text:
Login Register for an account
Arctic Warming
KEYWORDS — ARCTIC WARMING
Quicktabs: Keywords
"Arctic's strongest sea ice breaks up for first time on record" — Jonathan Watts — The Guardian — August 21, 2018
The oldest and thickest sea ice in the Arctic has started to break up, opening waters north of Greenland that are normally frozen, even in summer. This phenomenon – which has never been recorded before – has occurred twice this year due to warm winds and a climate-change driven heatwave in the northern hemisphere.
"Arctic winters are getting warmer" — Max Carol — Treehugger — July 14, 2017
A rise in Arctic storms has more than doubled the number of winter warming events, which could greatly impede ice growth.
"Nitrous Oxide Poses Fresh Threat to the Arctic" — Scott Waldman — Scientific American — May 30, 2017
The thawing of the Arctic permafrost is releasing a potent greenhouse gas into the atmosphere that has rarely been considered as a threat despite its tremendous potential to drive global warming.
"Arctic stronghold of world's seeds flooded after permafrost melts" — Damian Carrington — Guardian — May 19, 2017
It was designed as an impregnable deep-freeze to protect the world's most precious seeds from any global disaster and ensure humanity's food supply forever. But the Global Seed Vault, buried in a mountain deep inside the Arctic circle, has been breached after global warming produced extraordinary temperatures over the winter, sending meltwater gushing into the entrance tunnel.
"Arctic's Summer Sea Ice Could Be Gone in 20 Years" — Mia Bennett — Maritime Executive — May 01, 2017
A new report from Arctic Monitoring and Assessment Programme suggests that the Arctic may be ice-free in summer within twenty years. The 90 authors of the rigorously peer-reviewed AMAP report state, "Extrapolations of recent observed data suggest a largely ice-free summer ocean by the late 2030s, which is earlier than projected by most climate models."
"Major Report Prompts Warnings That the Arctic Is Unraveling" — Jeff Tollefson — Scientific American — April 28, 2017
The Arctic is warming more than twice as fast as the rest of the planet, suggests a huge assessment of the region. The warming is hastening the melting of Arctic ice and boosting sea-level rise.
"The Arctic Ocean Is Becoming More Like the Atlantic Ocean" — Brian Kahn — Scientific American — April 07, 2017
The Arctic is undergoing an astonishingly rapid transition as climate change overwhelms the region. New research sheds light on the latest example of the changes afoot, showing that parts of the Arctic Ocean are becoming more like the Atlantic. Warm waters are streaming into the ocean north of Scandinavia and Russia, altering ocean productivity and chemistry. That's making sea ice recede and kickstarting a feedback loop that could make summer ice a thing of the past.
"China's 'airpocalypse' linked to Arctic sea ice loss" — Matt McGrath — BBC News — March 15, 2017
The air pollution that lingered over eastern China for nearly a month in 2013 has been linked to the loss of Arctic sea ice the previous autumn.
Arctic Warming, China
"What If We Installed Millions Of Wind-Powered Pumps In The Arctic To Grow More Ice?" — Adele Peters — Fast Company CoExist — February 28, 2017
A new study offers an extreme solution to shrinking Arctic ice sheets -- if we covered ice in tiny wind-powered pumps, say researchers from Arizona State University, we could potentially pump seawater onto the surface of the ice during the winter. As the water freezes and thickens, it could help the ice survive the next summer.
"Beyond the extreme: Scientists marvel at 'increasingly non-natural' Arctic warmth" — Jason Samenow — Washington Post — February 01, 2017
The Arctic is so warm and has been this warm for so long that scientists are struggling to explain it and are in disbelief. The climate of the Arctic is known to oscillate wildly, but scientists say this warmth is so extreme that humans surely have their hands in it and may well be changing how it operates.
Global warming is causing a rapid melting of the Arctic, which could be ice-free by 2030
As a result, the Arctic is warming twice as fast as the rest of the globe. Specifically, scientists are observing retreating sea ice, melting glaciers, and shrinking snow and permafrost areas.6 The summer ice cap is estimated to be only half the size it was fifty years ago.7 Sea-ice extent in the Arctic has decreased steadily since the 1950s and in September 2007 reached a record low that was 39 percent below the 1979–2000 mean. September 2008 experienced the second-lowest Arctic ice extent on record, at 34 percent below the 1970–2000 mean. In September 2009, when the Arctic reached its minimum ice extent for the year, it was recorded at the third-lowest extent since 1979 satellite measurements began, further demonstrating the declining trend in summer sea ice over the past thirty years (see the figure).8
Although estimates for when the Arctic will experience ice-free conditions in the summer range from 2013 to 2060, the consensus of most models and researchers is that the Arctic will experience ice-free conditions for a portion of the summer by 2030.9 It is important to point out that no research or model simulations indicate that winter sea-ice cover of the Arctic Ocean will disappear during this century. This reinforces the point that the Arctic will still be a very challenging environment in which to operate.
[ Page 36 ]
Titley, Rear Admiral David W., U.S. Navy, and Courtney C. St. John. "Arctic Security Considerations and the U.S. Navy's Roadmap for the Arctic ." Naval War College Review. Vol. 63, No. 2 (Spring 2010): 35-48. [ More (3 quotes) ]
Arctic ice is melting faster than anyone predicted
The ice was never supposed to melt this quickly. Although climate scientists have known for some time that global warming was shrinking the percentage of the Arctic Ocean that was frozen over, few predicted so fast a thaw. In 2007, the Inter- governmental Panel on Climate Change estimated that Arctic summers would become ice free beginning in 2070. Yet more recent satellite observations have moved that date to somewhere around 2035, and even more sophisticated simulations in 2012 moved the date up to 2020. Sure enough, by the end of last summer, the portion of the Arctic Ocean covered by ice had been reduced to its smallest size since record keeping began in 1979, shrinking by 350,000 square miles (an area equal to the size of Venezuela) since the previous summer. All told, in just the past three decades, Arctic sea ice has lost half its area and three quarters of its volume.
Borgerson, Scott G. "The Coming Arctic Boom: As the Ice Melts, the Region Heats Up ." Foreign Affairs. Vol. 92, No. 4 (July/August 2013): 76-89. [ More (7 quotes) ]
Implications of warming Arctic are creating a national security imperative for the U.S to ratify UNCLOS
Imperative impetus for this change in the US ocean policy comes from the ongoing climate change in the Arctic region and its potential implications for the US. Indeed, receding ice in the Arctic provides new opportunities to the US to secure its energy security and to gain economically by extracting hitherto inaccessible offshore Arctic resources and utilizing navigable Northwest Arctic Passage for commercial shipping. However, its legal status as a non-Party to the LOS Convention has kept the US ―hobbled on the Arctic's geopolitical sidelines‖ and acts as a stumbling block in its active participation in important international policymaking bodies- CLCS and ISBA. The US has no say in the CLCS commission with the authority to validate its national claims for extended continental shelf, which may adverserly affect the US Arctic interests. Further, non-participation in the ISBA authority may also marginalise the US interests in the deep seabed mining in the Area, beyond national jurisdiction. All these factors have point out the need for a change in the US ocean policy in recent time. These imperatives emphasize on the need to accede to the LOS Convention to advance US economic and strategic interests in the contemporary world.
Kumar, Sunil Agarwal. Prospects of a Paradigm Shift in the American Policy Towards UN Convention on the Law of the Sea: Potential Implications . National Maritime Foundation: , April 15, 2011. [ More (7 quotes) ]
Warming of the arctic is a compelling reason for U.S. to pursue ratification of UNCLOS at earliest opportunity
The United States has basic and enduring national interests in the oceans. These diverse interests—security, economic, scientific, dispute settlement, environmental, and leadership—are best protected through a comprehensive, widely accepted international agreement that governs the varying (and sometimes competing) uses of the sea. Although the United States has lived outside the Convention for 30 years, climate change in the Arctic provides the current Administration with a new and urgent incentive to re-engage the Senate and urge that body to provide its advice and consent to U.S. accession to the treaty at the earliest opportunity. As a nation with both coastal and maritime interests, the United States would benefit immensely from becoming a party to UNCLOS—accession will restore U.S. oceans leadership, protect U.S. ocean interests and enhance U.S. foreign policy objectives, not only in the Arctic, but globally.
[ Page 774-775 ]
Pedrozo, Raul. "Arctic Climate Change and U.S. Accession to the United Nations Convention on the Law of the Sea ." International Law Studies. Vol. 89. (2013): 757-775. [ More (10 quotes) ]
Arctic warming at an accelerating rate and could be ice-free by 2013
The Arctic has always experienced cooling and warming, but the current melt defies any historical comparison. It is dramatic, abrupt, and directly correlated with industrial emissions of greenhouse gases. In Alaska and western Canada, average winter temperatures have increased by as much as seven degrees Fahrenheit in the past 60 years. The results of global warming in the Arctic are far more dramatic than elsewhere due to the sharper angle at which the sun's rays strike the polar region during summer and because the retreating sea ice is turning into open water, which absorbs far more solar radiation. This dynamic is creating a vicious melting cycle known as the ice-albedo feedback loop.
Each new summer breaks the previous year's record. Between 2004 and 2005, the Arctic lost 14 percent of its perennial ice -- the dense, thick ice that is the main obstacle to shipping. In the last 23 years, 41 percent of this hard, multiyear ice has vanished. The decomposition of this ice means that the Arctic will become like the Baltic Sea, covered by only a thin layer of seasonal ice in the winter and therefore fully navigable year-round. A few years ago, leading supercomputer climate models predicted that there would be an ice-free Arctic during the summer by the end of the century. But given the current pace of retreat, trans-Arctic voyages could conceivably be possible within the next five to ten years. The most advanced models presented at the 2007 meeting of the American Geophysical Union anticipated an ice-free Arctic in the summer as early as 2013.
Borgerson, Scott G. "Arctic Meltdown: The Economic and Security Implications of Global Warming ." Foreign Affairs. Vol. 87, No. 2 (March/April 2008). [ More (10 quotes) ]
Arctic warming is allowing access to more fossil fuels that will only exacerbate cause of warming
Ironically, the great melt is likely to yield more of the very commodities that precipitated it: fossil fuels. As oil prices exceed $100 a barrel, geologists are scrambling to determine exactly how much oil and gas lies beneath the melting icecap. More is known about the surface of Mars than about the Arctic Ocean's deep, but early returns indicate that the Arctic could hold the last remaining undiscovered hydrocarbon resources on earth. The U.S. Geological Survey and the Norwegian company StatoilHydro estimate that the Arctic holds as much as one-quarter of the world's remaining undiscovered oil and gas deposits. Some Arctic wildcatters believe this estimate could increase substantially as more is learned about the region's geology. The Arctic Ocean's long, outstretched continental shelf is another indication of the potential for commercially accessible offshore oil and gas resources. And, much to their chagrin, climate-change scientists have recently found material in ice-core samples suggesting that the Arctic once hosted all kinds of organic material that, after cooking under intense seabed pressure for millennia, would likely produce vast storehouses of fossil fuels.
Global warming is causing a dramatic decrease in Arctic ice
Global warming is most dramatic in the Arctic. n10 In Alaska and western Canada, average winter temperatures have increased by as much as seven degrees Fahrenheit in the past 60 years.n11 Scientists agree that atmospheric warming will continue for years to come, and that this warming will significantly affect ice coverage in the Arctic. Many experts believe the particularly sharp increase in warming and melting throughout the last few decades can be attributed to both human and natural causes.n12 Because ice and snow are white, they have what is known as a "high albedo" and reflect most solar energy.n13 Albedo is a measure of how strongly an object reflects light from sources such as the sun. Water is darker and thus has a "low albedo" that absorbs most solar radiation. This creates a condition known as a "positive feedback loop" and, as a consequence, the Arctic region essentially amplifies any sort of warming trend.n14 The ocean exposed by melting ice soaks up more heat, which melts more ice and exposes more sea.n15 In the most extreme scenario, the positive feedback loop could cause extreme deterioration of Arctic sea ice, leaving the Arctic Ocean more like the Baltic Sea, covered by only a thin layer of seasonal ice in the winter.n16 At the current pace of retreat, transArctic voyages could be possible within the next five to ten years, but it remains extremely difficult to make an accurate prediction.n17
[ Page 529 ]
Mendez, Tessa. "Thin Ice, Shifting Geopolitics: The Legal Implications Of Arctic Ice Melt ." Denver Journal of International Law and Policy. Vol. 38. (Summer 2010): 527-547. [ More (6 quotes) ]
Open Debate Engine Status Block
There are currently:
128 arguments
539 news articles
There are 200 orphaned quotes, or 15% of existing quotes.
About — Contact — Updates
Site powered by the Open Debate Engine.


text:
Der Haval H6 ist ein SUV des chinesischen Automobilherstellers Great Wall Motor, das seit 2011 in Tianjin und seit 2013 im bulgarischen Bahovitsa gebaut wird. Seit 2013 ist der Haval H6 das meistverkaufte SUV in China, bereits am 12. November 2015 wurde das millionste Fahrzeug hergestellt.

1. Generation (seit 2011)

Bis zur Einführung der eigenständigen Marke Haval im Jahr 2013 wurde das SUV als Great Wall Haval H6 vermarktet. Das Fahrzeug debütierte im April 2010 als Konzeptfahrzeug auf der Beijing Auto Show, ein Jahr später wurde auf der Shanghai Auto Show die Serienversion präsentiert. Diese kam am 25. August 2011 in China in den Handel. Im August 2013 kam das Fahrzeug in Bulgarien in den Handel, in Deutschland wird es nicht angeboten.

Die erste Generation wird mit eingeschränkter Angebotsvielfalt auch nach der Einführung der zweiten Generation im Mai 2017 weiterhin verkauft.

Technische Daten
Serienmäßig verfügt das SUV über Vorderradantrieb, optional ist ein Allradantrieb von BorgWarner erhältlich.

2. Generation (seit 2017)

Die zweite Generation des H6 wurde Anfang 2017 vorgestellt. Sie wird in China parallel zum weiterhin gebauten Vorgängermodell angeboten und ist wieder in einer "Red Label"-Variante und einer "Blue Label"-Variante erhältlich. Die Plattform teilt sich der neue H6 mit dem Wey VV5. Obwohl das Nachfolgemodell bereits im Handel war, präsentierte Haval im März 2021 eine überarbeitete Version der zweiten Generation.

Technische Daten

3. Generation (seit 2020)

Die dritte Generation der Baureihe wurde am 7. Juli 2020 präsentiert. Im August 2020 kam sie in China in den Handel. Den Antrieb übernahm zunächst der aus dem Vorgängermodell bekannte  starke 1,5-Liter-Ottomotor. Später folgte ein Zweiliter-Ottomotor mit  und ein 1,5-Liter-Ottomotor mit . Auf der Bangkok International Motor Show im März 2021 wurde die Baureihe auch mit einem Hybridantrieb präsentiert. Die Systemleistung wird mit  angegeben. Im Mai 2021 kam diese Antriebsvariante in den Handel. Eine Plug-in-Hybrid-Version mit einer elektrischen Reichweite von bis zu 201 km nach NEFZ debütierte im Dezember 2021 und kam im August 2022 auf den Markt.

Technische Daten

Haval H6 Coupé

Zwischen 2015 und 2021 hatte Great Wall das Haval H6 Coupé im Angebot. Dieses Fahrzeug basiert auf dem Haval H6, ist jedoch kürzer und flacher. 2021 wurde dieses Modell durch den Haval H6S ersetzt.

Einzelnachweise

Weblinks

 Offizielle Website (englisch)
 Offizielle Website (chinesisch)

Sport Utility Vehicle
Kombi
Hybridelektroautomodell

redpajama_set_name: RedPajamaWikipedia


Ingredients: purified deep sea fish oil (from anchovies and sardines), xylitol, purified water, sorbitol, gelatin, natural tutti frutti flavor, citric acid, trisodium citrate, paprika extract (for natural color), trace amounts of canola oil.
Suggested Use: For children 2 years and older, one fishie daily with food, or as directed by your health care professional. Please ensure child chews each fishie thoroughly.


text:
Regional Science Centre
Home / Odisha / Bhubaneswar / Regional Science Centre
Regional Science Centre has 84 interactive exhibits, which are spread over an 8-acre undulated landscape. In the centre, tourists can view exotic plants and cacti garden. By visiting the site, tourists can play with large exhibits and learn the underlying principles on which they function. Mathematics gallery of the centre aims at unravelling the world of mathematics and its application. In this gallery, tourists can also gain awareness about the Indian contribution towards this subject during the Vedic Period.
Motion gallery serves to enlighten tourists about the concept of motion and its all-pervading presence in the universe. Motions of varied items, ranging from deep within the molecules to celestial motion of planets are explained by numerous interactive exhibits. Gallery on fun science has many interactive exhibits like antigravity mirrors, vortex, jumping disc and others.
Science Gallery provides tourists a truly interactive experience by live action and enables them to discover for themselves why things happen the way they do. Prehistoric Life Park has three-dimensional figures of some extinct animals. Exhibits in this section enlighten tourists about the evolutionary process of birds, mammals and man.

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}
Biological  micro-structures such as synapses, dendritic spines,
{ subcellular domains}, sensor cells and many other structures,
are regulated by chemical reactions that involve only a
small number of molecules, that is, between a few and up to
thousands of molecules. A chemical reaction that involves only 10
to 100 proteins can cause a qualitative transition in the
physiological behavior of a { given} part of a cell. Large
fluctuations should be expected in a reaction if so few molecules
are involved, both in transient and persistent binding and
unbinding reactions. In the latter case large fluctuations in the
number { of} bound molecules should force the physiological
state to change all the time, unless there is a specific mechanism
that prevents the switch { and} stabilizes the physiological
state. Therefore, a theory of chemical kinetics of such reactions
is needed to predict the threshold at which switches occur and to
explain how the physiological function is regulated in molecular
terms at a sub-cellular level.

A physiological threshold can be determined in molecular terms,
for example, when the number of activated molecules exceeds a
certain value. The standard theory of chemical kinetics is
insufficient for the determination of the threshold value, because
it is based on the assumption that there is a sufficiently large
number of reactant molecules and it describes the time evolution
of only the average number of  molecules.
The standard theory of reaction-diffusion describes chemical reactions in terms of concentrations so that
fluctuations due to a small number of molecules are lost.

For example, in dendritic spines of neurons a flow of calcium
entering through the NMDA channels can induce a cascade of
chemical reactions. As calcium ions diffuse they can bind, unbind,
{and} leave the spine { without} binding. But if enough
calcium binds specific molecules, such as calmodulin, then {
certain proteins become activated, such as CaMK-II}, which are
involved in regulating synaptic plasticity \cite{Lisman94}. Now if
{ sufficiently many of them are activated at about the same time} and
thus the threshold is exceeded, additional changes can be induced at the
synapse level, affecting the physiological properties of a neuron.
In particular, such changes may include a modification of the
biophysical properties of some receptors and/or increase the
number of channels { at a specific area of the synapse, called
the postsynaptic density}. It is unclear how many CAMK-II are
needed {for crossing the threshold}, but the range is
somewhere between 5 to 50. It is remarkable that { as few as
5-50} molecules can control the synaptic weight \cite{Lisman94}.
The number of phosphorylated CAMK-II, activated after {a}
transient calcium flow, depends on the location of the proteins,
the location of the channels, the geometrical restrictions imposed
by the spine shape, and the state of the proteins. All of these
factors play a crucial role in determining the threshold.

The photoreceptor cells are another example, where fluctuations in
the number of bound molecules determine the physiological
limitations of the cell. Indeed, in the outer segment of cones and
rods of the retina, the total number of open channels fluctuates
continuously due to the binding and unbinding of specific gating
molecules to their receptors. These fluctuations are directly
converted into fluctuations of the membrane potential, which are
called ``dark noise'', and thus determine the signal to noise
ratio \cite{Baylor} for a photon detection. The fluctuation in the
number of open channels is regulated by the number of gated
molecules and depends on the geometry of the outer segment and the
distribution of channels. It is not clear what are the details of the biochemical
processes involved in regulating the number of open channels, but
interestingly, the signal due to a single photon event is
sufficient to overcome the noise amplitude in rods, but not in
cones, although their biochemical properties are similar. { As
the binding molecules diffuse in the cell they can bind and unbind
to channels, thus causing fluctuations in the number of open
channels. The fluctuation depends on the arrival time of the
binding molecules, called cGMP, to the channel binding sites}.

{In the mathematical description of the binding and unbinding
reactions, we model the diffusion of the particles as Brownian
motion and binding occurs when a particle reaches a binding site.}
The { binding} probability depends on the geometry of the
domain and on the distribution of the channels. A channel opens
when it binds to two or three cGMP molecules and in the absence of
light, the number of open channels is small, approximatively 6 to
10 per micro-domain in a mammalian rod, when there are only 60
cGMP molecules. Due to the random binding and unbinding of the
molecules to the channels the number of open channels is a
stochastic process.

In this paper we start with the stochastic dynamics of the
reactant molecules in a micro-domain and derive a master-diffusion
equation for the joint probability density of the mobile reactant
and the number of bound immobile substrate { molecules}. We use
the equation to calculate the fluctuations in the number of bound
substrate molecules as a function of initial reactant
concentration. We apply the present theory to the computation of
the mean and variance of the fluctuation in the number of open
channels, and find their dependence on the initial number of the
mobile reactant, the geometry of the { compartment}, and the
distribution of channels.

Our model can predict {the} fluctuation { intensity} produced by
binding and unbinding to channels in confined compartments, such
as the a compartment (the space between two consecutive disks) of
cone and rod outer segment, or any other sub compartment of a
sensor cell. Such a prediction can clarify part of the noise
generation. At the present time the noise in a confined
micro-domain cannot be directly measured. Instead, excised patch
measurements are done \cite{Yau,Koko}, in which the cell structure is destroyed.
Thus computation and simulation of mathematical models are the
only tools for studying noise in { this} biological context.

\section{A Stochastic model of a non-Arrhenius chemical reaction}\label{firstM}

\subsection{Chemical reaction}
We consider two species of reactants, the mobile reactant $M$ that diffuses
in a bounded domain $\Omega $, and the stationary
substrate $S$ (e.g., a protein). The boundary $\partial \Omega $ of the
domain $\Omega $ is partitioned into an absorbing part $\partial \Omega _{a}$
(e.g., pumps, exchangers, another substrate that forms permanent bonds with $%
M $, and so on) and a reflecting part $\partial \Omega _{r}$ (e.g., a cell
membrane). In this model the volume of $M$ is neglected. We assume that
there are binding sites on the substrate. In terms of traditional chemical
kinetics the binding of $M$ to $S$ follows the law
\begin{eqnarray}  \label{CRE}
M+S_{free}
\begin{array}{l}
k_{f} \\
\rightleftharpoons \\
k_{b}%
\end{array
MS,  \label{reaction}
\end{eqnarray}
where $k_{f}$ is the forward binding rate constant, $k_{b}$ is the backward
binding rate constant, and $S_{free}$ is the unbound substrate.

However, when only a small number of reactant and substrate
molecules are involved in the reaction, as is the case in a
micro-domain in a biological cell, this reaction has to be
described by a molecular model, rather than by concentrations. The
description of this reaction on the molecular level begins with
the following definitions:

\begin{itemize}
\item $M(t)$ = number of unbound $M$ particles at
time $t$
\item $S_{\Delta}(\mbox{\boldmath$x$},t)$ = number of free sites in volume $\Delta\mbox{\boldmath$x$}$ at
time $t$
\item $S(t)$ = number of unbound binding sites at time $t$
\item $MS(t)$ =$M(0)-M(t)$ = number of bound $M$
particles at time $t$.
\item $s_0(\mbox{\boldmath$x$})$ = initial density of substrate\\
\item $S_{\mbox{max}}=\displaystyle\int_{\Omega}s_0(\mbox{\boldmath$x$})\,d\mbox{\boldmath$x$}$ = total number of
binding sites in $\Omega$.
\end{itemize}
The $M(0)$ reactant particles are initially distributed with
probability density $m_0(\mbox{\boldmath$x$})$. The initial density $s_0(\mbox{\boldmath$x$})$
integrates to $S(0)$. We assume that the $M$ particles diffuse and
we denote by $\mbox{\boldmath$x$}(t)$ the random trajectory of an unbound $M$
particle. We consider a small volume $\Delta \mbox{\boldmath$x$}$ about $\mbox{\boldmath$x$}$, that
contains initially $s_0(\mbox{\boldmath$x$})\,\Delta\mbox{\boldmath$x$}$ free binding sites and
$m_0(\mbox{\boldmath$x$})\,\Delta\mbox{\boldmath$x$}$ unbound $M$ particles.

The joint probability of an $M$ trajectory and the number of bound
sites in the volume $\Delta \mbox{\boldmath$x$}$  is
 \begin{eqnarray}
p(\mbox{\boldmath$x$},S,t\,|\,\mbox{\boldmath$y$})\,\Delta\mbox{\boldmath$x$}=
\Pr\left\{\mbox{\boldmath$x$}(t)\in\mbox{\boldmath$x$}+\Delta\mbox{\boldmath$x$},\,S_{\Delta}(\mbox{\boldmath$x$},t)=S\,|\,\mbox{\boldmath$x$}(0)=\mbox{\boldmath$y$}\right\}.
\label{jpdf}
 \end{eqnarray}
The function $p(\mbox{\boldmath$x$},S,t\,|\,\mbox{\boldmath$y$})$ is the joint probability density
to find an $M$ particle and $S_{\Delta}(\mbox{\boldmath$x$},t)$ free binding sites
at $\mbox{\boldmath$x$}$ at time $t$, conditioned by the initial position $\mbox{\boldmath$y$}$ of
the $M$ particle.

The marginal probability density of an $M$ trajectory is
\begin{eqnarray*}
p(\mbox{\boldmath$x$},t\,|\,\mbox{\boldmath$y$}) = \sum_{\mbox{free $S$}} p(\mbox{\boldmath$x$},S,t\,|\,\mbox{\boldmath$y$}),
\end{eqnarray*}
where the sum is over all free binding sites in the volume $\Delta
\mbox{\boldmath$x$}$ . The number of free $M$ molecules in the volume $\Delta \mbox{\boldmath$x$}$
at $\mbox{\boldmath$x$}$ is
\begin{eqnarray*}
M_{\mbox{free}}(\mbox{\boldmath$x$},t) =M_0\,\Delta\mbox{\boldmath$x$}\,\sum_{\mbox{free
$S$}}\int_\Omega p(\mbox{\boldmath$x$},S,t|\mbox{\boldmath$y$})m_0(\mbox{\boldmath$y$})\,d\mbox{\boldmath$y$}.
\end{eqnarray*}
The joint probability density function of $\mbox{\boldmath$x$}$ and $S$ is
 \[p(\mbox{\boldmath$x$},S,t)=\int_\Omega p(\mbox{\boldmath$x$},S,t|\mbox{\boldmath$y$})m_0(\mbox{\boldmath$y$})\,d\mbox{\boldmath$y$}.\]
The two-dimensional process
$\left(M_{\mbox{free}}(\mbox{\boldmath$x$},t),S(\mbox{\boldmath$x$},t)\right)$ is Markovian. The
evolution of $p(\mbox{\boldmath$x$},S,t)$, is governed by the diffusion of $M$
particles in and out of the volume $\Delta \mbox{\boldmath$x$}$ , and their binding
and unbinding inside the volume $\Delta \mbox{\boldmath$x$}$ . The influx in the
time interval $[t,t+\Delta t]$ is
 \begin{eqnarray*}
&&\oint_{\displaystyle\partial\mbox{Volume $\Delta
\mbox{\boldmath$x$}$}}\mb{J}(\mbox{\boldmath$x$},S,t\,|\,\mbox{\boldmath$y$})\cdot\mb{n}(\mbox{\boldmath$x$}) \,dS_{\mbox{\boldmath$x$}}\Delta
t=\\
&&\\
&&\int_{\mbox{Volume $\Delta \mbox{\boldmath$x$}$}}[p(\mbox{\boldmath$x$},S,t+\Delta
t\,|\,\mbox{\boldmath$y$})-p(\mbox{\boldmath$x$},S,t\,|\,\mbox{\boldmath$y$})]\,d\mbox{\boldmath$x$},
\end{eqnarray*}
which represents diffusion with coefficient $D$. Additional change
in the contents of the volume $\Delta \mbox{\boldmath$x$}$  is due to the binding
and unbinding of $M$ particles to the substrate. When there are
$S$ free binding sites in the volume $\Delta \mbox{\boldmath$x$}$ (see Figure \ref{inside}), the probability
that one $M$ particle binds to a free site in the volume $\Delta
\mbox{\boldmath$x$}$  { in the time interval} $[t,t+\Delta t]$ is proportional
both to $S$ and to the number $M_{\mbox{free}}(\mbox{\boldmath$x$},t)$ of free $M$
particles in the volume $\Delta \mbox{\boldmath$x$}$ . It is given by
 \begin{eqnarray*}
k_1M_0p(\mbox{\boldmath$x$},S,t)S \Delta \mbox{\boldmath$x$} \, \Delta t.
\end{eqnarray*}
The probability that one particle unbinds in the volume $\Delta
\mbox{\boldmath$x$}$  in this time interval is proportional to the number
$s_0(\mbox{\boldmath$x$})\Delta\mbox{\boldmath$x$}-S$ of bound sites in the volume $\Delta \mbox{\boldmath$x$}$ ,
given by
 \[k_{-1}[s_0\Delta\mbox{\boldmath$x$}-S] \Delta t.\]
Thus the probability of $S$ free sites when no change occurred in
the number of free sites is
 \begin{eqnarray*}
 p(\mbox{\boldmath$x$},S,t) \Delta \mbox{\boldmath$x$}\left\{1-k_1M_0 p(\mbox{\boldmath$x$},S,t)
 S\,\Delta \mbox{\boldmath$x$}\,\Delta t-
 k_{-1}[s_0(\mbox{\boldmath$x$})\Delta\mbox{\boldmath$x$}-S] \Delta t\right\} .
 \end{eqnarray*}
The number of free sites can change to $S$ at the end of the
interval $[t,t+\Delta t]$ if it was $S+1$ at the beginning and one
bond was formed, or if it was $S-1$ and one particle was unbound.
The probability of this event, as calculated above, is
 \begin{eqnarray*}
k_1M_0\,\Delta \mbox{\boldmath$x$}\, (S+1) p(\mbox{\boldmath$x$},S+1,t)\,\Delta
t+k_{-1}[s_0(\mbox{\boldmath$x$})\Delta\mbox{\boldmath$x$}-S+1)]\Delta t.
\end{eqnarray*}
The probability of $\mbox{\boldmath$x$},S$ at time $t+\Delta t$ is therefore
\begin{eqnarray*}
&&p(\mbox{\boldmath$x$},S,t+\Delta t) \Delta \mbox{\boldmath$x$}=\\
&&\\
 &&-\nabla\cdot\mb{J}(\mbox{\boldmath$x$},S,t\,|\,\mbox{\boldmath$y$})\,\Delta\mbox{\boldmath$x$}  +\\
 && \\
 &&p(\mbox{\boldmath$x$},S,t) \Delta \mbox{\boldmath$x$}\{1-M_0k_1\Delta \mbox{\boldmath$x$}\,S p(\mbox{\boldmath$x$},S,t)\Delta
 t-k_{-1}[s_0(\mbox{\boldmath$x$})\Delta\mbox{\boldmath$x$}-S] \Delta t\}+\\
&&\\
&& M_0k_1(S+1)  p^2(\mbox{\boldmath$x$},S+1,t)\,(\Delta \mbox{\boldmath$x$})^2\Delta t+\\
&&\\
&&  k_{-1}[s_0(\mbox{\boldmath$x$})\Delta\mbox{\boldmath$x$}-S+1]p(\mbox{\boldmath$x$},S-1,t) \Delta \mbox{\boldmath$x$}\Delta t.
 \end{eqnarray*}
{ for $S=0,1,2,\dots,S_{\mbox{max}}$, the coupled partial
differential equations}
 \begin{eqnarray}
\frac{\partial p(\mbox{\boldmath$x$},S,t) }{\partial t} &=& -\nabla\cdot\mb{J}(\mbox{\boldmath$x$},S,t)
 - K_1 p^2(\mbox{\boldmath$x$},S,t)S -k_{-1}[S_0(\mbox{\boldmath$x$})-S]p(\mbox{\boldmath$x$},S,t) \nonumber\\
&&\label{eqff}\\
&+& K_1
(S+1)p^2(\mbox{\boldmath$x$},S+1,t)+k_{-1}[S_0(\mbox{\boldmath$x$})-S+1]p(\mbox{\boldmath$x$},S-1,t)\nonumber,
 \end{eqnarray}
where by definition $\mb{J}(\mbox{\boldmath$x$},S,t)$ is the joint probability flux
at position $\mbox{\boldmath$x$}$ at time $t$, and $S$ proteins are free. It is
defined in the diffusion case by
 \begin{eqnarray} \mb{J}(\mbox{\boldmath$x$},S,t) = -D \nabla
p(\mbox{\boldmath$x$},S,t).
 \end{eqnarray}
{ The} new (forward) binding rate is
\begin{eqnarray*}
K_1 =M_0 k_1 \Delta\mbox{\boldmath$x$}
\end{eqnarray*}
and
 \[S_0(\mbox{\boldmath$x$})=s_0(\mbox{\boldmath$x$})\,\Delta\mbox{\boldmath$x$}=\mbox{total number
of binding sites in the volume $\Delta \mbox{\boldmath$x$}$ }.\] Thus $K_1$ is the
probability flux into the binding sites. The boundary conditions
on $S$ are
 \begin{eqnarray}
 p(\mbox{\boldmath$x$},S,t)=0\quad\mbox{for $S<0$ and $S>S_0(\mbox{\boldmath$x$})$}.\label{SBC}
 \end{eqnarray}

 \vspace{3mm}

{\bf \noindent Remark 1.} By summing equation (\ref{eqff}) over
$0\leq S\leq S_0(\mbox{\boldmath$x$})$ and using the boundary conditions
(\ref{SBC}), we obtain
 \begin{eqnarray}\label{eqffM}
  \frac{\partial M_{\mbox{total}}(\mbox{\boldmath$x$},t) }{\partial t} = -\nabla\cdot\mb{J}_M(\mbox{\boldmath$x$},t),
 \end{eqnarray}
for the marginal density of the $M$ particles. It means { that}
all bound and free $M$ particles effectively diffuse.\\

{\bf \noindent Remark 2.} When at specific locations there can be
at most one binding site, the system (\ref{eqff}) reduces to the
coupled equations
 \begin{eqnarray}\label{eqff2}
\frac{\partial p(\mbox{\boldmath$x$},0,t) }{\partial t} =
D\Delta p(\mbox{\boldmath$x$},0,t) -k_{-1}p(\mbox{\boldmath$x$},0,t) + K_1 p^2(\mbox{\boldmath$x$},1,t)\nonumber  \\
\\
\frac{\partial p(\mbox{\boldmath$x$},1,t) }{\partial t} = D\Delta p(\mbox{\boldmath$x$},1,t) +k_{-1}p(\mbox{\boldmath$x$},0,t)
-K_1 p^2(\mbox{\boldmath$x$},1,t).\nonumber
 \end{eqnarray}
Here $S_0(\mbox{\boldmath$x$})$ can take the values 0 or 1. When no molecules can
escape from a bounded domain, the flux associated with $p(\mbox{\boldmath$x$},S,t)$
satisfies the reflective boundary condition
 \begin{eqnarray}
 \left.\mb{J}\cdot \mbox{\boldmath$n$}\right|_{\partial \Omega} =0.\label{reflect}
 \end{eqnarray}

The initial condition, when no substrate is bound, is given by
$p(\mbox{\boldmath$x$},0,0) =m_{0}(\mbox{\boldmath$x$})$, hence $p(\mbox{\boldmath$x$},1,0) =0$. When the total
number of $M$ particles stays constant (i.e., no particles leave
the domain), adding equations (\ref{eqff2}) gives in the steady
state
 \begin{eqnarray}
 p(\mbox{\boldmath$x$},0)+p(\mbox{\boldmath$x$},1)=\frac1{|\Omega|}.\label{conserve}
 \end{eqnarray}

If the $M$ particles can escape the compartment, e.g., by being
absorbed in a part of the boundary $\partial\Omega_a$, the condition
(\ref{reflect}) should be changed to
 \begin{eqnarray}
 \left.\mb{J}\cdot \mbox{\boldmath$n$}\right|_{\partial\Omega-\partial \Omega_a} =0\label{reflecta}
 \end{eqnarray}
and
 \begin{eqnarray}
 \left.p(\mbox{\boldmath$x$},S,t)\right|_{\partial\Omega_a}=0.\label{absorba}
 \end{eqnarray}
In this case (\ref{conserve}) no longer holds.\\

\noindent {\bf Remark 3:} Obviously, $S_0(\mbox{\boldmath$x$})$ takes only integer
values. We assume that its discontinuities are located on smooth
interfaces. The density $p(\mbox{\boldmath$x$},S,t)$ and the normal component of
the flux $\mb{J}(\mbox{\boldmath$x$},S,t)\cdot\mb{n}(\mbox{\boldmath$x$})$ are continuous across the
interfaces for all $S=0,1,\dots,S_{\mbox{max}}$.

\subsection{Moments of the pdf}
Statistical moments of the pdf can be computed from  equation
(\ref{eqff}). The average and the standard deviation of the number
of bound proteins are evaluated for equation (\ref{eqff2}). The
mean of the number of  bound proteins at time $t$ is given by
 \begin{eqnarray}
\langle S_b(t)\rangle = \int_{\Omega} S_0(\mbox{\boldmath$x$})p(\mbox{\boldmath$x$},1,t) d\mbox{\boldmath$x$}.
 \end{eqnarray}
The standard deviation is given by
 \begin{eqnarray} \label{sd}
\sigma^2(t) = \langle S_b^2(t)\rangle-\langle S_b(t)\rangle^2  =
\int_{\Omega} S^2_0(\mbox{\boldmath$x$})p(\mbox{\boldmath$x$},1,t)\, d\mbox{\boldmath$x$} - \left(\int_{\Omega}
S_0(\mbox{\boldmath$x$})p(\mbox{\boldmath$x$},1,t)\, d\mbox{\boldmath$x$}\right)^2.
 \end{eqnarray}
When the proteins are uniformly distributed on a subset $\Omega'
\subset \Omega$, the distribution $S_0(\mbox{\boldmath$x$})$ is given by the
characteristic function of the subset $\Omega'$,
 \begin{eqnarray}
S_0(\mbox{\boldmath$x$}) = \chi_{\Omega'}(\mbox{\boldmath$x$})\frac{N_0}{\mid \Omega' \mid},
 \end{eqnarray}
where the total number of binding sites is
 \[N_0=\int_{\Omega} S_0(\mbox{\boldmath$x$})\, d\mbox{\boldmath$x$}.\]
We obtain the standard deviation of bound sites from the
expression (\ref{sd}) as
 \begin{eqnarray}\label{sigma2}
  \sigma^2(t) &=&
\langle S_b^2(t)\rangle-\langle S_b(t)\rangle^2\\
&&\nonumber\\
&=& \left(\frac{N_0}{\mid \Omega' \mid}\right)^{2}\left(
\bar{S_0}(t)-\bar{S_0}^2(t)\right)= \left(\frac{N_0}{\mid \Omega'
\mid}\right)^{2} \bar{S_0}(t)(1-\bar{S_0}(t)),\nonumber
 \end{eqnarray}
where

 \begin{eqnarray} \label{Pro}
\bar{S_0}(t) = \int_{\Omega'} p(\mbox{\boldmath$x$},1,t)\, d\mbox{\boldmath$x$}
 \end{eqnarray}
is the fraction of bound sites. Note that
 \[\bar{S_0}(t) \leq 1.\]

\subsection{Standard deviation of the number of bound protein in some cases}
We consider the one-dimensional case where $\Omega=[0,L]$ and
$S_0(x)$ is either 0 or 1 in intervals. In the steady state the
system (\ref{eqff2}) is
  \begin{eqnarray}
0&=& D\Delta p(x,0) -k_{-1}S_0(x) p(x,0) + K_1 p^2(x,1)\nonumber  \\
&&\nonumber\\
0&=& D\Delta p(x,1) +k_{-1}S_0(x) p(x,0) -K_1 p^2(x,1)\label{eqff3}\\
&&\nonumber\\
\frac{1}{L} &=&  p(x,0)+ p(x,1), \nonumber
 \end{eqnarray}
which reduces to
 \begin{eqnarray}\label{eqq4}
 D p''(x,1)
+k_{-1}S_0(x)\left[\frac{1}{L}-p(x,1)\right] -K_1 p^2(x,1)=0.
 \end{eqnarray}
We convert to densities by setting
 \[c_M(x,1)=M_0p(x,1).\]
Then (\ref{eqq4}) becomes
 \begin{eqnarray}\label{eqq5}
 D c_M''(x,1)
+k_{-1}S_0(x)\left[\frac{M_0}{L}-c_M(x,1)\right]
-\frac{K_1}{M_0}c_M^2(x,1)=0.
 \end{eqnarray}
The function $c_M(x,1)$ is supported where the protein are
located.

When the boundary conditions are reflective for the $M$ trajectories, using the uniqueness of the solution, then
at a point $x$, where $S_0(x)$ is supported is
 \begin{eqnarray*}
c_M(x,1) =\frac{2k_{-1}S_0(x)M_0/L}{k_{-1}S_0(x)+ \sqrt{
(k_{-1}S_0(x))^2+4 K_1k_{-1}S_0(x)/L }}.
 \end{eqnarray*}
In particular, If the substrate is uniformly distributed in intervals $S_0(x)=N_S/L$ and $M_0\ll N_S$, then the fraction of bound $M$ particles is
 \begin{eqnarray}
 p_M=\frac1{M_0}\int_0^Lc_M(x,1)\,dx=\frac{2}{1+ \sqrt{
1+4\displaystyle \frac{M_0k_1\Delta x}{N_Sk_{-1}}}}\sim 1,\label{bound}
 \end{eqnarray}
which means that practically all $M$ particles are bound. If
$M_0\gg N_S$, then  (\ref{bound}) gives
 \begin{eqnarray}
  p_M=\frac1{M_0}\int_0^Lc_M(x,1)\,dx\sim\sqrt{\frac{N_Sk_{-1}}{M_0k_1\,\Delta
  x}}\ll1.
  \end{eqnarray}

In this case, the variance of the fluctuations in the number of
bound $M$ particles, which is the same as the number of bound
sites, as a function of $M_0$ and $N_S$, is given by
 \begin{eqnarray}
 \sigma^2_S(M_0)=p_M(1-p_M)=\frac{2}{1+ \sqrt{
1+4\displaystyle \frac{M_0k_1\Delta x}{N_Sk_{-1}}}}\left(1-\frac{2}{1+
\sqrt{ 1+4\displaystyle \frac{M_0k_1\Delta
x}{N_Sk_{-1}}}}\right).\label{Var}
 \end{eqnarray}
The graph of $\sigma^2_S(M_0)$ vanishes for both $M_0\to0$ and
$M_0\to\infty$ and has a unique finite maximum, as in figure \ref{variance1}

\subsection{General equations when binding proteins are located on the boundary of the domain}
We consider the same problem, but with binding sites located on
the boundary $\partial \Omega$ with surface density $S_0(\mbox{\boldmath$x$})$ (see figure \ref{boundary}). This may
represent for example binding to gated channels, which open when
they bind agonist molecules. Some channels may need to bind
several agonist molecules to open, however, we consider here the
case that a single agonist molecules opens the channel upon
binding. We assume that the agonist molecule cannot escape
$\Omega$.

The initial $M_0$ agonist molecules diffuse in $\Omega$ and are
reflected at $\partial\Omega$ at non binding sites, but can bind to a
free binding site on a protein channel in the membrane with a
certain forward binding rate $k_1$. When this occurs the channel
opens and stays open as long as the agonist is bound. The bound
agonist is released from the bound state at a backward rate
$k_{-1}$. Then the number of open channels is the number of
missing $M$ particles in $\Omega$. We write
 \begin{eqnarray*}
 M_0=\int_\Omega
 c(\mbox{\boldmath$x$},t)\,d\mbox{\boldmath$x$}+\int_{\partial\Omega}S_{\mbox{bound}}(\mbox{\boldmath$x$},t)\,dS_{\mbox{\boldmath$x$}.}
 \end{eqnarray*}
The equations for the density of $M$ particles in $\Omega$ and in
$\partial\Omega$ is
 \begin{eqnarray}
\frac{\partial c(\mbox{\boldmath$x$},t)}{\partial t}& = &D\Delta c(\mbox{\boldmath$x$},t)\nonumber\\
&&\nonumber\\
-D\frac{\partial
c(\mbox{\boldmath$x$},t)}{\partial\mbox{\boldmath$n$}}&=&\mbox{\boldmath$J$}(\mbox{\boldmath$x$},t)\cdot\mbox{\boldmath$n$}(\mbox{\boldmath$x$}){\Bigg{|}}_{\mbox{\boldmath$x$}\in\partial\Omega}=\dot S_{\mbox{bound}}(\mbox{\boldmath$x$},t)\nonumber\\
&&\nonumber\\
&=& k_1
c(\mbox{\boldmath$x$},t)[S_0(\mbox{\boldmath$x$})-S_{\mbox{bound}}(\mbox{\boldmath$x$},t)]-k_{-1}S_{\mbox{bound}}(\mbox{\boldmath$x$},t).\label{dotsb}
 \end{eqnarray}

The probability of $k$ bound sites on the boundary at time $t$
satisfies the equations
 \begin{eqnarray}
\dot P_k(t)&=&-P_k(t)K_1\oint_{\partial\Omega}c(\mbox{\boldmath$x$},t)\left[S_0(\mbox{\boldmath$x$})-
S_{\mbox{bound}}(\mbox{\boldmath$x$},t)\right]\,dS_{\mbox{\boldmath$x$}}
-kP_k(t)k_{-1}\nonumber\\
&&\nonumber
 \\&&+P_{k-1}(t)K_1\oint_{\partial\Omega}c(\mbox{\boldmath$x$},t)\left[S_0(\mbox{\boldmath$x$})
-S_{\mbox{bound}}(\mbox{\boldmath$x$},t)\right]\,dS_{\mbox{\boldmath$x$}}
+(k+1)P_{k+1}(t)k_{-1},\label{Pkdot},
 \end{eqnarray}
\begin{eqnarray}
\dot P_0(t)&=&-P_0(t)K_1\oint_{\partial\Omega}c(\mbox{\boldmath$x$},t)\left[S_0(\mbox{\boldmath$x$})-
S_{\mbox{bound}}(\mbox{\boldmath$x$},t)\right]\,dS_{\mbox{\boldmath$x$}} +P_{1}(t)k_{-1},\label{Pkdot0},
 \end{eqnarray}
 \begin{eqnarray}
\dot P_{S_0}(t)&=&-P_{S_{0}}(t)K_1\oint_{\partial\Omega}c(\mbox{\boldmath$x$},t)\left[S_0(\mbox{\boldmath$x$})-
S_{\mbox{bound}}(\mbox{\boldmath$x$},t)\right]\,dS_{\mbox{\boldmath$x$}}
-S_0P_{S_0}(t)k_{-1}\nonumber\\
&&\nonumber
 \\&&+P_{S_0-1}(t)K_1\oint_{\partial\Omega}c(\mbox{\boldmath$x$},t)\left[S_0(\mbox{\boldmath$x$})
-S_{\mbox{bound}}(\mbox{\boldmath$x$},t)\right]\,dS_{\mbox{\boldmath$x$}}
 \end{eqnarray}
where, according to eq.(\ref{dotsb})
 \begin{eqnarray*}
 \dot S_{\mbox{bound}}(\mbox{\boldmath$x$},t)= k_1
c(\mbox{\boldmath$x$},t)[S_0(\mbox{\boldmath$x$})-S_{\mbox{bound}}(\mbox{\boldmath$x$},t)]-k_{-1}S_{\mbox{bound}}(\mbox{\boldmath$x$},t).
 \end{eqnarray*}
Here $0\leq k\leq S_0$, where
 \begin{eqnarray*}
 S_0=\oint_{\partial\Omega}S_0(\mbox{\boldmath$x$})\,dS_{\mbox{\boldmath$x$}},
 \end{eqnarray*}
that is, $P_{-1}(t)=P_{S_0+1}(t)=0$.

The moments of the number of bound sites are
 \begin{eqnarray*}
 \langle S_b(t)\rangle=\sum_{k=1}^{S_0}kP_k(t),\quad \langle
 S_b^2(t)\rangle=\sum_{k=1}^{S_0}k^2P_k(t).
 \end{eqnarray*}
The variance in the number of bound sites is
 \begin{eqnarray*}
 \sigma^2(t)=\langle
 S_b^2(t)\rangle-\langle S_b(t)\rangle^2.
 \end{eqnarray*}

\section{The fluctuation in the number of particles in a push-pull chemical reaction}
A push-pull chemical reaction consists of a source that produces particles at a given rate and a sink or a killing term that destroy or remove the particles from the system with its rate. When the movement of the particles is driven by diffusion and the source and the sink are not uniformly distributed, the number of particles fluctuates.  In this section, we propose an approach to estimate the mean number and the fluctuations. A permanent regime imposes that the rate of the sink and the source satisfy some specific conditions. At equilibrium, in the limit of a large number particles,  the
push-pull mechanism reaches a steady state and the number of particles does not fluctuate and can be computed using the rate constant. But for a small number, the analysis requires to study separately the dynamics of the particles and especially the law of injection by the source.

In the neurobiological context, many biochemical reactions are based on push-pull mechanisms: at a given location, a source produces molecules and somewhere else, an enzyme modeled as a sink, destroys the molecules with a certain efficassy. In general, sinks and sources are not uniformly distributed, which induces fluctuations.  As an example, gating molecules that can open channels are produced by such push-pull mechanism, and the fluctuations in the number of molecules induce a fluctuation in the number of open  channels, which is reveals at the cellular level. It is of particular interest to examine the situation  where molecules move by diffusion, the enzyme sink is represented by a single molecule and the sources are uniformly distributed. For example, this occurs inside a compartment of photo-receptors for the regulation of cGMP molecules.

\subsection{The push-pull mechanism}\label{ppm}
The sink of the push-pull mechanism is modelled by a killing term
located at a single point. The killing term is a Dirac function,
({ see \cite{HMS} for the exact mathematical interpretation}).
The source is assumed to be uniformly distributed and molecules
are produced at a constant rate $\gamma$.

We assume that the molecules are independent and their movement
{ in a domain $\Omega$ can be described by the stochastic
differential equations
 \begin{eqnarray} \dot{\mbox{\boldmath$x$}}_k = \mb{b}(\mbox{\boldmath$x$}_k)
+\sqrt{2D}\, \dot{\mb{ w}}_k, \quad k=1,\ldots,N,
 \end{eqnarray}
where $\mb{ b}(\mbox{\boldmath$x$})$ is a drift vector}. Since the molecules cannot
escape, reflecting boundary conditions are imposed at the boundary
$\partial\Omega$. If a particle is injected in $\Omega$ at time $t=0$,
its pdf $p(\mbox{\boldmath$x$},t)$ satisfies the Fokker-Planck equation
 \begin{eqnarray} \label{dynb}
 \frac{\partial p(\mbox{\boldmath$x$},t)}{\partial t} &=&-\nabla\cdot{\mbox{\boldmath$J$}}(\mbox{\boldmath$x$},t)-k_1\delta(\mbox{\boldmath$x$}-\mbox{\boldmath$x$}_1) p(\mbox{\boldmath$x$},t)
\nonumber \\
& &\\
 \left.\mbox{\boldmath$J$}\cdot \mbox{\boldmath$n$}\right|_{\partial\Omega}& =&0,\nonumber
 \end{eqnarray}
where the probability flux { density vector} is given by
 \begin{eqnarray}
 \mbox{\boldmath$J$}(\mbox{\boldmath$x$},t)=-D\nabla p(\mbox{\boldmath$x$},t)+\mb{b}(\mbox{\boldmath$x$}) p(\mbox{\boldmath$x$},t).\label{pflux}
  \end{eqnarray}
At any moment of time, when there are $N$ particles inside
$\Omega$, the concentration $c(\mbox{\boldmath$x$},t)$ is given by
 \begin{eqnarray}
 c(\mbox{\boldmath$x$},t) = Np(\mbox{\boldmath$x$},t).
  \end{eqnarray}
Particles are injected with a Poisson stream at a rate $\gamma$,
{ that is, at independent and identically distributed
inter-injection times, whose pdf is}
 $$f(t) =\gamma e^{-\gamma t}.$$
{The mean inter-injection time is}  $E(T)=\gamma^{-1}$. The
probability that a molecule injected at time $t=0$ at a point $\mbox{\boldmath$y$}$
survives at time $t$ is
 \begin{eqnarray}
 S_{\mbox{\boldmath$y$}}(t) =Pr\{ \mbox{\boldmath$x$}(t) \in \Omega \} = \int_\Omega p(\mbox{\boldmath$x$},t\,|\,\mbox{\boldmath$y$})\,d\mbox{\boldmath$x$}.
 \end{eqnarray}
When $\mbox{\boldmath$y$}=\mb{0}$, we denote $S(t)=S_{\mb{0}}(t)$. To compute the
mean and the variance of the number of molecules $N(t)$ surviving
in $\Omega$ at time $t$, we use here the renewal equation
\cite{kleinrock,karlin,amit},
\begin{eqnarray}
\Pr\{N(t)=0 \} & = & \Pr\{ \mbox{\boldmath$x$}(t) \notin \Omega \}\cdot \left[
\int_0^
tf(s)\Pr\{N(t-s)=0\}\,ds + \int_t^\infty f(s)\,ds\right] \\
&&\nonumber\\
\Pr\{N(t)=1 \} & = & \Pr\{\mbox{\boldmath$x$}(t)\in \Omega \}\cdot \left[ \int_0^t
f(s)\Pr\{N(t-s)=0\}\,ds + \int_t^\infty f(s)\,ds \right] \nonumber \\
&&\nonumber\\ & & + \Pr\{\mbox{\boldmath$x$}(t) \notin \Omega\}\cdot \int_0^t
f(s)\Pr\{N(t-s)=1\}\,ds. \\
&&\nonumber\\ \Pr\{N(t)=n\} & = & \Pr\{\mbox{\boldmath$x$}(t)\in \Omega \}\cdot
\int_0^t
f(s)\Pr\{N(t-s)=n-1\}\,ds \nonumber \\
&&\nonumber\\ & & + \Pr\{\mbox{\boldmath$x$}(t) \notin \Omega\}\cdot \int_0^t
f(s)\Pr\{N(t-s)=n\}\,ds, \quad n > 1.
\end{eqnarray}
The expected number $EN(t)$ { of molecules surviving in
$\Omega$ at time $t$} is
\begin{eqnarray}
EN(t) & = & \sum_{n=1}^\infty n\Pr\{N(t)=n\} \nonumber \\
&&\nonumber\\
 & = & \Pr\{\mbox{\boldmath$x$}(t)\in \Omega \}\cdot \left( \int_0^t
f_L(s)EN(t-s)\,ds + \int_0^\infty f(s)\,ds \right) \nonumber
\\
&&\nonumber\\
 && + \Pr\{\mbox{\boldmath$x$}(t)\notin \Omega\}\cdot \int_0^t
f(s)EN(t-s) \,ds \nonumber \\
&&\nonumber\\ & = & \Pr\{\mbox{\boldmath$x$}(t)\in \Omega \} + \int_0^t
f(s)EN(t-s)\,ds.\label{ieq}
\end{eqnarray}
The integral equation (\ref{ieq}) is solved by the Laplace
transform as
\begin{eqnarray}
\bar n(\tau) = \frac{\bar{S}(\tau)}{1-\bar{f}(\tau)},
\end{eqnarray}
where $\bar{S}(\tau)$ is the Laplace transform of $\Pr\{\mbox{\boldmath$x$}(t)\in \Omega \}$ when the initial position of insertion is 0.
\begin{eqnarray}
\bar{f}(\tau) = \frac{\gamma}{\gamma +\tau}
\end{eqnarray}
and $\bar{S}(\tau)$ is the Laplace transform of the survival
probability. Therefore the Laplace transform of $EN(t)$ is given
by
\begin{eqnarray}
\bar n(\tau) = \frac{(\gamma+\tau) \bar{S}(\tau)}{\tau}.
\end{eqnarray}

\subsubsection{Computation of  $\bar{S}(\tau)$ in a driftless
one-dimensional model}

We consider the one-dimensional equation (\ref{dynb}) in
$\Omega=[0,L]$ and $\mb{b}(\mbox{\boldmath$x$})=\mb{0}$. We assume, for simplicity,
that $L=\pi$ and $D=1$.

The survival probability given by $S(t) = \displaystyle\int_0^L p(x,t\,|\,y)
dx$ is also (see \cite{HMS}) equal to\begin{eqnarray} S(t) =
1-k_1\int^t_0 p(x,t\,|\,y) dx.
\end{eqnarray}
The Laplace transform is
\begin{eqnarray}
\bar{S}(\tau) = \frac{1-k_1\bar{p}(x_1,\tau\,|\,y)}{\tau}.
\end{eqnarray}
$\bar{p}(x_1,\tau\,|\,y)$ can be computed using the Green's
function for the Neumann problem for (\ref{dynb}), given by
\begin{eqnarray}
G(x,t\,|\,y) =1+\frac{2}{\pi} \sum_{n=1}^\infty e^{-n^2t}\cos nx\cos ny .
\end{eqnarray}
Following the computations of \cite{HMS}, an integral representation of the solution
 of equation \ref{dynb} is given by
\begin{eqnarray}\label{dynb2}
p(x,t\,|\,y) = -k_1 \int_{0}^{t} p(x_1,s\,|\,y) G(x,t-s\,|\,x_1) \,ds+\,G(x,t\,|\,y).
\end{eqnarray}
The Laplace transform of equation (\ref{dynb2}) is given by
\begin{eqnarray}
\bar{p}(x,\tau\,|\,y) = -k_1 \bar{p}(x_1,\tau\,|\,y)
\bar{G}(x,\tau\,|\,x_1) \,ds+\,\bar{G}(x,\tau\,|\,y).
\end{eqnarray}
Thus,
\begin{eqnarray}
\bar{p}(x_1,\tau\,|\,y) = \frac{ \bar{G}(x_1,\tau \,|\,y)}{1+k_1
\bar{G}(x_1,\tau \,|\,x_1)}
\end{eqnarray}
and
\begin{eqnarray}
\bar{p}(x,\tau\,|\,y) = \frac{1}{k_1} +\frac{\bar{G}(x_1,\tau \,|\,y)-\bar{G}(x_1,\tau \,|\,x_1)-k^{-1}_{1}}{1+k_1
\bar{G}(x_1,\tau \,|\,x_1)}.
\end{eqnarray}
With
\begin{eqnarray}
\bar{G}(x,\tau \,|\,y) = \frac{1}{\tau}+\frac{2}{\pi}
\sum_{n=1}^\infty \frac{\cos nx\cos ny}{n^2 +\tau},
\end{eqnarray}
{ for $x,y \in (0,\pi)$, and to first order in $\tau$,  we
obtain}
\begin{eqnarray*}
\bar{p}(x,\tau\,|\,y)&=& \frac{1}{k_1} +\frac{\displaystyle\frac{2}{\pi}
\sum_{n=1}^\infty \left[\displaystyle\frac{\cos nx\cos ny}{n^2}
-\displaystyle\frac{\cos^2 nx}{n^2}\right]
-k^{-1}_1}{k_1+\displaystyle\frac{2k_1}{\pi} \sum_{n=1}^\infty \displaystyle\frac{\cos
nx\cos ny}{n^2}}\tau +o(\tau)\\
&&\\
&=&\frac{1}{k_1} +\frac{\displaystyle\frac{1}{2\pi}(x^2-y^2)
-k^{-1}_1}{k_1\left(1+\displaystyle\frac{2}{\pi}\left(\displaystyle\frac{\pi^2}{6}
-\displaystyle\frac{\pi}{2}x +\frac{x^2}{2}\right) \right)}\tau+o(\tau).
\end{eqnarray*}
Hence
\begin{eqnarray*}
\bar{S}(\tau)& =& \frac{1-k_1\bar{p}(x_1,\tau \,|\,y)}{\tau}\\
&&\\
&=& \frac{\displaystyle\frac{-1}{2\pi}\left(x^2-y^2\right)
+k^{-1}_1}{\left(1+ \displaystyle\frac{2}{\pi}\left(\displaystyle\frac{\pi^2}{6}
-\displaystyle\frac{\pi}{2}x +\displaystyle\frac{x^2}{2}\right) \right)}+o(1).
\end{eqnarray*}

Using the normalization condition that $S(0)=1$, {we find the
long time asymptotics}
\begin{eqnarray}
S(t) \sim_{r\rightarrow +\infty} \exp(-\alpha t),
\end{eqnarray}
where
\begin{eqnarray}\label{alphap}
\alpha^{-1} = \frac{ \displaystyle\frac{-1}{2\pi}\left(x^2-y^2\right)
+k^{-1}_1}{\left(1+\displaystyle\frac{2}{\pi}\left(\displaystyle\frac{\pi^2}{6}
-\displaystyle\frac{\pi}{2}x +\displaystyle\frac{x^2}{2}\right)\right)}.
\end{eqnarray}
The un-normalized time constant for $x,y \in [0,L]$ is
\begin{eqnarray}\label{alpha}
\alpha^{-1} = \frac{ \displaystyle\frac{-\pi^3}{2D}\left(x^2-y^2\right)
+k^{-1}_1}{\left(1+\displaystyle 2\pi\left(\displaystyle\frac{1}{6}
-\displaystyle\frac{1}{2L}x +\displaystyle\frac{x^2}{L^2}\right)\right)}.
\end{eqnarray}
\subsubsection{The steady state limit}
{In the steady state, the mean number of molecules $N(t)$
surviving in $\Omega$ at time $t$ is}
\begin{eqnarray}
\lim_{t\rightarrow \infty} EN(t) = \lim_{\tau\rightarrow 0} \tau \bar{n}(\tau) = \gamma  \bar{S}(0)
\end{eqnarray}
Note that
 \begin{eqnarray*}
\bar{S}(0) =\int_{0}^\infty \int_0^L  p(x,t|0)\, dx.
 \end{eqnarray*}
If at time $t=0$ the particle is injected randomly at a point $y$,
we index $S$ by $y$ and { write}
 \begin{eqnarray*}
\bar{S_{y}}(0) =\int_{0}^\infty \int_0^L  p(x,t|y) dx = \int_0^L G(x|y) dy
 \end{eqnarray*}
{ where} $G(x\,|\,y)$ is { the} solution of
\begin{eqnarray*}
-\delta(x-y) &=& D\frac{\partial^2 G}{\partial x^2} -k \delta(x-x_1) G\\
&&\\
 \frac{\partial G(0\,|\,y)}{\partial x} &=& \frac{\partial G(L\,|\,y)}{\partial x} = 0.
\end{eqnarray*}
Thus,
\begin{eqnarray*}
G(x\,|\,y)= -\frac{\theta(x-y)}{D} +\frac{\theta(x-x_1)}{D}
+\frac{1}{k_1},
\end{eqnarray*}
{ where $\theta(x)$ is the integral of the Heaviside unit step function. It
follows that}
\begin{eqnarray*}
\bar{S}_y(0)=\int_0^L G(x\,|\,y) dy =  -\frac{(L-y)^2}{2D}
+\frac{(L-x_1)^2}{2D} +\frac{L}{k_1}.
\end{eqnarray*}
At steady state, the mean number of molecules surviving in
$\Omega$ at time $t$ is given by
\begin{eqnarray}
N_y(\infty)= \gamma \bar{S}_y(0)= \gamma (-\frac{(L-y)^2}{2D} +\frac{(L-x_1)^2}{2D} +\frac{L}{k_1})
\end{eqnarray}
{ If the particle is initially uniformly distributed}, $\Pr\{ y
\in[x,x+dx]\}=\displaystyle\frac{dx}{L}$, then the steady state { mean
number of surviving molecules} is given by
\begin{eqnarray}\label{mean}
N(\infty)=\gamma E_y[\bar{S_{y}}(0)]=\gamma \left( -\frac{L^2}{6D}
+\frac{(L-x_1)^2}{2D} +\frac{L}{k_1} \right).
\end{eqnarray}

\subsubsection{Variance}
The second moment of the total { number of surviving particles
is defined as}
\begin{eqnarray*}
EN^2(t)=\sum_{n=1}^\infty n^2\Pr\{N(t)=n\}.
\end{eqnarray*}
We will now compute such number using the renewal equations. We have
\begin{eqnarray*}
EN^2(t)&=&\Pr\{x(t)\in \Omega \}\int_0^t f(s) k^2
\Pr\{N(t-s)= k-1\}\,ds \\
&&\\
&&+\Pr\{x(t)\notin [0,L] \}\int_0^t f(s) k^2
\Pr\{N(t-s)=k\}\,ds+\Pr\{x(t)\notin  \Omega\}\int_t^{\infty}
f(s)\,ds,
\end{eqnarray*}
which leads to
\begin{eqnarray*}
EN^2(t)&=&\Pr\{x(t)\in [0,L] \}\int_t^{\infty}
f(s)\,ds+\Pr\{x(t)\in \Omega \}\int_0^t EN^2(t-s) f(s)\,ds +\\ &
&\Pr\{x(t)\notin [0,L] \} \int_0^t EN^2(t-s) f(s)\,ds \\&+&
\sum_k\Pr\{x(t)\in \Omega \} \int_0^t(2k+1)Pr\{ N(t)=k-1 \}
f(s)\,ds.
\end{eqnarray*}
Thus,
\begin{eqnarray*}
EN^2(t) & = & \int_0^t EN^2(t-s) f(s)\,ds +\Pr\{x(t)\in \Omega \}\int_t^{\infty} f(s)\,ds+\\
& & 2\Pr\{x(t)\in \Omega\}\int_0^t E(N(t-s) f(s)\,ds +\Pr\{x(t)\in
\Omega\} \int_0^t f(s)\,ds
\end{eqnarray*}
and
\begin{eqnarray} \label{fluct}
EN^2(t) =  \int_0^t EN^2(t-s)  f(s)\,ds + \Pr\{x(t)\in \Omega\} +
2\Pr\{x(t)\in \Omega \}\int_0^t f(s) E(N(t-s)) ds.
\end{eqnarray}

{\bf \noindent Remark.}\par
\noindent The variance can also be written in the form
\begin{eqnarray*}
&&\sigma^2_N(t)=EN^2(t)- [E(N(t))]^2 = \Pr\{x(t)\in  \Omega\} - \Pr\{x(t)\in  \Omega\}^2 +\\
&&\\
&&\int_0^t EN^2(t-s)  f(s)\,ds -\left( \int_0^t E(N(t-s)) f(s)\,ds
\right)^2.
\end{eqnarray*}

\subsubsection{The steady state variance}
To compute the variance in the steady state limit, as $t
\rightarrow \infty$, { we} Laplace transform equation
(\ref{fluct}),
\begin{eqnarray*}
\bar{n^2}(\tau) = \frac{\bar{S}(\tau)}{1-\bar{f}(\tau)}
+\frac{2}{1-\bar{f}(\tau)} \int^\infty_{0} \left[S(t)\int_{0}^t
f(t-s)n(s)\,ds\right] e^{-t\tau}\,dt.
\end{eqnarray*}
{It follows that}
\begin{eqnarray*}
EN^2(\infty)=\lim_{t\rightarrow \infty} EN^2(t)=  \lim_{\tau
\rightarrow 0}\tau\bar{n^2}(\tau).
\end{eqnarray*}
{ Writing}
\begin{eqnarray*}
\tau\bar{n^2}(\tau)= I(\tau) +II(\tau),
\end{eqnarray*}
{ we obtain the short $\tau$ approximation}
\begin{eqnarray*}
I(\tau)= \tau \frac{\bar{S}(\tau)}{1-\bar{f}(\tau)} \sim \gamma
\bar{S}(0).
\end{eqnarray*}
To { evaluate}
\begin{eqnarray*}
II(\tau)=\tau\frac{2}{1-\bar{f}(\tau)} \int^\infty_{0}
\left[S(t)\int_{0}^t f(t-s)n(s)\,ds\right] e^{-t\tau}\,dt,
\end{eqnarray*}
{ we recall} that $f(t) = \gamma e^{-\gamma t}$ and {
approximating $S(t)$ by $e^{-\alpha t}$ and interchanging the
order of integration, we obtain}
\begin{eqnarray*}
II(\tau)&=&\tau\gamma \frac{2}{1-\bar{f}(\tau)} \int^\infty_{0} n(s)e^{\gamma s}
\int_{s}^\infty  e^{-t(\gamma+\tau+\alpha )}dt, \\
&&\\
&=&\tau\gamma \frac{2}{1-\bar{f}(\tau)}\frac{1}{\gamma+\tau+\alpha } \int^\infty_{0} n(s)
e^{-t(\tau+\alpha )}dt,\\
&&\\
&=&\tau\gamma \frac{2}{1-\bar{f}(\tau)}\frac{1}{\gamma+\tau+\alpha }\bar{n}(\tau+\alpha )\\
&&\\
 &\sim& 2 \gamma^2\frac{1}{\gamma+\alpha }\bar{n}(\alpha ).
\end{eqnarray*}
Finally,
\begin{eqnarray*}
EN^2(\infty)=\gamma \bar{S}(0)+ 2 \gamma^2\frac{1}{\gamma+\alpha
}\bar{n}(\alpha )
\end{eqnarray*}
and
\begin{eqnarray} \label{noise}
\sigma^2 =EN^2(\infty)- \left(EN(\infty)\right)^2=\gamma
\bar{S}(0)-\gamma^2 \bar{S}^2(0) + \frac{2 \gamma^2}{\gamma+\alpha
}\bar{n}(\alpha ),
\end{eqnarray}
where  $\bar{S}(0)= \displaystyle\frac{N(\infty)}{\gamma}$, { the time}
$\alpha$ depends only on $k_1$, the diffusion constant and the
length $L$, as given in formula (\ref{alpha}), and $\bar{n}(\alpha
) =\displaystyle \frac{\gamma+ \alpha}{\alpha} \bar{S}(\alpha)$, where
$\bar{S}(\alpha )$ is the Laplace transform of the survival
probability at time $\alpha$.

\subsection{Push-pull chemical reaction in the continuum limit}
When the number of reacting molecules is large enough, regulated
by a  push-pull mechanism of hydrolysis and synthesis, the
reaction-diffusion equations is sufficient to  describe the
evolution of such system.  We assume that the molecules diffuse
inside a domain $\Omega$, but the synthesis occurs only uniformly
in a domain $\Omega' \subset \Omega$ while the hydrolysis is
performed by isolated enzymes. The hydrolysis can be modelled as a
killing measure $k(\mbox{\boldmath$x$})$ (see \cite{HMS}). When there are a
discreet number $N_h$ of killing sources, the killing measure
$k(\mbox{\boldmath$x$},t)$ is the sum of $\delta$ measures given by \begin{eqnarray*} k(\mbox{\boldmath$x$}) =
\sum_{1}^{N_h} k_1\delta(\mbox{\boldmath$x$}-\mbox{\boldmath$x$}_k), \end{eqnarray*} where $k_1$ is the
arrival rate constant to the killing sources. If the boundary of
the domain is divided into an absorbing part $\partial\Omega_a$ and a
reflective part $\partial\Omega_b$, the concentration $c(\mbox{\boldmath$x$},t)$ can be
defined as the probability density function $p(\mbox{\boldmath$x$},t)$ of the
system of N particles as \begin{eqnarray*} c(\mbox{\boldmath$x$},t)= Np(\mbox{\boldmath$x$},t), \end{eqnarray*} and \begin{eqnarray*}
\frac{\partial c(\mbox{\boldmath$x$},t)}{\partial t} &=& D \Delta c(\mbox{\boldmath$x$},t) -k(\mbox{\boldmath$x$},t) c(\mbox{\boldmath$x$},t) +\gamma \chi_{\Omega'} \\
&&\\
\frac{\partial c(\mbox{\boldmath$x$},t)}{\partial n}\bigg|_{\partial \Omega_r }  &=& 0 ,\\
&&\\
c(\mbox{\boldmath$x$},t)\bigg|_{\partial \Omega_a}&=& 0, \\
&&\\
c(x,0) &=& c_0(x),
 \end{eqnarray*}
where
 \begin{eqnarray*} \int_{\Omega} c_0(\mbox{\boldmath$x$}) d\mbox{\boldmath$x$} = N.
  \end{eqnarray*}
The mean number of particles $N(t)$ at any moment of time is given
by
 \begin{eqnarray*} N(t) = \int_{\Omega} c(\mbox{\boldmath$x$},t) dx.
  \end{eqnarray*}
When the particles are contained inside a fixed domain and cannot
escape,$\Omega_a $ is empty. { The steady state density
$c(\mbox{\boldmath$x$})=\lim_{t\to \infty}c(x,t)$ is a solution of}
 \begin{eqnarray*}
0 &=& D \Delta c(\mbox{\boldmath$x$}) -k(\mbox{\boldmath$x$}) c(\mbox{\boldmath$x$}) +\gamma \chi_{\Omega'} \\
&&\\
0 &=& \frac{\partial c(\mbox{\boldmath$x$})}{\partial n}\bigg|_{\partial \Omega_r }.
 \end{eqnarray*}
{ In one dimension} $\Omega'=\Omega=[0,L]$ and the killing
measure is reduced to a single point. The steady state equation is
\begin{eqnarray}\label{sst}
0 &=& D\frac{\partial^2 c(x)}{\partial x^2} -k_1 \delta(x-x_1) c(x) +\gamma,\\
&&\nonumber\\
0 &=&\frac{\partial c(x)}{\partial x}\bigg|_{x=0} =\frac{\partial c(x)}{\partial
x}\bigg|_{x=L}, \nonumber
 \end{eqnarray}
{ where} $k_1$ is the forward rate for one active site ({ it
has units of length per time}),  $\gamma$ is the uniform injection
rate of particle due to synthesis ({ it is the number of
injected particles per second per unit length}). Two integrations
of equation (\ref{sst}) lead to the solution
 \begin{eqnarray}
c(x) = \gamma \left[\frac{L\theta(x-x_1)}{D} -\frac{x^2}{2D}
+\left(\frac{L}{k_1} +\frac{x_1^2}{2D}\right)\right],
 \end{eqnarray}
where $\theta$ is the integral of the Heaviside function:
\begin{eqnarray*}
\theta(x)=\left\{
\begin{array}{lll}
0&&\hbox{for $x \leq 0$} \\
&&\\
x& &\hbox{for $x >0$}
\end{array}
\right.,
\end{eqnarray*}
Finally, the total number of free molecules is given by
\begin{eqnarray*}
N_0 =\int_{0}^L c(x)dx =\frac{\gamma L}{2D}(L-x_1)^2 -
\frac{\gamma L^3}{6D} +\gamma L \left(\frac{L}{k}
+\frac{x^2_1}{2D}\right),
\end{eqnarray*}
which should be compare to the steady state mean number obtained
in the discrete case (\ref{mean}).
\section{Markovian model of cell noise}\label{Markov}
We propose in this section an alternative model for computing the fluctuation of the number of bound molecules, based on a Markovian approach. We consider a domain $\Omega\subset\hbox{\bb R}^3$ with $M$ mobile agonist
molecules and $S$ receptors, embedded in the boundary $\partial\Omega$,
that open channels when they bind a single mobile agonist
molecule. We assume that the receptors occupy a small portion of
the surface area of $\partial\Omega$. The agonists diffuse in $\Omega$
independently of each other. Bound agonists are released
independently of each other at exponential waiting times with rate
$k_{-1}$.

For a single receptor and a single agonist the time to binding is
the first passage time {to diffuse} to a small portion of the
boundary, $\partial\Omega_a$, which is absorbing and represents the
active surface of the receptor, whereas the remaining part of
$\partial\Omega$ is reflecting. It can be shown \cite{HS} that the
probability distribution of the first passage time to $\partial\Omega_a$
is approximately exponential with rate
 \begin{eqnarray*}
 \lambda_1=\frac{1}{\langle\tau_1\rangle},
 \end{eqnarray*}
where $\langle\tau_1\rangle$ is the mean first passage time to
$\partial\Omega_a$.

When there are $S$ channels, $k(t)$ of which are free at time $t$,
and $M$ agonists (gating molecules),
 \begin{eqnarray}
 N(t)=(M-S+k(t))^+\label{Nk}
 \end{eqnarray}
of which are free to diffuse in $\Omega$ at time $t$, where
\begin{eqnarray*}
 x^+=\max\{0,x\}.
 \end{eqnarray*}
Thus $$(S-M)^+\leq k(t)\leq S.$$

We assume that the pdf of the time for the next receptor to bind
is well approximated by the exponential pdf with instantaneous
rate
 \begin{eqnarray*}
 \lambda_k(t)=\frac{N(t)k(t)}{\langle\tau_1\rangle}.
 \end{eqnarray*}
This assumption is justified if the total area of the absorbing
boundary (the channels) is small relative to the surface area of
the {reflecting} boundary \cite{HS}.

It follows that when $N(t)=N$ and $k(t)=k$, the mean time to bind
is
 \begin{eqnarray*}
\lambda_k=\frac{Nk}{\langle\tau_1\rangle}=\frac{k(M-S+k)^+}{\langle\tau_1\rangle}.
 \end{eqnarray*}
The number of bound receptors at time $t$ is a birth-death process
with states $0,1,2,\dots,\min\{M,S\}$ and transition rates
 \begin{eqnarray*}
 \lambda_{k\to k+1}=\lambda_k,\quad\lambda_{k\to
 k-1}=\mu=k_{-1}.
 \end{eqnarray*}
The boundary conditions are
 \begin{eqnarray*}
 \lambda_{S\to S+1}=0,\quad \lambda_{0\to-1}=0.
 \end{eqnarray*}

Setting
 \begin{eqnarray*}
 P_k(t)=\Pr\{k(t)=k\},
 \end{eqnarray*}
we have  \cite{Saaty}
 \begin{eqnarray*}
\dot P_{(S-M)^+}(t)&=&-k_{-1}SP_{(S-M)^+}(t)+\lambda_{1}P_{(S-M)^++1}(t) \\
&&\\
 \dot
P_k(t)&=&-\left[\lambda_{k}+k_{-1}(S-k)\right]P_k(t)+\lambda_{k+1}P_{k+1}(t)+k_{-1}(S-k+1)P_{k-1}(t)\\
&&\\
&&\quad\mbox{for}\quad
k=(S-M)^++1,\dots,S-1\nonumber\\
&&\nonumber\\
\dot P_S(t)&=&-\lambda_SP_S(t)+k_{-1}P_{S-1}(t).
 \end{eqnarray*}
The initial condition is $P_{k,q}(0) = \delta_{k,S} \delta_{q,0}$.
In the steady state the average number of open channels is
 \begin{eqnarray*}
 \langle k_\infty\rangle=\sum_{j=(S-M)^+}^{S}jP_j,
 \end{eqnarray*}
where $P_j=\displaystyle\lim_{t\to\infty}P_j(t)$, and the stationary
variance in the number of open channels is determined from the
second moment
 \begin{eqnarray*}
 \langle k^2_\infty\rangle=\sum_{j=(S-M)^+}^{S}j^2P_j
 \end{eqnarray*}
by
 \begin{eqnarray*}
 \sigma^2(M,S)=\langle k_\infty^2\rangle-\langle k_\infty\rangle^2.
 \end{eqnarray*}

\subsection{The steady state approximation}
In the steady state
    \begin{eqnarray*}
0&=&-k_{-1}SP_{(S-M)^+}+\lambda_{1}P_{(S-M)^++1} \\
&&\\
0&=&-\left[\lambda_{k}+k_{-1}(S-k)\right]P_k+\lambda_{k+1}P_{k+1}+k_{-1}(S-k+1)P_{k-1}\\
&&\\
&&\quad\mbox{for}\quad
k=(S-M)^++1,\dots,S-1\nonumber\\
&&\nonumber\\
0&=&-\lambda_SP_S+k_{-1}P_{S-1},
 \end{eqnarray*}
which gives for $(S-M)^+\leq k\leq S$
 \begin{eqnarray*}
 P_{S-1}&=&P_S\frac{\lambda_S}{k_{-1}}\\
 &&\\
 P_{S-k}&=&P_S\frac{\prod_{i=S-k+1}^S\lambda_i}{k!k_{-1}^k}=P_S\frac{\prod_{i=S-k+1}^S i(M-S+i)^+}{k!(\langle\tau_1\rangle k_{-1})^k}\\
 &&\\
 P_{(S-M)^+}&=&P_S\frac{\prod_{i=(S-M)^++1}^S\lambda_i}{(S-(S-M)^+)!k_{-1}^{S-(S-M)^+}}
 =P_S\frac{\prod_{i=(S-M)^++1}^S i(M-S+i)^+}{(S-(S-M)^+)!(k_{-1}\langle \tau_1
 \rangle)^k}.
 \end{eqnarray*}
{ The} constant $P_0$ is determined from the normalization
condition
 \begin{eqnarray*}
 \sum_{k=(S-M)^+}^{S}P_k= \sum_{k=S-(S-M)^+}^{0}P_{S-k}    =1.
 \end{eqnarray*}
Thus,
 \begin{eqnarray} \label{moments}
 P_S &=& \frac{1}{1+\sum_{k=1}^{S-(S-M)^+} \displaystyle\frac{\prod_{i=S-k+1}^S \lambda_i}{k!k_{-1}^k}}=\displaystyle
\frac{1}{1+\displaystyle\sum^{S-(S-M)^+}_{k=1}\displaystyle\frac{\prod_{i=S-k+1}^S
i(M-S+i)^+ }{k!(\langle\tau_1\rangle k_{-1})^k }}
\nonumber\\
& & \nonumber \\
\langle k_\infty \rangle &=&   \sum_{k=S-1}^{(S-M)^+} (S-k)^+ P_{S-k} =  P_S  \sum_{k=S-1}^{(S-M)^+} (S-k)^+ \frac{\prod_{i=S-k+1}^S i(M-S+i)^+}{k!(\langle\tau_1\rangle k_{-1})^k}
\nonumber  \\
& &\nonumber\\
 \langle k^2_\infty\rangle&=&  \sum_{k=S-1}^{(S-M)^+} [(S-k)^+]^2 P_{S-k} =  P_S  \sum_{k=S-1}^{(S-M)^+} [(S-k)^+]^2 \frac{\prod_{i=S-k+1}^S i(M-S+i)^+}{k!(\langle\tau_1\rangle k_{-1})^k} \nonumber \\
& &\nonumber\\
\sigma^2_S(M) &=&  \langle k^2_\infty\rangle - \langle
k_\infty\rangle^2.
 \end{eqnarray}
The graph of $\sigma^2_S(M)$ {\em vs} $M$ is given  in Figure
\ref{varianceMarkov}.

\subsection{The steady state mean number of open channels}
Using the above Markov model an explicit formula can be
obtained for the mean number of open channels ( that bind an
agonist molecule). It is assumed here that a channel can bind only
a single agonist molecule at a time. This approach does not give
any information about the fluctuations.

{ The steady state fraction $n^*_1$ of bound agonist molecules
in a bounded domain $\Omega$ that contains $S$ channels and $M$
gating agonists is}
 \begin{eqnarray}
n^*_1 & =&\frac{k_1}{k_1+k_{-1}} M,\quad \\
& & \label{equil}
\\
n_1   & =&\frac{k_{-1}}{k_1+k_{-1}}M,
 \end{eqnarray}
where $k_1$ is the forward binding rate in solution and $k_{-1}$
is the backward rate. If $S_{ch}$ is the total surface occupied by
the channels, then $S_{ch} = S S^1_{ch}$, where $S^1_{ch}$ is the
effective surface occupied by a single channel.  $S^1_{ch}$ is
defined by the area where the electrostatic force of the channel
is { sufficient to bind an agonist}. Recall that $k_1$ is by
definition { the mean} number of { agonist molecules}
arriving at the channel per unit of time. Using the result of the
previous section, { we can write}
 \begin{eqnarray}\label{iden} k_1
=\frac{S_{ch}}{\tau_{n_1 }(M-S+n_1)^+}.
 \end{eqnarray}
Thus, {using equations (\ref{equil}) and (\ref{iden}), we find
that the mean number of bound agonist, which is the same as that
of bound channels}, is given by
 \begin{eqnarray}
n^*_1&=&\frac{1}{1+\displaystyle\frac{\tau_1
k_{-1}}{S_{ch}}(M-n^*_1)(S-n^*_1)}M,
 \end{eqnarray}
{ with
 \begin{eqnarray}
 \langle \tau_1 \rangle =\frac{|\Omega|}{\pi D} \log
\frac{S_{ch}}{|\partial \Omega|},
 \end{eqnarray}
where $|\Omega|$ is the volume of the $\Omega$} and $|\partial \Omega|$
is the surface area { of its boundary} (see \cite{HS}). For
$n^*_1$ small relative to $M$ and $S$,
 \begin{eqnarray}
n^*_1 & \sim &\frac{1}{1+\displaystyle\frac{\tau_1 k_{-1}}{S_{ch}}MS}M\\
&&\\
& = &\frac{1}{1+\displaystyle\frac{\tau_1 k_{-1}}{S^1_{ch}}M}M,
 \end{eqnarray}
which gives explicitly the number of bound channels as a function
of the geometrical parameters. A similar formula can be derived,
when a single channel can be bound to several gating molecules.

\subsection{The Michaelis-Menten law in micro-structures}
The rate of production of a product $P$ from a substrate $M$ by a
catalytic enzyme $E$ is usually described  in text books by the Michaelis-Menten
law for reactions in solution. In confined micro-domains, the above analysis can be used to estimate the
number of $P$ produced in the reaction.

The  chemical reaction is described by
\begin{equation}
M+E
\begin{array}{l}
k_{f1} \\
\rightleftharpoons \\
k_{b1}%
\end{array}%
ME
\begin{array}{l}
\overset{k_{b2}}{\rightharpoonup } \\
\end{array}
E+P.  \label{MMR2}
\end{equation}
and a master equation for the joint probability that the number of
$P$ molecules produced is $k$ and $q$ enzymes are bound { at
time $t$}, $ P_{k,q}(t) = Pr \{P(t)=k, E(t)=q \}$, can be derived
{ as above}. In particular, {the kinetics of the reaction
(\ref{MMR2}) are}
 \begin{eqnarray*}  \label{MM1}
\dot{P}_{k,q}(t) &=& (1-q(k_{-1}+k_2)-\lambda_{M,q} )P_{k,q}(t) +
k_2 (q+1)
P_{k-1,q+1}(t) +k_{-1}(q+1)P_{k,q+1}(t)\\
&&\\
&&+ \lambda_{M,q-1} P_{k,q-1}(t),\quad\hbox{ for }  1\leq q, k \leq E_0-1 \\
&&\\
\dot{P}_{0,q}(t) &=& (1-\lambda_{M,q} )P_{0,q}(t) +k_{-1}(q+1)P_{0,q+1}(t)+
\lambda_{M,q-1} P_{0,q-1}(t),\\
&&\\
\dot{P}_{k,E_0}(t) &=& (1-k(k_{-1}+k_2) )P_{k,E_0}(t) + k_2 q P_{k-1,E_0}(t)
+ \lambda_{M,E_0-1} P_{k,q-1}(t),\\
&&\\
\dot{P}_{M_0,0}(t) &=& (1-M_0(k_{-1}+k_2) )P_{M_0,0}(t) + k_2
P_{M_0-1,0}(t),
  \end{eqnarray*}
{with the initial condition ${P}_{k,q}(0) = \delta_{M-0,0}$
and}
 \begin{eqnarray*}
\lambda_{M,q} &=& \frac{M(E_0-k)}{\tau_1},\\
&&\\
M &=& (M_0-k-q)^+.
 \end{eqnarray*}
{Here} $M_0$ is the initial number of substrate {
molecules}, $M$ is the number of { unbound} substrate
molecules, and $E_0$ is the total number of enzyme available. The
mean  number of $P$ produced at time $t$ is
 \begin{eqnarray}
\langle P(t) \rangle = \sum_k k Pr\{ P(t)=k \} =  \sum_k k \sum_q Pr\{ P(t)=k,E(t)=q \}.
 \end{eqnarray}
The only steady state solution of equation (\ref{MM1}) is zero as any steady state.

\subsection{Fluctuations in a push-pull system with binding}
We consider now a push-pull system, where gating molecules can
also bind and unbind to some proteins. {We consider two main
approaches to the description of this dynamics}. The first
approach is based on the procedure used in the first model in {Section} \ref{firstM}
and consists in deriving an equation for the
joint probability of a trajectory, the number of bound sites in
the volume $\Delta \mbox{\boldmath$x$}$, and the total number of { molecules} in
$\Omega$,
 \begin{eqnarray}
p(\mbox{\boldmath$x$},S,M,t\,|\,\mbox{\boldmath$y$})\,\Delta\mbox{\boldmath$x$}= \Pr\left\{\mbox{\boldmath$x$}(t)\in\mbox{\boldmath$x$}+\Delta\mbox{\boldmath$x$},\,
S_{\Delta}(\mbox{\boldmath$x$},t)=S\,, M |\,\mbox{\boldmath$x$}(0)=\mbox{\boldmath$y$}\right\}.
\label{jpdfp}
 \end{eqnarray}
The function $p(\mbox{\boldmath$x$},S,M,t\,|\,\mbox{\boldmath$y$})$ is the joint probability density
to find an agonist and $S$ free binding sites at $\mbox{\boldmath$x$}+\Delta \mbox{\boldmath$x$}$ at
time $t$, and $M$ agonists in the domain, conditioned by the
initial position $\mbox{\boldmath$y$}$ of the agonist. An analysis { similar to
that of Section  \ref{firstM} leads to}
 \begin{eqnarray} \frac{\partial p(\mbox{\boldmath$x$},S,M,t)
}{\partial t} &=& -\nabla\cdot\mb{J}(\mbox{\boldmath$x$},S,M,t)
 - K_1 Mp^2(\mbox{\boldmath$x$},S,M,t)S -k_{-1}[S_0(\mbox{\boldmath$x$})-S]p(\mbox{\boldmath$x$},S,M,t) \nonumber\\
&&\nonumber\\
&+& K_1 M
(S+1)p^2(\mbox{\boldmath$x$},S+1,M,t)+k_{-1}[S_0(\mbox{\boldmath$x$})-S+1]p(\mbox{\boldmath$x$},S-1,M,t)+\nonumber\\
&&\nonumber\\
 & &\gamma p(\mbox{\boldmath$x$},S,M,t) -(M-S)^+
K_{-1}p(\mbox{\boldmath$x$},S,M,t)\label{eqfff},
 \end{eqnarray}
where by definition $\mb{J}(\mbox{\boldmath$x$},S,M,t)$ is the joint probability flux
at position $\mbox{\boldmath$x$}$ at time $t$, and $S$ proteins are free for $M$ agonists. It is
defined in the diffusion case by
 \begin{eqnarray} \mb{J}(\mbox{\boldmath$x$},S,M,t) = -D \nabla p(\mbox{\boldmath$x$},S,M,t).
 \end{eqnarray}
{The} new (forward) binding rate is
\begin{eqnarray*}
K_1 = k_1 \Delta\mbox{\boldmath$x$},
\end{eqnarray*}
$\gamma$ is the uniform production rate, and $K_{-1}$ is the
uniform killing rate for the $(M-S)^+$ free {agonist
molecules. The moments can be calculated from $p(\mbox{\boldmath$x$},S,M,t)$, as
above.}

The second procedure consists in adding directly the push-pull
effect in the Markovian model. Defining the joint probability that
$k$ channels are free and $M(t)$ are in the reacting domain at
time $t$,
 \begin{eqnarray} P_{k,q}(t)=\Pr\{k(t)=k,M(t)=q \},
  \end{eqnarray}
{ neglecting} the distribution of the sources and sinks
responsible for the fluctuation of the number of molecules, {
we consider} only the case { that} hydrolysis and synthesis
{ occur at exponential waiting times (i.e., are Poissonian),
with rates $K_{-1}$ and $\gamma$, respectively}. Following the
same steps as in Section \ref{Markov}, the master equation for
$P_{k,q}(t)$ becomes
 \begin{eqnarray*}
\dot P_{(S-M)^+,q}(t)&=&-k_{-1}SP_{(S-M)^+,q}(t)+\lambda_{1,q}(q-S+1)^{+}P_{(S-M)^++1,q}(t) \\
&&\\
\dot P_{k,q}(t)&=&-(\lambda_{k,q}+(S-k)k_{-1} +\gamma+K_{-1}q)P_{k,q}(t)+\\&&\\
& &\lambda_{k+1,q}P_{k+1}(t))+k_{-1}(S-k+1)P_{k-1,q}(t)+ \gamma P_{k,q-1} +K_{-1}(q+1)P_{k,q+1},\\&&\\
&&\mbox{for}\quad k=(S-M)^++1,\dots,S-1,\nonumber
 \end{eqnarray*}
where we recall that
\begin{eqnarray*}
\lambda_{k,q} =\frac{Nk}{\langle\tau_1\rangle}=\frac{k(M-q+k)^+}{\langle\tau_1\rangle}.
 \end{eqnarray*}
The initial condition is $P_{k,q}(0) = \delta_{k,S} \delta_{q,0}$.

Finally, if the push-pull rate is much slower than the binding
rate, the previous results of the Markovian model can { be}
used directly and the mean number of free channels, $N_\infty(t)$,
is directly computed using the moments (\ref{moments}) and
 \begin{eqnarray*}
N_\infty(t) = \sum_{M} \langle k_\infty\rangle(M) \Pr\{N(t)=M \},
\end{eqnarray*} where $\langle k_\infty\rangle(M)$ is the mean number of
bound channels when there are $M$ gating molecules in { the
domain,}
 \begin{eqnarray*}
\langle k_\infty\rangle(M) = \sum_{k=S-1}^{(S-M)^+} (S-k)^+
P_{S-k}(q).
 \end{eqnarray*}
{ Here, by definition, the dependence of $P_{S-k}$ on $q$ is
that obtained} in equation (\ref{moments}), and $\Pr\{N(t)=m \}$
is the probability that $M$ molecules are in the reacting domain
at time $t$ (see Section \ref{ppm}). In the limit $t\to\infty$
 \begin{eqnarray*}
\langle N_\infty(\infty)\rangle &=& \sum_{M} \langle k_\infty\rangle(M) \Pr\{N(\infty)=M \}\\
&&\\
 \langle N^2_\infty(\infty)\rangle &=& \sum_{M} \langle
k_\infty\rangle^2(M) \Pr\{N(\infty)=M \}. \end{eqnarray*}

\section{Conclusion and biological implications}

\subsection{Comparison of the models}
We presented here two models that describes fluctuation due to
binding and binding of agonist to some fixed  proteins.  In the
Markovian model, the only { geometric feature of the cell that
enters the model is the cell's volume}. { The distribution of
the channels, as well as other geometric features are ignored}. The
{ advantage} of such approach is that it gives explicit
estimates of the mean and the variance (see equation \ref{moments}). In the first model, more
details of the geometry and organization of the channel { are captured, however, the computation of the moments requires the solution of a} system of partial differential equations. The expression of the variance
is given in general by equation \ref{sd} and in steady state by expression \ref{Var}.

\subsection{The Forward binding rate}
The Markovian model does not rely on the forward binding rate and
it is replaced here by the mean  arrival time to the channels
$\langle\tau_1\rangle$. The initial forward binding { rate} per
molecule is the arrival rate of a gating molecule { to any one
of the binding sites on the $N$ proteins}, and can be defined as
 \begin{eqnarray} K_1 =
\frac{1}{\langle\tau_1\rangle N}.
 \end{eqnarray}
{ The number $N$ can be, for example, the number of proteins in
a given volume. It} can be computed from the concentration $[N]$.
In that case $K_1$ can be rewritten as
 \begin{eqnarray}
K'_1 = \frac{1}{\langle\tau_1\rangle[N]}.
 \end{eqnarray}
The Markovian approach proves that the traditional forward binding
rate has to be { abandoned} and a dynamical rate has to be used
{ instead}. The first model uses another definition of the
forward binding rate, that can be related to the traditional one.

\subsection{Biological implications}
The mean and the variance of the number
of bound molecules were derived in the first model for a { finite system of
molecules.} In particular formula (\ref{sigma2}) proves that the
variance depends { on the total number of bound molecules at
time} $t$ $\bar{S_0}(t)$ ({ see} formula (\ref{Pro})). {
Note} that $\bar{S_0}(t)$ is the integral of the joint probability
{ density function} $p(x,1,t)$, which is a solution of a
partial differential equation. Thus the number of bound molecules
is a complicated function of the geometry, the distribution of the
substrate molecule and the interaction between the molecules
through the number of binding sites.

As seen in the experimental data \cite{Yau,Koko,matthews,sigworth}, the variance
(\ref{sigma2}) does not only reflect binding to proteins, but {
also} a dynamical process of binding and unbinding in a
microstructure, which involves the geometry, the distribution of
the protein and the diffusion process. Only at high concentration
of binding molecules, a molecule that leaves a binding site will be
immediately replaced by another gating molecule. In the regime
{ of high concentration}, when the protein represents a gating
channel located on the { cell} membrane, the fluctuation of the
current represents { an} intrinsic property of the channel.
The fluctuations, then, are proportional to the gating property of
the channel. But  for  many sensor cells, the concentration of
gating molecules is not high, the small number of gating molecules
is the cause of perpetual fluctuations due to binding and unbinding.

The graphs \ref{variance1} and \ref{varianceMarkov} of the current noise variance {\em vs} $\bar{S_0}(t)$
has almost the shape of an inverted parabola, which implies that there it has unique maximum point at a specific location. In the steady state limit, it is obtained in case of the first model for $\bar{S_0}(t)=1/2$. Biological microsystems, such as photo-receptors, usually operate at low noise levels, far from the maximum of the graph.

Controlling the membrane voltage fluctuation for sensor cells is a
crucial issue for the transduction process that consist in
transmitting a molecular signal. Channels fluctuation is not
responsible only for the noise of the cell, but the biochemistry
underneath and the organization of the cell play a crucial role.
In a set of experiments \cite{Yau,Koko,matthews,sigworth}, the
membrane noise was measured but the purpose was to identify the
properties of the channel rather than the molecular dynamics. Using
the results of the previous mathematical analysis, it is a hard
problem to convert the data of fluctuation obtained for a detached
patch experiment to the fluctuation inside a single cell. A
separate mathematical analysis has to be performed to identify
such fluctuation.

In the experimental data, the variance of the fluctuation has been
related \cite{sigworth} to the total current, when the probability
of the channel to be in an open state is independent from the
other. When the gating molecule is not as abundant, channels share
such resource and when a gating molecule is bound to a channel,
this resource is not available for the neighboring channels. This
effect coupled the probability of the channel to be in open state
through the number of gating molecules. It is useful to compare the
 experimental result \ref{variance1Koko}, with the simulations of the theoretical model,
 obtained in figures  \ref{variance1}  and \ref{varianceMarkov}. This comparison  suggests
that the tail distribution of the variance cannot be approximated everywhere by a parabola,
as it  was done, under the assumption that the opening of the channels is independent. This
assumption breaks at high  concentration, where the channels are coupled through
the gating molecules.

In a system composed by a cell membrane containing channels and
gating molecules only, two time scales should be dominant. The
first time scale is related to the time it takes for a gating
molecule to find a channel. This time depends on the geometry of
the domain, the number of gating molecules and the number of
channels: for few molecules, this time can be approximated by the
mean time it takes for a gating molecule to find a small absorbing
boundary. This problem has been treated in \cite{HS} and this
approximation was used in the Markovian model of the paper to
estimate the number of open channels. This
assumption is valid because channels or binding proteins occupy a
small portion of the boundary surface. The second time scale is
related to the backward binding constant of the gating molecule to
the channel. The backward binding constant depends on the property
of the channel only and thus does not depend on the statistical
property of the system.

\noindent {\bf Acknowledgments:} We would like to thank J. Korenbrot and R. Nicoll for stimulating discussions. D. H thanks the Sloan-Swartz foundation for the financial support. Part of this work was done while he was visiting UCSF.

redpajama_set_name: RedPajamaArXiv


Izumi was moved to Thailand as 2-year-old baby and brought to the Kiryugaoka Amusement Park Zoo in Japan in 1964. She went on to spend the rest of her life ALONE at this zoo: 53 years. Izumi died in April 2017 at the age of (estimated) 61 years and 11 months. The zoo is investigating the cause of her death.
With Izumi's death in April 2017, the zoo should close its elephant exhibit.
Izumi stands alone in her barren enclosure, a few months before her death. She spent over 50 years here like this.


text:
Lachen ist eine Wüstung auf der Gemarkung Unterjesingen im Landkreis Tübingen.

Lage
Die genaue Lage ist bisher unbekannt. Lachen lag vermutlich am Schönbuchrand. Das Landesdenkmalamt vermutet, der Weiler habe möglicherweise am "Ausgang des Rosecker Tälchens 200 m westlich der Ortskirche" von Unterjesingen gelegen, das heißt am Enzbach. Der abgegangene Weiler ist möglicherweise identisch mit dem Ort, auf den der Bachname Weilersbach zurückgeht.

Geschichte
Lachen war der namengebende Sitz der Niederadligen von Ihlingen und von Lachen im 13. Jahrhundert. Die Herren von Lachen sind später abgewandert.

Bis zum kritischen Datum von 1342, als die Güter der Pfalzgrafen von Tübingen vor allem durch Verkauf an Württemberg oder durch Schenkung an das Kloster Bebenhausen kamen, sind in der Gegend von Tübingen mehrere Orte verlassen worden und abgegangen, zum Beispiel Hindebach und Wennfeld. Dadurch kam es zu einer Einkommensminderung der Pfalzgrafen und deren Ministerialen sowie zu einem durch die Landflucht ausgelösten Bevölkerungsanstieg in Tübingen.

Urkundliche Erwähnungen

Im Landesarchiv Baden-Württemberg gibt es darüber fünf Urkunden unter Allgemeines zur Geschichte, Päpstliche Urkunden und Jesingen:

Bereits um 1204 wird es als "in Lache" urkundlich erwähnt.

Als Papst Gregor IX. am 8. März 1229 das Kloster Bebenhausen in seinen Schutz nahm, bestätigte er unter anderem dessen rechtmäßig erworbenen Güterbesitz in Lachen und weitere aufgezählte Rechte und Freiheiten.

Eberhard der Scherer verkaufte am 23. April 1289 die Vogtei über die in Lachen gelegenen Güter des Albert Munsær von Tübingen im Wert von 4 Pfund Heller und die des Albert von Lachen im Wert von 8 Pfund Heller an das Kloster Bebenhausen.

Bekannte Personen

Bentz von Lachen war 1363 ein Zeuge bei Götz III., und im Jahr 1415 wird ein Hainz von Lachen erwähnt. Im Jahr 1419 war Konrad von Lachen Richter in Weil im Schönbuch.

Einzelnachweise

Wüstung im Landkreis Tübingen
Geschichte (Tübingen)
Ersterwähnung im 13. Jahrhundert


Paradise Beach Weddings works in the East Coast of Central Florida, Volusia County. Primary beaches are New Smyrna Beach, Daytona Beach and Ponce Inlet. This includes Port Orange, Wilbur by the Sea, and Ormond Beach.
While we predominately perform destination and beach weddings, we also do location weddings in parks, at bed & breakfast locations, barns, private residences and more.


This is not exactly a feature by itself (it is part of the Filter feature [core]), but I felt like there was a need for a specific page to talk about this feature.
The token replacement feature is a way to speak of the capability for the system to provide a set of variables that can be used as replacement of tokens found in pages.
There is one main reason for making this feature part of core: we want ONE scheme that works for all the plugins in a seamless manner.
name ::= [_a-zA-Z] ( [:_a-zA-Z0-9]* [_a-zA-Z0-9] )?
Some tokens do not require the parenthesis, for example [year] simply prints out the current year. Many tokens are defined by other plugins in which case the name will be qualified. For example, [users::picture] is used to display the picture of a user registered with a Snap! website.
This example would first replace [images::name] with an image name, say that is "happy", then the system will call [versions::plugin("happy")] and replace that with the corresponding version.
2) No plug-in know about that token in the current context.
The context is determined by the data being parsed. It is not always a page per se.
The available tokens and parameters vary depending on the context. For example, you cannot print out the name of the page if you are outputting something else than a page.
The context is determined by the parameters offered to the token transformation capability. For example, the Snap! Websites offers a way to define a default URL using the title of a page, the parent page(s) of the current page, etc. When this process happens, the page and website objects are both passed along the request allowing all the variables available in those two objects to be accessed by the different plugins to generate the necessary data.
To continue on the title of a page and parent of a page, in a book, you could get a URL that makes use of the parent page URL and appends the current page title to that URL and use that new full path as the path to the new page. Without the book feature, access to the parent URL would not be availabe.
Similarly, the book plugin offers tokens that output a link to the Next Page, Previous Page, Top Page (home/index of the book), Index Page, Glossary Page, Author Page, Copyright Page, etc.


BeautyBb1 01HOTPINOYBIGCUM livejasmin. TamilaCam. KellyLollipop. Zarinochkka.
MyriamBlakeNaamyXJadaSkyeNikaBents .CRISHMONTENEGRONaamyXsosmalldickTakeMissHelga .GRANSEXYamiewoodsCandiceMillerSweetiEva .SHIELBOYHOTHotAsianCummerxEbonyExtaxisPaulAdonis .dulcemaria12FetishQTAcel4uAfroditaXJulian .amiewoodsDiamondvonDCuteCalindaVlastaFlexiX .CandywitchMuscleDiamondPaulAdonissw8sexyNnaughty .amiewoodsEvaClaireeEvaClaireeBawdy .JewelCarterAcel4uTakeMissHelgamarkyhot18 .KymRibeiroEbonyExtaxisKymRibeiromaggicflower .LilyForYouextremeboy4uZackXremeNavaAndYou .
AfroditaXJulianJastinCrazyHotPaulAdoniseloy09 .NikaBentsAbbyChantalRenataSmithKimmyPerez .JastinCrazyHotZackXremeHotBlondSpikeamiewoods .EllyHeatcrystalstanleySHIELBOYHOTSublimeSarah .PrettyCuteAsiaKatnissVannJoyfulSophiaCandiceMiller .NavaAndYouMelanyxBlackZarinochkkaCuteCalinda .AbzholidayCandywitchSexyAndroCandiceMiller .tiffanireyPandoraFemDomBRUTS2Zarinochkka .SilviaSexyyAcel4uZackXremeGRANSEXY .AlejandraLopezCsovushkaaWonderfulFranciXlustdaniela .extremeboy4ulustdanielasosmalldickShantia .


text:
Germany Donates €15 million To Support Provision Of Quality ,Equitable Education In Adamawa And Yobe
6 March 2019 National News News
The German government has announced a new contribution of 15 million euros to Nigeria government through UNICEF to support the provision of quality and equitable basic education and strengthen the resilience of the most vulnerable communities, especially children, in Adamawa and Yobe states.
In a statement made available to newsmen by UNICEF, Chief of Communication, Eliana Drakopoulos that the contribution was provided through the German Government's Development Bank – Kreditanstalt für Wiederaufbau (KfW),
According to the statement this will enable UNICEF support the Government to provide 1,060,000 conflict-affected children and adolescents with increased access to quality education in a safe and healthy learning environment.
The statement says It will also provide 8,200 teachers with increased capacity to deliver quality learning at pre-primary and primary levels and support 500 school-based management committees to plan,maintain a safe and healthy learning environment in schools, with increased community participation.
UNICEF Country Representative, Mohamed Fall. observed that the investment was timely, as it comes at a moment when the countryand UNICEF was doubling its efforts to improve both access and quality of education for conflict-affected children.
Mohamed Fall said all children, regardless of their situation, must be supported to access all their basic rights
"All children, regardless of their situation, must be supported to access all their basic rights."
"This investment is timely, as it comes at a moment when we are redoubling our efforts to improve both access and quality of education for conflict-affected children."
He said the contribution would also support UNICEF's work to link humanitarian needs and long-term development opportunities in target conflict-affected communities in north-east Nigeria.
The German Government is a long-standing partner of UNICEF Nigeria, having contributed over 21 million euros towards the humanitarian and development response in Nigeria since 2016.
Modupe.aduloju Germany, KAPITAL929.FM, UNICEF


text:
First Released Apr 29, 2008
Essential Check out the review
Latest on Grand Theft Auto IV
The Evolution Of Grand Theft Auto's Main Characters
Best GTA Games: Counting Down The Grand Theft Auto Series From Worst To Best
With GTA 6 seemingly on the horizon, we're looking back and ranking Rockstar's celebrated open-world series.
All Of GameSpot's 10/10 Reviews And How Those Scores Are Decided
Here is every single game that earned a 10/10 review from GameSpot, and how we decided upon those scores.
Every Grand Theft Auto Game, Reviewed: GTA 5, San Andreas, And More
Going strong for more than two decades, the Grand Theft Auto series has seen big changes over its time as the premiere open-world game. Here's every game we've reviewed in ...
GTA Remasters? Take-Two CEO Says That's A Great Question
Strauss Zelnick likes the question and says it's an "encouraging" thought.
GTA 4 PC Update Restores Cut Music But Renders The Game Unplayable For Some
GTA 4 on PC update reportedly added cut music but also corrupted users' saves, while an unrelated bug has since made the game unplayable.
GTA 4 Returns To Steam--With A Big Feature Missing
The beloved entry in the Grand Theft Auto canon will soon be playable on Steam again, sans multiplayer.
GTA 4 Not Available To Buy On Steam Due To Game For Windows Live Issue
2008's Grand Theft Auto IV is no longer available to buy on Steam.
These 11 Heartbreaking Game Moments Will Make You Weep Like a Baby
You may have to put down your controllers and pick up your hankies, because here are some of the most emotional gaming moments of all time. Warning: Spoilers ahead, people!
Remembering 2008: Biggest Games Turning 10 This Year
Check out the releases that made 2008 an exciting year for gaming, which includes big names like Metal Gear Solid 4, Grand Theft Auto IV, Fallout 3, and more.
Oct 4, 2018 12:14pm
Grand Theft Auto 4's Open World Is Still Fantastic 10 Years Later
On the tenth anniversary of GTA IV, we take a look back on Rockstar Games' seminal open world game set in Liberty City.
Evolution of Grand Theft Auto: Milestones From Every GTA
From its humble beginnings on PS1 and PC to the staggeringly persistent sales of Grand Theft Auto V, here's a look at the evolution of GTA and how it transformed ...
GTA 4 Update Removes Some Great Songs
A new update for the 10-year-old game cuts out some classic rock songs.
Some GTA 4 Songs Being Removed, But New Ones Will Take Their Place
Due to licenses expiring, several tracks from GTA IV's soundtrack are being taken out of the game.
Happy 10th Anniversary To The Biggest Games Of 2008
GameSpot looks back at some of the more notable games of 2008. It was packed with high-profile sequels including GTA IV, Fallout 3, and Metal Gear Solid 4.
Check out the releases that made 2008 an exciting year for gaming, which includes big names like Metal Gear Solid 4, Grand Theft Auto IV, and more.
Grab GTA 4, Max Payne 3, And More Rockstar Games Cheap To Support Charity Right Now
You can snag a bunch of great Rockstar games on PC on a steep discount right now through Humble Bundle.
The Evolution of Grand Theft Auto
Grand Theft Auto IV builds on the series' rock solid foundation with an awe inspiring recreation of Liberty City.
72054 Rating(s)
Sign In to rate Grand Theft Auto IV
Rockstar Toronto
Rockstar North
CyberFront
Synergex
Blood, Intense Violence, Partial Nudity, Strong Language, Strong Sexual Content, Use of Drugs and Alcohol
Announced DLC
Jun 5, 2017 11:41am
Grand Theft Auto IV: The Lost and Damned
GTA 4 Backwards Compatibility Livestream
GS News Update: GTA 4 Is Now Playable on Xbox One
The Top 10 News Stories of January
Exploring The Real Liberty City - The Point
5 Of The Worst PC Ports Ever - The Gist
The Best Gaming Anti-Piracy Methods In 2 Minutes
First Released Oct 7, 2021
First Released Jan 20, 2021


A single molecule of water is the reason certain mutations are more responsive to a 'targeted' cancer therapy, according to molecular-scale simulations carried out at the A*STAR Bioinformatics Institute.
It turns out that the critical difference was a mere molecule of water. Afatanib binds to a pocket on EGFR that is missing five amino acids in the EGFR19del mutant. Molecular dynamics simulations carried out at the Bioinformatics Institute and the National Supercomputing Centre, Singapore, showed that this deletion creates a more physically constrained structure, which can snugly accommodate both afatanib and a single water molecule in a very stable arrangement. In contrast, the mutant EGFRL858R, also found to characterize lung cancer, contains just a single amino acid substitution, and is predicted to hold two water molecules in its pocket. However, these are in a weaker and relatively less stable arrangement, resulting in weakened interactions with afatanib that likely reduce the drug's affinity and hence effectiveness.
"Our hypothesis is based on understanding the physics of the system and could offer a compelling rationale for the clinical observations with afatanib," says Kannan. Intriguingly, he and Verma were also able to home in on other binding pocket mutations (single nucleotide polymorphisms, or 'SNPs') that occur alongside EGFR19del that might likewise influence the efficacy of afatanib.
The researchers are now examining whether these predictions hold in real-life experiments and see this as a promising general strategy for predicting patient response to targeted treatments. "We are trying to develop a robust pipeline to examine the structural effects of mutations on drug interactions," Kannan says. "This will be a very valuable approach to complement our engagement with the national precision medicine efforts."