable states were observed, which corresponded to the upper and lower intersection points in Figure 7 B. There is a small offset between the two calculations because Figure 7 C reports all forms of CaMKII in the PSD, whereas Figure 7 B refers only to Aut-CaMKII.

We evaluated the robustness of the CaMKII model (model 3) using the same approach as for the AMPAR model (model 1). We found that the CaMKII bistability was highly robust with respect to variation of parameters ( Figure 7 D). Many parameters could be varied from 0.1 to ten times the reference value without losing bistability. Most of the remaining parameters could be varied from 0.5 to two times the reference value, and only PP1 and PKA were more sensitive. This sensitivity reflects the key role of PP1 in dephosphorylating CaMKII, and the role of PKA in controlling the activity of PP1.

We checked the robustness of model 3 in synaptic volumes by simulating it using stochastic numerical methods. Model 3 was resistant to spontaneous switching and we did not observe any switches to either state in over 300 cumulative hours of simulation time. A 33-h sample of the high and low states is shown in Figure 7 E. This state stability turned out to be an artefact of our reduced model for CaMKII, which only used the final active concentration of PKA as one of the key regulatory inputs. The PKA pathway model output was quite noisy in small volumes [ 36 ]. When we incorporated the full PKA pathway into model 3, we found it introduced a considerable amount of additional stochasticity into the system and did lead to bidirectional state flips ( Figure 7 F). We repeated these simulations 50 times and found that the off state endured for 17.3 ± 2.5 h, and the on state for 37.2 ± 6.9 h ( Figure 7 G). Thus, there may be a marked reduction in bistable state lifetimes when noisy inputs are taken into account. This issue is considered in the Discussion.

Overall, our CaMKII translocation model (model 3) also exhibited bistability coupled with translocation, such that the active state led to accumulation of CaMKII in the PSD. Under stochastic conditions, the lifetime of stable states in the model was sensitive to noise from the PKA input pathway.

Bistability Interactions: Tight Coupling

At this point we had a reasonably constrained model of AMPAR and CaMKII trafficking, and had shown that under certain conditions both molecules could be bistable. In the final part of the study we considered interactions between these two forms of bistability. We first made model 4 by merging model 1 and model 3, while sharing the same PP1 molecule in the PSD ( Figure 8 A). This scenario assumes that the PP1 is free to move between its CaMKII and AMPAR substrates while remaining in the PSD. Thus, there is a tight coupling between the two forms of bistability, mediated both by PP1 and by CaMKII phosphorylation of Ser831 of AMPAR, as in model 1. We asked how the system would respond to stimuli designed to activate the AMPAR and CaMKII switches independently.

In our first test we stimulated the model 4 with Ca 2+ (2.7 μM, 500 s) then allowed the system to settle for 3 h, then stimulated it with cAMP (108 nM, 2000 s) ( Figure 8 B). Following the Ca 2+ stimulus, the CaMKII switch turned on transiently, but soon returned to baseline. The AMPAR switch did not turn on until the cAMP stimulus was applied, and at this time CaMKII also turned on. When the cAMP stimulus was applied first, it rapidly turned on both the AMPAR and CaMKII switches ( Figure 8 C). Together, these simulations show that in model 4 the two forms of bistability function in lockstep. That is, sustained activation of CaMKII is contingent upon the activity of AMPAR. If AMPAR is activated, it saturates PP1, and this leads to activation of the CaMKII bistable switch. As shown in Figure 8 C, CaMKII is also activitated by the cAMP stimulus independently of the PP1-mediated cross-activation, leading to rapid turn on. This is because cAMP activates PKA, which relieves the PP1 inhibition of CaMKII. We ran separate simulations (not shown) that showed that even in the absence of this cAMP activation of CaMKII, the activation of AMPAR caused CaMKII to turn on as well.

We ran model 4 using stochastic methods to test its propensity to spontaneously change state. The off state was very stable though it did exhibit occasional transient spikes of activity ( Figure 8 D). The model spontaneously turned on only once in a cumulative total of over 1000 h of simulation time. As before, we tested high-state durations by applying an initial cAMP stimulus at about 1 h to turn the system on, and then ran the simulation for about 33 h. We repeated these runs 24 times to obtain the distribution. The high state was not as stable, and was subject to occasional spontaneous flips to the off state with an average time of 101 ± 79 h (mean ± standard error of the mean). As expected from the lockstep mechanism, both CaMKII and AMPAR underwent a state flip at nearly the same time ( Figure 8 E).

Bistability Interactions: Weak Coupling

In our final model (model 5), we considered the situation where CaMKII and AMPAR interacted only weakly. This is in contrast to model 4, where CaMKII and AMPAR were tightly coupled through a shared pool of PP1. In model 5, we combined the CaMKII and AMPAR bistable models (models 1 and 3) while keeping an independent pool of PP1 for each ( Figure 9 A). Such a scenario might arise if PP1 were bound to distinct scaffold proteins for each of its targets, and were restricted in its mobility across targets. The CaMKII-coupled pool of PP1 was treated as independent of PKA activity. There was one indirect form of coupling still present from CaMKII to the AMPAR bistability, since CaMKII phosphorylates AMPAR on Ser831. While this does not alter traffic rates directly, the Ser831 is a substrate for the AMPAR-associated pool of PP1 in model 5. Thus, CaMKII activity did contribute to the saturation of the AMPAR-associated PP1.

As before, we examined the interactions between the CaMKII and AMPAR switches using stimuli designed to turn each switch on independently of the other. When we first turned on the CaMKII switch alone using a Ca 2+ stimulus, we observed a slow activation of the AMPAR switch ( Figure 9 B). The Ca 2+ stimulus indirectly increased AMPAR insertion through the following steps: Ca 2+ → CaM → CaMKII → phosphorylation of Ser831 → saturation of AMPAR-specific PP1 → reduced endocytosis of AMPAR. We did not model any changes in recycling or internalisation rates due to receptor phosphorylation on Ser831.

A particularly interesting effect was seen when the AMPAR switch was activated first using cAMP ( Figure 9 C). This stimulus turned on the AMPAR switch without affecting the state of CaMKII. At about 4 h we applied a Ca 2+ stimulus that turned on the CaMKII switch as well. Thus, in model 5, the two switches were able to coexist in three combinations of states: both off, only AMPAR on, or both on. The fourth possible combination, of CaMKII on and AMPAR off, was not stable because of weak coupling between CaMKII and AMPAR, which slowly turned the latter on as shown in Figure 9 B. The weak coupling is due to the phosphorylation of GluR1 on Ser831 by CaMKII. While this does not directly affect translocation, it does engage the AMPAR-specific pool of PP1, leading to eventual phosphatase saturation and turn on of the AMPAR switch.

Overall, this model exhibited nested bistability. The fundamental switch took place when AMPAR turned on or off. Nested within this was the capacity for CaMKII to turn on or off. A possible physiological outcome of the nesting of CaMKII activation is that the phosphorylation state and hence the conductance of the synapse can settle to three levels: (1) no AMPAR, (2) AMPAR with low levels of Ser831 phosphorylation, and (3) AMPAR with high Ser831 phosphorylation and consequently a higher conductance ( Figure 9 C). As shown in Figures 3 and 4 , the equivalent conductance is expressed in terms of the number of unphosphorylated AMPAR channels that would have the same conductance.

The three states of the model were quite stable under stochastic conditions ( Figure 9 D). The only state that showed any transitions over the entire cumulative duration of simulations tested was that in which AMPAR was on and CaMKII was off (time to switch was 24.8 ± 3.5 h). We were particularly interested in the long-term stability of the states with both switches off, and both switches on. To examine these we performed several hundred independent stochastic simulation runs on a cluster, each representing 120000 s (33 h) of simulated time. No state transitions were observed in either direction, over a cumulative duration of over a year of simulated time for each state.

Transient Responses of Models

How does the model respond to stimuli that induce changes in synaptic efficacy? We tested two synaptic plasticity protocols on each of the models 1, 3, 4, and 5 ( Figures S1 and S2 ). These tests were only qualitative, as the models in the current study were parameterized using steady-state rather than transient experiments. Nevertheless, they are useful in showing model behaviour under transient conditions. The first protocol had been used to elicit long-term potentiation (LTP) of synaptic efficacy and consisted of three bursts of 100 impulses at 100 Hz, each separated by 600 s. The second protocol was used to induce long-term depression (LTD) at the synapse, and consisted of 900 impulses at 1 Hz. We represented each stimulus as a computed calcium transient with an exponential build-up and decay of Ca 2+ using the formulation of Zhabotinsky [ 34 ] ( Figure S1 ). The LTP stimulus gave calcium peaks of 12 μM, and the LTD stimulus had peaks of 0.5 μM. We found that the LTP stimulus was able to cause a switch to the on state only in the CaMKII model (model 3) that incorporated the PKA activation pathway ( Figure S2 ). This is interesting, as it suggests that even in its current form the CaMKII model is reasonably sensitive to physiological stimuli that may play a role in synaptic plasticity. The LTD stimulus did not turn off any of the models, indicating that the current models are missing some key interactions. These tests highlight some of the unknowns in our models, in particular, the specific transient interactions that are needed to trigger the steady-state effects we have analysed. As discussed below, more experimental detail will be needed to extend the models to include transient response characteristics.

Summary of Bistable Behaviour

In summary, we analysed four bistable synaptic trafficking models (models 1, 3, 4, and 5). The remaining two models in this study (models 0 and 2) were used to characterise model 1. Model 3 included CaMKII alone, but models 1, 4, and 5 included both AMPAR and CaMKII. As described above, we performed a number of steady-state and transient tests to characterise the stable states of each model. Properties of these models are summarised in Table 1 .

Discusión

In this study we have developed a model of the movement of a glutamate receptor (AMPAR) and a calcium-activated kinase (CaMKII) to and from the synaptic membrane, using steady-state trafficking rates as a major experimental constraint. We find that the AMPAR trafficking cycle may lead to a form of switching or bistability where the presence of sufficient receptors at the synapse leads to recruitment of more receptors. This process may be a candidate for the transition from silent to active synapses, and early phases of their subsequent maintenance. When combined with a previously proposed mechanism for a CaMKII molecular switch, we observe interesting interactions between these two forms of synaptic bistability. Depending on the degree of coupling between these switches, we predict that AMPAR recruitment and CaMKII activation may either occur in a tightly coordinated manner, or nearly independent of each other. The latter may give rise to multiple stable synaptic states. Stochastic calculations suggest that these stable states persist for many hours, and in some cases over a year, despite biochemical fluctuations due to small numbers of molecules in the synapse.

Structural Bistability

Long-term storage of information at the synapse is intimately connected to structural changes [ 8 , 15 ]. Such changes arise from the insertion, removal, and reorganisation of synaptic molecules. For example, many types of glutamatergic synapses initially lack postsynaptic AMPARs and are unresponsive to moderate synaptic input. These “silent synapses” become active when AMPARs are inserted. The change from silent to active synapses is a major mechanism for increases in synaptic efficacy [ 8 ]. Similarly, formation of dendritic spines is facilitated by the presence of the GluR2 subunit of the AMPAR [ 29 ]. Several signalling events are known that may affect these structural processes, but it is not always clear how persistent changes may be maintained. A single brief pulse of insertion of molecules, or formation of a synaptic spine, will have a transient effect unless some self-sustaining mechanism is also available to keep the changes in place. Formally, bistable systems are a possible mechanism for such structural memory, as such systems can withstand molecular turnover [ 2 ]. The AMPAR model analysed in the current paper shows that synaptic membrane insertion (a form of structural change) can be self-sustaining even when molecular turnover and noise are considered.

There are several proposed forms of synaptic bistability, and each has some structural correlates. The classical form of synaptic bistability is the CaMKII autophosphorylation system (reviewed in [ 3 ]). This is a biochemical bistability that relies on autophosphorylation leading to self-activation of CaMKII. This self-activation leads to bistability if PP1 is present at sufficiently low levels that it can be saturated. The involvement of PP1 saturation is also an important aspect of our model for AMPAR bistability. CaMKII activation has several structural correlates, including translocation to the PSD and formation of complexes with NMDA and other PSD proteins [ 33 ]. There is also recent evidence that CaMKII activation leads to an increase in AMPAR numbers [ 37 ]. Another synaptic biochemical bistability involving a MAPK feedback loop has been proposed [ 4 ] and tested in a fibroblast model system [ 38 ]. MAPK is known to be important in synaptic plasticity and may also play a role in structural changes at the synapse [ 39 , 40 ]. However, the MAPK bistable feedback mechanism is vulnerable to biochemical noise and may not be plausible in synaptic volumes [ 7 ]. A more recent proposal for a self-sustaining synaptic plasticity mechanism involves the mTOR system and local protein synthesis. mTOR phosphorylation increases protein synthesis at the dendrites, and the synthesis machinery itself is one of the products. Several other proteins have also been identified that may participate in such a feedback loop [ 6 ]. Such a protein-synthesis-dependent bistability would be very interesting for structural change at the synapse, as it could account for increased availability of many synaptic and PSD proteins.

In the current study we propose a novel form of synaptic bistability involving self-recruitment of AMPARs to the synapse. The mechanism is particularly interesting for the synapse in three ways: (1) it has parallels with the conversion of silent to active synapses, (2) it works at basal levels of activity of synaptic enzymes, and (3) it intimately involves a translocation and synaptic membrane insertion process. As analysed in Figures 5 and 6 , this form of bistability is a function of the number of molecules of receptor in the synapse, rather than biochemical activation. The bistability involves phosphatase saturation of PP1 due to its action on AMPAR in the PSD. This is a specific prediction of this study. However, the prediction of AMPAR-specific PP1 activity is yet to be tested in detail. We discuss the issue of PP1 access to other substrates below.

It should be stressed that this mechanism for synaptic state maintenance is by no means exclusive. We discuss several other possible mechanisms above. There are also important details about AMPAR cycling that invite further analysis. For example, the AMPARs in Ser831/Ser845 double phosphomutant mice would not sustain this form of bistability. Nevertheless, these animals form synapses and retain some degree of synaptic plasticity [ 12 ]. Furthermore, our model considers activity-dependent changes in GluR12, and does not account for a presumed hand-over of synaptic state to some long-term process involving GluR23 insertion. Mechanisms for maintenance of GluR23 levels are still poorly understood, but we speculate that this too involves a self-recruiting bistable process.

Phosphatase saturation is a key part of our self-recruitment model. This has parallels with a distinct form of bistability analysed for the MAPK system by Markevich et al. [ 41 ]. In this MAPK model, there are two stable states of MAPK activity. The high state is sustained in part because of the saturation of phosphatases that reverse its activity. The distinction, again, is that the AMPAR bistability involves translocation without sustained biochemical activation whereas the Markevich model involves biochemical activation without intrinsic structural effects.

In a broader context, the AMPAR structural bistability could be generalised as a state-dependent translocation of molecules coupled to a saturable interconversion between these states. Many cellular trafficking events have a similar general form, including the Rab-mediated system of small GTPases and nuclear transport control. By our analysis in Figures 5 and 6 , translocation bistability should exhibit two clearly separated regimes in which traffic occurs into one of the compartments, and a regime in which addition of the translocated molecule actually decreases its number in one of the cellular compartments. This might be an experimentally accessible signature for such behaviour in the cell.

Stochasticity and Robustness

The typical synapse has a volume of 0.1 fl [ 16 ], and contains rather small numbers of key signalling molecules. This introduces fluctuations in reactions taking place in the synapse. We have previously analysed small-volume signalling for several pathways using simple assumptions about scaling and diffusion, and find that stochastic effects are so severe that some conventional signalling mechanisms simply do not work in these volumes [ 7 , 36 ]. Stable retention of synaptic state is a potential victim of stochasticity, as biochemical noise can lead to spontaneous state transitions. Bialek [ 17 ] and Miller et al. [ 42 ] have previously analysed synaptic bistability and suggest that in principle an autophosphorylation mechanism can give molecular stability of the order of hundreds of years even with a small number of CaMKII holoenzymes. Our current study is both coarser and more detailed than these analyses. Our representation of CaMKII does not consider individual holoenzymes, but on the other hand we explicitly represent translocation and several important signalling interactions at the synapse. In some models, and for some states, we did not observe any transitions over a year of simulation time. For other states the predicted lifetime is of the order of tens of hours ( Table 1 ). Thus, in terms of state stability, our models are quite robust.

While it is heartening that our models are stable for many hours, it is clear that this may change in either direction if additional interactions are considered. For example, we ignore the diffusive exchange of most spine molecules with the PSD and dendrite. Over the time scales of our simulations, all the spine molecules would in fact exchange or degrade. This introduces further challenges to synaptic stability mechanisms that are beyond the scope of this study. A further illustration of the importance of context on stability comes from the CaMKII analysis in Figure 7 . Here the inclusion of a more detailed PKA signalling model reduces the predicted lifetime of CaMKII states from hundreds of hours to tens. Additional input pathways may contribute to increased noise, but scaffold anchoring of enzymes such as PKA may create local signalling environments that could potentially reduce stochastic effects. Direct experimental measurements of synaptic stochasticity are difficult to perform [ 43 ], but such measurements would be invaluable for comparison with our simulated estimates of stochasticity.

Another commonly used measure of robustness is to ask whether the model retains some critical attribute over a wide range of parameters. In our analysis, we use bistability as the attribute of interest. We perform parameter sensitivity analyses by varying key reaction rates and enzyme parameters over a 100-fold range. Our parameter sensitivity analyses for the AMPAR and CaMKII systems suggest that our models are fairly robust by this measure. Most parameters can be varied at least 0.5- to 2-fold their original values without the model losing bistability (see Figures 5 and 7 ).

A particularly stringent measure of the robustness of a mechanism is to see how well essential features are preserved as reaction details are changed. We have performed a rather severe pruning of the AMPAR bistable model to a bare skeleton (see Figure 6 ) and with very little parameter tuning the simplified model is also bistable. Thus, by several measures, the bistable processes we have analysed are resistant to stochasticity, parameter uncertainty, and even changes in reaction mechanisms.

Coupled Bistable Switches

Individual synapses are likely to exist in many states [ 18 ]. Given the short life of synaptic molecules discussed above, it seems possible that one mechanism for stabilising such states might be to associate bistable switches with them. Multiple states may be achieved if the individual switches are coupled loosely, so that combinations of states become possible. Here we have shown (model 5; Figure 9 ) that distinct forms of bistability may coexist to give rise to three possible synaptic states. The AMPAR switch is the major one, as it brings the receptors to the synaptic membrane in the first place. The CaMKII switch is nested within this as it can fine-tune the conductance of the receptors. This situation of nested bistable states is possible if the mobility of PP1 at the synapse is limited so that each PP1 molecule has exclusive access either to CaMKII or to AMPAR. The alternate assumption (model 4) is that PP1 is mobile within the PSD, and can access both CaMKII and AMPAR. This assumption causes the two forms of bistability to function in lockstep, where the activation of the AMPAR switch causes the CaMKII switch to turn on. This occurs because the two switches share the same pool of PP1 enzymes, and phosphatase saturation resulting from the AMPAR switch activates CaMKII. It is likely that the biological situation involves different degrees of PP1 mobility between multiple possible synaptic targets, and may even differ for the same synapse in different contexts. In our study, we obtain distinct outcomes for two cases of PP1 mobility and targeting. This sensitivity of synaptic state to PP1 mobility is a testable prediction and highlights the possible importance of subtle details of PSD anchoring on synaptic function.

An alternative proposal for multiple levels of synaptic activation is that the CaMKII–NMDAR complex may act in a highly modular manner (reviewed by Lisman and McIntyre [ 3 ]). In this scenario, each CaMKII–NMDAR complex can independently persist in a high or low state of activity. This situation would make it possible for an individual synapse to present graded levels of conductance depending on the number of active CaMKII complexes. Our model of CaMKII is at the bulk rather than holoenzyme level, and is too coarse-grained to address this possibility. The CaMKII phosphorylation of AMPAR plays two roles in our study. First, it directly increases the conductance of individual receptor tetramers. Second, it indirectly leads to an increase in the number of receptors at the synapse, by producing additional phosphorylation states of AMPAR for PP1 to act on. In the model, this leads to further saturation of the phosphatase and ultimately to an increase in surface AMPAR. There is recent evidence that CaMKII activity may also affect AMPAR numbers [ 37 ]. This would provide another mechanism for coupling between our proposed bistable mechanisms.

In addition to forming multiple synaptic states, our simulations show that coexisting bistable mechanisms may function to “hand-over” information about synaptic state from one switch to another. For example, in model 5 (weakly coupled synaptic switches), a rather brief Ca 2+ input is sufficient to activate CaMKII, which can then turn on the AMPAR switch over a timescale of hours ( Figure 9 B). This is loosely analogous to different forms of computer information storage, where information is initially stored in fast but volatile form (eg, RAM) and is later transferred to slow but stable forms of memory (eg, hard disk).

Our study illustrates how two mechanisms for synaptic bistability may coexist to give rise to multiple possible synaptic states. We propose that the synapse may exhibit a combinatorial set of states through the interactions of several molecular switches. These may include local protein synthesis feedback loops involving mTOR, self-assembly processes at the synapse, or presynaptic switches. From the cell-biological perspective, we have considered synaptic recruitment mechanisms for only two of the hundreds of postsynaptic molecules. All these molecules undergo turnover, and many experience regulated movement similar to that in the switches we have analysed. We suggest that there are many forms of self-recruitment, coordinated self-assembly, and other potentially switch-like processes that contribute to the maintenance of different constituents and states of the synapse.

Materials and Methods
Model development.

Our model was developed to closely tie with experimental observations and to build on existing, well-documented, and experimentally constrained models. Two molecular trafficking cycles form the core of the model: (1) the trafficking of AMPARs, and in particular GluR12, between internal vesicular pools and the synaptic membrane associated with the PSD and (2) the movement of CaMKII to and from the spine cytosol and the PSD (see Figure 1 ). As elaborated below, we developed the models using published experimental observations on these trafficking processes, and considerable specific data on the biochemistry of the phosphorylation of these molecules. A few regulatory pathways were also modelled to provide signalling input and context. Reactions in the model take place in two compartments: the PSD and the bulk cytosolic volume of the spine. The receptors are membrane-associated, so the PSD-associated synaptic membrane is included in the PSD compartment. Likewise, the internalised, vesicular pool of receptors is included in the cytosolic compartment. The PSD volume is taken as 0.01 fl, and the spine head volume as 0.09 fl. There is a third, dendritic compartment of 5 fl that is occupied only by diffusible cAMP and by a bulk AMPAR pool. The bulk AMPAR pool is assumed to be at a steady level and is meant to represent synthesis and degradation of the receptor. No reactions occur in the dendritic compartment as it is meant only to couple diffusively with the spine.

AMPAR model.

Due to a combinatorial proliferation of states, the reaction diagram of the AMPAR steps appears complex. As described below, we modelled 16 phosphorylation states of the receptor each in the internal and synaptic-membrane-associated pools. However, most of the reactions involving these 16 states were symmetric as they involved independent phosphorylation sites. This simplifies the model definition. We assumed that symmetric reactions had the same rates, so our model relies on only a few trafficking and phosphorylation parameters. AMPARs occur as tetrameric structures [ 44 ] with most AMPARs composed of two subunits each of GluR1 and GluR2 (GluR12) or GluR2 and GluR3 (GluR23). GluR23 receptors show constitutive trafficking and are responsible for basal synaptic transmission whereas GluR12 receptor insertion can be altered by stimuli [ 10 , 20 ]. We considered only GluR12 receptors in the model to focus on activity-stimulated events. The dynamics of GluR12 AMPAR trafficking were determined by kinase/phosphatase activities at the Ser845 sites of the two GluR1 subunits in the tetrameric GluR12 complex. Dephosphorylation of these sites by phosphatases triggers endocytosis whereas phosphorylation by PKA is required for synaptic membrane targeting [ 10 , 11 ]. We modelled the phosphorylation/dephosphorylation as a two-step reaction, where phosphorylation or dephosphorylation of both GluR1 subunits is necessary for synaptic membrane targeting or internalisation, respectively. Through simulations we found that basal activities of PP1 and PKA can account for the constitutive cycling of receptors in our model, consistent with experimental studies [ 8 , 45 ]. We assumed that PKA and PP1 were the relevant enzymes, but the model does not exclude the possibility that the same cycling effects might be mediated by other phosphorylation enzymes.

There is evidence that the membrane-associated PP1 dephosphorylates AMPARs only in the PSD, as loading neurons with active PP1 does not alter basal synaptic strength transmission [ 46 ]. Hence, we assumed that PP1 acts on GluR1 only in the PSD whereas PKA phosphorylates GluR1 in both compartments. In both compartments, Ser845 of GluR1 was also dephosphorylated by PP2B [ 10 , 47 , 48 ], which itself is inactive at basal Ca 2+ concentrations.

Phosphorylation of Ser831 of the GluR1 subunit by CaMKII alters channel properties of the receptor in that the phosphorylation increases channel conductance approximately 2-fold [ 49 ]. As for the Ser845 sites, we modelled Ser831 phosphorylation of GluR1 so that both sites of a tetrameric complex could be phosphorylated individually by CaMKII. Dephosphorylation of Ser831 was modelled to occur only in the PSD, as internalisation was reported not to alter the phosphorylation state of AMPARs at Ser831 [ 10 ].

In the bistable models we explicitly modelled protein turnover through activity-dependent degradation [ 1 ]. The activity dependence was introduced by restricting the turnover to the doubly Ser845-phosphorylated states in the internal pool of AMPARs (see Figure 5 A). There are around 150 receptors in an active dendritic spine [ 50 ]. We represented this constraint in the model as an anchor protein (possibly GRIP [ 8 ]) required for AMPAR insertion into the synaptic membrane.

The synaptic AMPAR conductance is a function both of the number of synaptic membrane receptors, and of their phosphorylation state. We assumed that if a single GluR1 subunit was phosphorylated on the CaMKII site (Ser831), the channel conductance was 1.5 times the basal level, and if two GluR1 subunits were phosphorylated the channel conductance doubled. In the figures, conductances are expressed as percent maximal conductance. We obtained the maximal conductance by considering that all the anchor protein was occupied, and that all the AMPARs were doubly phosphorylated and hence had double the basal conductance.

CaMKII model.

The CaMKII model was derived closely from a previously developed single-compartment model of CaMKII activity [ 4 , 51 ]. This model is duplicated for the cytosol and the PSD and trafficking steps included. There is evidence that PP2A dephosphorylates CaMKII in the cytosol and PP1 in the PSD [ 5254 ]. Because of limited data about the PP2A activity, we represented the cytosolic dephosphorylation step as involving a distinct phosphatase from the PP1 in the PSD, but using the same kinetics as PP1.

Binding of Ca 2+ /CaM is necessary and sufficient for the kinase to translocate to the PSD [ 55 ], where it binds to the NMDAR [ 56 , 57 ]. As we lacked direct association constants between CaMKII and NMDAR, we used time course information to constrain translocation of CaMKII to the PSD [ 14 ]. NMDAR was modelled as a putative binding site within the PSD [ 58 ]. Robust translocation away from the PSD occurs upon removal of the Ca 2+ stimulus, and phosphorylation of Thr305 is required in this process [ 14 ]. However, only simultaneous dephosphorylation at Thr286 is sufficient for effective dissociation of CaMKII from the PSD [ 14 , 59 ].

Other pathways.

There are numerous regulatory inputs, which are taken from a pre-existing library of signalling pathway models ([ 19 ]; the DOQCS database [ http://doqcs.ncbs.res.in ]). The parameters of these models are substantially the same, with the exception of PP2B (calcineurin), the cAMP pathway, and some scaling of phosphatase activities.

In the case of PP2B we did not vary any rates, but we eliminated the catalytic activity of two substates (Ca2.CaM.Ca4.CaN and Ca3.CaM.Ca4.CaN) as their contribution to the total was small (data not shown ), and since the inclusion of these additional phosphorylation steps for all states of AMPAR would have substantially increased the number of reactions in the model.

In the case of cAMP we increased cyclase concentrations by a factor of approximately 4-fold, to get integral numbers of molecules in the model and to compensate for the reduction in assumed ATP concentrations from earlier model values of 5 mM to 2 mM. Phosphodiesterase concentrations have also been scaled up to maintain effective cAMP concentrations. A diffusion step is modelled for cAMP exchange with the dendrite, using a diffusion constant from frog olfactory neurons [ 60 ].

The phosphatase rates were scaled to obtain correct steady-state phosphorylation levels of inhibitor 1 of PP1 and CaMKII. PP1 rates and concentrations were also scaled, as described in the Results section, for the model of CaMKII bistability.

Most molecules in the simulation were modelled as independent pools for the PSD and cytosol. Only PKA was assumed to have access to both the cytosolic and PSD volumes. The adenylyl cyclase pathway was modelled only in the cytosolic volume. Because of its rapid diffusion, cAMP was modelled as exchanging between the spine head cytosol and the dendrite. We make an implicit assumption that the concentration of spine head constituents is maintained over the long periods of our simulations, through unspecified trafficking or other processes.

Computations.

Simulations were performed on Linux workstations and on a Linux cluster (Atipa Technologies, Lawrence, Kansas, United States) for stochastic calculations. Models were implemented using Kinetikit/GENESIS [ 61 ], and solved using the Exponential Euler method [ 61 ]. Enzyme reactions were modelled with an explicit enzyme–substrate complex, with the exception of the adenylyl cyclase activity (see Figure 1 G), which used the Michaelis-Menten form to improve numerical stability.

Stochastic calculations were done using an adaptive stochastic method [ 62 ] and using the Gillespie exact stochastic method as implemented in GENESIS 3/MOOSE [ 27 ]. The exact stochastic calculations used the Mersenne Twister random number generator [ 63 ]. When using the exact stochastic method, the entire model was simulated with the Gillespie method. Thus, all reactions led to integral changes in the numbers of the variable molecules. A few molecules in the model are buffered. The numbers of these buffered molecules were folded into the corresponding rate terms for efficiency. For example, if we have the reaction A <==> B and A is buffered, then the propensity of formation of B is dn dn B /dt = kf. n A − kb. n B , where n A and n B are the numbers of molecules of A and B respectively. Since A is buffered, the value of n A is fixed and we replaced this equation with dn dn B /dt = kf′ − kb. n B , where kf′ = kf. n A. This substitution also meant that it was possible to use nonintegral numbers for buffered molecules. This was meant to represent situations where the chemical buffering system on average gave rise to a nonintegral number of molecules.

Stochastic transition time calculations.

Several lengthy stochastic runs were performed to estimate transition times between states of the models. In order to obtain longer samples, we set off many independent simulations in parallel on a cluster using distinct random number seeds, typically for a simulation time of 120000 s (approximately 33 h) each. Transition times were estimated for a set of independent simulations as follows. Let T be the time to first transition in a given run, or total time of the run if there were no transitions. Let N be the number of runs where there were transitions. Then transition time for the entire sample is ΣT ΣT / N . In some cases there were no transitions at all, even for a large sample of runs. In these cases N was zero, so we could only set a lower bound to the transition time to be of the order of ΣT ΣT .

In some cases (eg, Figure 7 G) we estimated individual transition times by summing T for successive runs until the first run that had a transition. This sum gave the estimated transition time. Then the sum was reset to zero and the process repeated for the next transition.

Estimation of thresholds (unstable fixed points).

Thresholds for transition between lower and upper states of AMPAR in the spine were estimated using an iterative bisection method. The range of possible values was known from the upper and lower steady states of the bistable model (model 1). These were set as the upper and lower limits U and L, respectively. The first estimate E of the threshold was halfway between U and L: E = ( U + L )/2. The model was equilibrated at E receptors by blocking the AMPAR exchange with the bulk and AMPAR degradation. Then the exchange and degradation reactions were unblocked, and the model was run out for 10000 s. Depending on whether E was above or below the actual threshold point, the model settled toward U or L, respectively. If E was high, then U was reassigned to E . If E was low, then L was reassigned to E . This process was repeated seven times to obtain an approximately 1% accurate estimate of the threshold.

Model and simulator availability.

Complete model parameters and reaction schemes are presented in Protocol S1 . All models, demonstration simulations, and the GENESIS/Kinetikit simulator are freely available at http://www.ncbs.res.in/~bhalla/AMPAR_switch/index.html . Models 0 to 5 (including the buffered PKA version of model 3) are deposited in the DOQCS database ( http://doqcs.ncbs.res.in ) as accession numbers 59 to 65.

Supporting Information
Transient Stimuli and Responses of Model 0 with the Total Number of AMPARs Set to 80
(51 KB PDF)
Responses of Different Bistable Models to Transient Inputs
(60 KB PDF)
Model Equations and Parameters
(240 KB PDF)

We acknowledge many helpful discussions with Mike Ehlers and Dennis Bray. AH received support from the Conseil Regional Alsace, USB from the Wellcome Trust and NCBS/TIFR. The development of GENESIS 3/MOOSE was supported in part by Biophase Systems.