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\title{Reaction and Diffusion on Fractal Sets}
\author{%
{\otherblue M.R. Muldoon, D. S. Broomhead, }\\
{\otherblue Caroline Riley and J. P. Huke}\\
\\
{\otherblue The School of Mathematics}\\
and \\
{\otherblue Centre for Interdisciplinary Computational and Dynamical Analysis (CICADA)} \\
{\otherblue The University of Manchester.}   
}% End of author

\begin{document}

%---------------------------------------------------------------------- SLIDE -
\maketitle
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{wideslide}{Modelling Cellular Processes}
	\vspace{-0.5cm}
	When modelling chemical processes in cells, it is common to build
	models that are based on mass action kinetics and have no spatial
	structure beyond simple compartmentalisation.
	\medskip
	
	These observations prompt the questions:
	\begin{description}
		\item[{\otherblue$\bullet$}] Is the mass action assumption appropriate? Is `concentration' a useful concept?
		\item[{\otherblue$\bullet$}] Is spatial structure important?  Experimental techniques are beginning to resolve spatial detail---perhaps dynamically in the not too distant future
		\item[{\otherblue$\bullet$}] It turns out that these questions are not independent. 
		\item[{\otherblue$\bullet$}] How should we model spatio-temporal dynamics within cells?
		Two possibilities I'll discuss this afternoon are
		\begin{description}
			\item[{\darkred$\bullet$}] Gillespie-like lattice gas simulations on fractal domains
			\item[{\darkred$\bullet$}] Models with continuous concentrations, but on fractal domains.
		\end{description}
	\end{description}
\end{wideslide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{wideslide}[toc=]{ Transport in Cells}
	\vspace*{-0.5cm}
	\dualslide{lcolwidth=0.6\linewidth,rcolwidth=0.4\linewidth}{%
		\begin{description}
			\item[{\otherblue$\bullet$}] The cytoplasm is a very crowded place (5 -- 40\% of volume is occupied by macromolecules)
			\item[{\otherblue$\bullet$}] The available volume through which a given molecule can move depends on its size and shape
			\item[{\otherblue$\bullet$}] Small molecules can diffuse but may be effectively confined to low-dimensional spaces
			\begin{description}
				\item[{\darkred$\bullet $}] 1D pores, 
				\item[{\darkred$\bullet $}] Highly ramified (fractal) spaces
			\end{description}
			\item[{\otherblue$\bullet $}] Large molecules are transported in a dynamic microtubule network by motor proteins (kinesins and dyneins) 
			\item[{\otherblue$\bullet$}] We need both computational and mathematical models of this.
		\end{description}
	}{%
		\begin{center}
			\includegraphics[scale=0.25]{figures/SchnellTurner_Fig1.eps}
		\end{center}
		{\tiny \hfill Medalia et al. (2002) \emph{Science} \textbf{298} 1209--1213.}
		\vspace{.1cm}
		\begin{center}
			\includegraphics[scale=0.25]{figures/SchnellTurner_Fig2.eps}
		\end{center}
		{\tiny \hfill Schnell, Turner (2004) \emph{Prog. Biophys. \&amp; Mol. Biol.} \textbf{85} 235--260.}
	}
\end{wideslide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{slide}{A model system from chemistry}
	\vspace*{-0.75cm}
	\includegraphics[width=9cm]{Figures/Kopelman_JStatPhys42_citation.eps}
	\smallskip
	
	\includegraphics[width=9cm]{Figures/Kopelman_JStatPhys42_title.eps}
	\smallskip
	
	\hrule
	\bigskip
	
	{
		\small
		\textbf{From the Abstract:} \newline 
		Heterogeneous kinetics are shown to differ drastically from 
		homogeneous kinetics. For the elementary reaction 
		\begin{displaymath}
			A + A \rightarrow \mbox{products} 
		\end{displaymath}
		we show that the diffusion-limited reaction rate is proportional to
		$t^{-h}[A]^2$ $\dots$ where $h = 1 - d_s/2 \; \dots$
		and $d_s$ is the effective spectral dimension.
	}
	
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{wideslide}{The experimental system}
	\vspace*{-0.5cm}
	\dualslide{lcolwidth=0.6\linewidth,rcolwidth=0.4\linewidth}{%
	  \begin{description}
		 \item[{\otherblue$\bullet$}] Experiments involved naphthalene
		 crystals grown from a mixture of two forms of the molecule. The
		 bulk of the mixture consisted of the isomer
		 $\mbox{C}_{10}\mbox{D}_{8}$, in which all the hydrogens have been
		 replaced by deuterium, combined with a mole fraction of $\sim 8\%$
		 of ordinary $\mbox{C}_{10}\mbox{H}_{8}$.

		 \item[{\otherblue$\bullet$}]  Certain atomic
		 excitations---so-called \emph{triplet excitons}---are mobile in the
		 undeuterated component, which forms a percolating cluster (see
		 right) within the crystal.

		 \item[{\otherblue$\bullet$}]  When pairs of triplets arrive at the
		 same lattice site they can fuse to form a singlet exciton: the
		 basic reaction is:
		 \begin{displaymath} 
		 	T + T \longrightarrow S
		 \end{displaymath}
		 
	\end{description}
	}{%
	%\vspace{-1.5cm}
		\begin{center}
			\includegraphics[scale=0.5]{Figures/Kopelman_Science241_Fig2.eps}
		\end{center}
		\emph{\small
			A two-dimensional percolation cluster: the set shown above is a
			``giant'' connected cluster produced when 59.3\% of the sites
			are coloured black at random.
		}
		\medskip
		
		{\tiny \hfill R. Kopelman (1988) \emph{Science} \textbf{241} 1620--1626.}
		
	}
\end{wideslide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{wideslide}[toc=Time-varying rate coef.]{A time-dependent rate coefficient}
	\vspace*{-0.5cm}
	\dualslide{lcolwidth=0.6\linewidth,rcolwidth=0.4\linewidth}{%
	  \begin{description}
		 \item[{\otherblue$\bullet$}] It's possible to use a bright
		 flash of light to produce many triplet excitons throughout
		 the whole of a crystal.
		 
		 \item[{\otherblue$\bullet$}]  Both kinds of exciton decay via 
		 the emission of photons, so one can monitor their 
		 concentrations via fluorescence:
		 green for triplets, ultraviolet for singlets.
		 
		 \item[{\otherblue$\bullet$}] 	 Na\"{i}ve mass action kinetics
		 suggest that the rate of triplet fusion will be is proportional to
		 the square of the triplet concentration:
		 \begin{displaymath}
		 	\frac{d T}{dt} = - \kappa T^2
		 \end{displaymath}
		 
	 \item[{\otherblue$\bullet$}]  Experimentally, however, the `constant'
	 $\kappa$ decays in time as a power law {\otherblue $\kappa(t)\sim t^{-h}$}

	\end{description}
	}{%
	%\vspace{-1.5cm}
		\begin{center}
			\includegraphics[scale=0.5]{Figures/Kopelman_JStatPhys42_Fig4.eps}
		\end{center}
		\emph{\small
			Time courses of $\kappa(t)$ for $\mbox{C}_{10}\mbox{H}_{8}$ fractions
			of 16, 14, 12, 10, 8, 6, 4, and 2\%.
		}
		\medskip
		
		{\tiny \hfill R. Kopelman (1986) \emph{J. Stat. Phys.} \textbf{42} 185--200.}
	}
\end{wideslide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{slide}[toc=]{Spectral dimension}
	\begin{description}
		 \item[{\otherblue$\bullet$}] At the critical
		 $\mbox{C}_{10}\mbox{H}_{8}$ fraction, 8\%, the estimated power-law
		 exponent for $\kappa(t)$ is $h = 0.35 \pm 0.03$.
		 
		 \item[{\otherblue$\bullet$}] This is in good agreement with a
		 theoretical prediction {\otherblue $h = 1 - d_s / 2$}, where $d_s$
		 is the spectral dimension of the underlying fractal. The
		 percolation clusters that arise in a random alloy of
		 $\mbox{C}_{10}\mbox{H}_{8}$ and $\mbox{C}_{10}\mbox{D}_{8}$ have
		 $d_s = 4/3$, so the prediction is
		 \begin{displaymath}
		 	h \; = \; 1 - d_s/2 \; = \; 1 - 2/3  \; = \; 1/3.
		 \end{displaymath}
		 
		 \item[{\otherblue$\bullet$}] More generally, $d_s$ is related
		 to the
		 asymptotic time-dependence of the probability that a random
		 walker on the fractal will return to its original position after
		 a time $t$: 
		 \begin{displaymath}
		 	\mbox{P(return after $t$)} \sim t^{-d_s/2}
		 \end{displaymath}
	\end{description}

	
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{wideslide}[toc=Aside: on $\kappa(t)$]{Aside: time-dependence of $\kappa$ is weird}
	\vspace*{-0.5cm}
	\begin{quote}
		\small \em
		The replacement of the rate constant by a time-dependent rate
		coefficient $\kappa(t)$ has unexpected consequences. Assume that one
		can look at one container in which a reaction $T + T \rightarrow
		S$ is in progress, with instantaneous concentration $[T]$ at time
		$t = 16$s after preparation. Now imagine another container that
		simultaneously happens to pass through the same instantaneous
		concentration $[T]$, but has only been evolving for $t = 1$s. The
		second container was obviously prepared with a lower initial
		concentration, but both have the same concentration at the time of
		observation and both containers may be assumed to have the same
		thermodynamic conditions---temperature and pressure, 
		for example. \\ \ \\
		
		Classically, they should react at the 
		same rate, but if $\kappa(t) \sim t^{-h}$ and, say, $h = 1/2$,
		the second, younger system is reacting at a rate $16^{h} = 4$
		times faster than the first one.
	\end{quote}
	{\tiny \hfill Paraphrased from R. Kopelman (1988) \emph{Science} \textbf{241} 1620--1626.}
\end{wideslide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{wideslide}{The Sierpinski gasket}
	\vspace*{-0.5cm}
	\begin{center}
		\includegraphics[height=2.5cm]{Figures/SierpinskiGasketConstruction.eps}
	\end{center}
	\medskip
	
	The Sierpinski Gasket is a fractal in the spirit of the Cantor
	middle-third set: one can construct it as the limit of contraction
	mappings
	$F_j: \RR^2 \rightarrow \RR^2$ given by
	\begin{displaymath}
		F_j( {\mathbf x} ) = \left[ \begin{array}{cc} 
						1/2 &amp; -0 \\
						0 &amp; 1/2
					\end{array} \right]
					 \left[ \begin{array}{c} x_1 \\ x_2 \end{array} \right] 
					 + {\mathbf b_j}
	\end{displaymath}
	where ${\mathbf b_1} = (1/2, \, \sqrt{3}/2)^T$, 
	${\mathbf b_2} = (0, \, 0)^T$ and ${\mathbf b_3} = (1, \, 0)^T$. 
	We will mainly be interested in
	the sequence of graphs ${\otherblue (V_n,\, E_n)}$ whose edges are the
	edges of the triangles and whose vertices are the points at which triangles
	meet. 
\end{wideslide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{slide}[toc=]{More on the Sierpinski gasket}
	\vspace*{-0.5cm}
	
	\begin{center}
		\includegraphics[scale=0.6]{Figures/SGraphs2.eps}
	\end{center}

	\begin{description}
		 \item[{\otherblue$\bullet$}] The limiting gasket is an
		 uncountable, compact, self-similar subset of $\RR^2$

		 \item[{\otherblue$\bullet$}] Its Hausdorff dimension is
		 ${\otherblue d_H= \log 3/\log 2}$ and spectral dimension
		 ${\otherblue d_s=\log 9/\log 5 \approx 1.365 }$

		 \item[{\otherblue$\bullet$}]  The vertices in the sets $V_n$
		 defined recursively by \begin{displaymath} {\otherblue V_{n+1}=
		 F_1(V_n)\cup F_2(V_n)\cup F_3(V_n)} \end{displaymath} become dense
		 in the gasket.
	\end{description}
	
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{slide}[toc=Gillespie on gaskets]{Simulating chemistry on the gasket}
	\vspace*{-0.5cm}
	Here I'll formulate a discrete-state, continuous time Markov 
	process that will model exciton dynamics on the gasket.
	\medskip

	\begin{description}
		 \item[{\otherblue$\bullet$}] I'll use the
		 finite-graph approximations to the gasket. If we say $V_1$ is the
		 vertex set of the initial triangle then
		 \begin{displaymath}
		 	N_n \; \equiv \;  |V_n| \; = \;  3 \times (3^n - 1) /2
		 \end{displaymath}
		 
		 \item[{\otherblue$\bullet$}] I'll model 
		 neither the (comparatively slow) fluorescence processes,
		 nor the singlet excitons: triplets will simply annihilate in pairs,
		 $T + T \rightarrow \emptyset$.

		 \item[{\otherblue$\bullet$}] I'll imagine that 
		 the triplet excitons are
		 localized at vertices, so I'll 
		 describe a state of the Markov process by giving
		 a list of $N_n$ nonzero integers that specify the number
		 of excitons at each site.
		 
	\end{description}
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{wideslide}{Reactions and transitions}
	\vspace*{-0.5cm}
	\dualslide{lcolwidth=0.65\linewidth,rcolwidth=0.35\linewidth}{%
		Consider a pair of adjacent vertices---say $u$ and $v$. To specify
		the states of just these two vertices, we need only a 
		pair $(N_u, \, N_v) \in \N^2$. \newline
		
		\begin{center}
			\begin{tabular}{l|ccc}
				Reaction &amp; $\delta N_u$ &amp; $\delta N_v$ &amp; Rate \\
				\hline \hline
				Diffusive hop &amp;&amp; \\
				from $u$ to $v$ &amp; -1 &amp; 1 &amp; $k_d N_u$ \\
				\hline
				Diffusive hop &amp;&amp; \\
				from $v$ to $u$ &amp; 1 &amp; -1 &amp; $k_d N_v$ \\
				\hline
				Annihilation &amp; &amp; \\
				at $u$ &amp; -2 &amp; 0 &amp; $k_a N_u(N_u - 1)$ \\
				\hline
				Annihilation &amp; &amp; \\
				at $v$ &amp; 0 &amp; -2 &amp; $k_a N_v(N_v - 1)$ \\
			\end{tabular}
		\end{center}	
	}{%
		\begin{center}
			\includegraphics[width=4.0cm]{Figures/GillespieGraphForGasket.eps}
		\end{center}
		\medskip
		\emph{\small A small part of the state space for the vertices $u$ and 
			$v$: blue arrows correspond to diffusive hops, while the black
			arcs represent annihilation of triplet pairs.
		}
	}
\end{wideslide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{wideslide}{Dilute initial conditions}
	\vspace*{-0.5cm}
	\dualslide{lcolwidth=0.5\linewidth,rcolwidth=0.5\linewidth}{%
		\begin{description}
			\item[{\otherblue$\bullet$}] Simulation based on $V_9$.
			
			\item[{\otherblue$\bullet$}] Annihilation taken to be
			faster than diffusion: $k_a / k_d = 10$.
			
			\item[{\otherblue$\bullet$}] Initial triplet populations
			are independent draws from a Poisson distribution with $\lambda = 1$
			so $P(0) = P(1) = e^{-1} \approx 0.38$.
			
			
			\item[{\otherblue$\bullet$}] 100 simulations averaged together
			to obtain mean time course.
			
			\item[{\otherblue$\bullet$}] Blue dots show estimates of 
			\begin{displaymath}
				\kappa(t) = \frac{d[T]/dt}{[T]^2}
			\end{displaymath}
			where $[T]$ is the average exciton occupancy over the whole graph.
		\end{description}
	}{%
		\begin{center}
			\includegraphics[width=4.0cm]{Figures/agm_random_depth9_theta10_nZero1_nReps100.eps}
		\end{center}
		\medskip
		
		\emph{\small A linear model fit to the part of the curve with
			$20 \leq t \leq 500$ yields 
			\begin{displaymath}
				\kappa(t) \sim t^{-h}
			\end{displaymath}
			with $h \approx 0.317$. Agrees wells with
			$1 - d_s/2 = 1 - \log(3)/\log(5) \approx 0.317$.
		}
	}
\end{wideslide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{slide}[toc=]{Similar results for $V_8$, $V_{10}$}
	\vspace*{-0.5cm}
	\dualslide{lcolwidth=0.5\linewidth,rcolwidth=0.5\linewidth}{%
		\begin{center}
			\includegraphics[width=4.0cm]{Figures/agm_random_depth8_theta10_nZero1_nReps100.eps}
		\end{center}
	}{%
		\begin{center}
			\includegraphics[width=4.0cm]{Figures/agm_random_depth10_theta10_nZero1_nReps100.eps}
		\end{center}
	}
	\medskip
	
	In both cases, a line fit to the region $20 \leq t \leq 500$ yields
	$h \approx 0.322$.
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{wideslide}[toc=Large initial populations]{Larger initial concentrations}
	\vspace*{-0.5cm}
	\dualslide{lcolwidth=0.5\linewidth,rcolwidth=0.5\linewidth}{%
		\begin{description}
			\item[{\otherblue$\bullet$}] All vertices start with 
			100 excitons: one should perhaps think
			of vertices as representing small volumes containing 
			several molecules and capable of holding
			many excitons.
			
			\item[{\otherblue$\bullet$}]  Analysing the data as previously
			yields $h \approx 0.463 \neq 1 - d_s/2$.
			
			\item[{\otherblue$\bullet$}] Similar results for $V_{8}$ and $V_{10}$.
			
			\item[{\otherblue$\bullet$}]  Initially, dynamics are dominated 
			by annihilation and $\kappa(t)$ is essentially constant.
			
			\item[{\otherblue$\bullet$}] Peak for $\kappa(t)$ occurs when $[T] \approx 1$.
		\end{description}
	}{%
		\begin{center}
			\includegraphics[width=4.0cm]{Figures/agm_uniform_depth9_theta10_nZero100_nReps100.eps}
		\end{center}
		\medskip
		
		\emph{
			\small The vertical dashed line marks the time at which
			$[T]$ falls below 1.
		}
	}
\end{wideslide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{wideslide}[toc=Calculus on the Gasket]{Calculus on the Sierpinski Gasket}
	\vspace*{-0.5cm}
	\dualslide{lcolwidth=0.5\linewidth,rcolwidth=0.5\linewidth}{%
		\begin{description}
			\item[{\otherblue$\bullet$}] A Laplacian can be constructed
			as a limit of a suitably renormalized sequence of graph Laplacians
			
			\item[{\otherblue$\bullet$}]  Its spectrum, ${\otherblue
			\Lambda_N}$, can be found exactly using a decimation technique due
			to Fukushima and Shima.
			
			\item[{\otherblue$\bullet$}]  Asymptotics of the counting
			function ${\otherblue \rho(x)=\sharp\{\lambda\in \Lambda_N: 0&lt;
			\lambda\leq x\} \asymp x^{d_s/2}  }$
			
			\item[{\otherblue$\bullet$}] In fact, there are positive
			constants ${\otherblue C_\pm}$ such that for ${\otherblue x\geq
			\lambda_1}$ \[ {\otherblue C_- x^{d_s/2} &lt; \rho(x) &lt; C_+
			x^{d_s/2} }\] 
			
		\end{description}
	}{%
		\begin{center}
			\includegraphics[scale=0.3]{Figures/EigenvalueCountingFunctionLogPlot.eps}
		\end{center}
	}
\end{wideslide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{slide}[toc=]{Neumann boundary conditions}
	\vspace*{-0.5cm}
	\dualslide{lcolwidth=0.5\linewidth,rcolwidth=0.5\linewidth}{%
		\begin{description}
			\item[{\otherblue$\bullet$}]  A normal derivative
			(${\otherblue \partial_n}$) can be defined and a Gauss-Green
			theorem proved, from which follows: $${\otherblue \int_{\G}
			\nabla^2 A\; d\mu = \sum_{V_0} \partial_n A}$$
			
			\item[{\otherblue$\bullet$}] The upper panel at right
			shows a Neumann (zero derivative at the boundary) eigenfunction.
			
			\item[{\otherblue$\bullet$}] There are also highly 
			localized eigenfunctions, an
			example of which appears at lower right. 
		\end{description}
	}{%
		\begin{center}
			\includegraphics[scale=0.19]{Figures/Neumann_Eigenfunction1.eps}
		\end{center}
		\vspace{-.25cm}		
		\begin{center}
			\includegraphics[scale=0.19]{Figures/Localised_Eigenfunction.eps}
		\end{center}
		\vfill
	}
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{wideslide}{Reaction and diffusion} % 
	\vspace*{-0.5cm}
	We can solve the following reaction-diffusion 
	equation on the Sierpinski Gasket:
	\begin{displaymath}
	{\otherblue
 		\frac{\partial A}{\partial t}  =  D \nabla^2 A - k A^2  
	}
	\end{displaymath}
	To indicate that we are no longer necessarily discussing the naphthalene
	system, I have used $A$ to indicate the concentration of some 
	arbitrary chemical species.
	\bigskip
	
	
	\dualslide{lcolwidth=0.6\linewidth,rcolwidth=0.4\linewidth}{%
		In the numerical solution at right (with $k=0.1$ and $D=0.01$),
		the time course of 
		\begin{displaymath}
			\kappa(t) = \frac{d \overline{A}/dt}{\overline{A}^2}
		\end{displaymath}
		where $\overline{A}$ is the average of $A$ over the gasket, does 
		not show a power law behaviour---instead $\kappa(t)$ decays to $k$.
	}{%
		\begin{center}
			\includegraphics[scale=0.35]{Figures/AAReactionK.eps}
		\end{center}
	}
\end{wideslide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{slide}{No obvious power laws} % 
	\vspace*{-0.5cm}
	\dualslide{lcolwidth=0.6\linewidth,rcolwidth=0.4\linewidth}{%
		Given that $\kappa(t)$ tends to $k$ as $t \rightarrow \infty$:
		\begin{description}
			\item[{\otherblue $\bullet$}] The mass action 
			approximation becomes valid at long times.
			
			\item[{\otherblue$\bullet$}] It appears remotely possible
			that the difference
			${\otherblue(\kappa(t)-k)}$ decays in a power-law way for large $t$.
		\end{description}
	}{%
		\begin{center}
			\includegraphics[scale=0.30]{Figures/AAPlot2.eps}
		\end{center}
	}
	
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{slide}[toc=]{Mass action and averaging} % 
	\vspace*{-0.5cm}
	
	The underlying chemistry is $A + A \rightarrow \mbox{\sf product}$,		
	and so if we define the average $\overline{A}$ via 
	\begin{displaymath}
		{\otherblue \overline{A} = \int_{\G} A \, d\mu}
	\end{displaymath}
	the law of mass action--allowing for a time-varying rate 
	coefficient---suggests that the following ODE should 
	provide a good description of the kinetics:
	\begin{displaymath}
		{\otherblue \frac{d \overline{A}}{dt} = -\kappa(t)  \overline{A}^2}
	\end{displaymath}

\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{slide}{Relations between rates} % 
	\vspace*{-0.5cm}
	
	On the other hand, by integrating both sides of the  reaction-diffusion
	equation we obtain 
	\begin{eqnarray*}
		 \int_{\G} \frac{\partial A}{\partial t} \, d\mu &amp; = &amp; \int_{\G} \left[ D \nabla^2 A - k A^2 \right]\, d\mu \\
		 \frac{d \overline{A}}{dt} &amp; = &amp; \left[\int_{\G} D \nabla^2 A \, d\mu \right] -k  \int_{\G} A^2 \, d\mu \\
		 	&amp; = &amp; -k \overline{A^2}  
	\end{eqnarray*}
	where, in eliminating the integral over the diffusion term $D \nabla^2 A$,
	I have used both the Gauss-Green formula and the Neumann boundary conditions.
	
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{slide}[toc=]{A lower bound for $\kappa(t)$} % 
	\vspace*{-0.5cm}
		
	Setting the two ODEs equal gives
	\begin{displaymath}
		-\kappa(t)  \overline{A}^2 = -k \overline{A^2}
		\qquad \mbox{or} \qquad
	{\otherblue 
		\kappa(t) = \frac{\overline{A^2}}{\overline{A}^2} \; k
	}
	\end{displaymath}
	\medskip
	
	We also have the inequality (with equality iff $A$ is uniform) 
	\begin{displaymath}
		{\otherblue\overline{A^2}  \geq \overline{A}^2  \quad \Rightarrow  \quad \kappa(t) \geq k}
	\end{displaymath}
	Thus
	\begin{description}
		\item[{\darkred$\bullet$}] If, initially, the concentration of $A$
		is not uniform, the initial rate will exceed the intrinsic rate.
		
		\item[{\darkred$\bullet$}] The intrinsic rate is a lower bound on
		the observed rate---this excludes the possibility that ${\otherblue
		\kappa(t)\sim t^{-h}}$
	\end{description}

\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{slide}[toc=]{Power laws and fractals} % 
	\vspace*{-0.5cm}
		
	We have used only those properties that the fractal shares with 
	classical domains---the Gauss-Green formula for a suitably 
	defined Laplacian and normal derivative---and have been able to 
	rule-out the possibility of power-law behaviour. Given that such
	behaviour is observed in both the naphthalene system and in the
	Gillespie simulations, it appears that the discreteness of the 
	reacting entities plays a crucial role.

\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{wideslide}[toc=Traces of fractality]{Is there any trace of fractality?} % 
	\vspace*{-0.5cm}
	The fractal nature of the gasket does not seem to leave much of a trace
	on the dynamics of the reaction-diffusion system, but it is possible 
	to detect a subtle effect when $A(x,t)$ is very nearly spatially uniform.
	Consider a small perturbation $\epsilon \alpha(x,t)$ away from uniformity
	\begin{displaymath}
		A(x,t) = \overline{A}(t) + \epsilon \alpha(x,t) + O(\epsilon^2)
	\end{displaymath}
	were $\epsilon \alpha(x,t)$ and all higher order terms have spatial mean zero.
	\medskip
	
	Substituting this into the PDE $\partial_t A = D \nabla^2 A - k A^2$ and
	equating powers of $\epsilon$ leads to the conclusions that
	\begin{displaymath}
		{\otherblue
		\overline{A}(t) = \frac{\overline{A}(0)}{1 + \overline{A}(0) kt}
		} \qquad \mbox{and} \qquad { \otherblue
		\partial_t \alpha = D \nabla^2 \alpha - 2 k \overline{A}(t) \alpha.
		}
	\end{displaymath}	
\end{wideslide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{slide}[toc=]{Decay of perturbations} % 
	\vspace*{-0.5cm}
	Expanding $\alpha(x,t)$ in the orthonormal, Neumann eigenfunctions of
	the Laplacian 
	\begin{displaymath}
		\alpha(x,t) = \sum_{n=1}^\infty a_n(t) u_n(x)
	\end{displaymath}
	leads to a system of ODEs,
	$\dot{a}_n = -\left(D\lambda_n + 2 k \overline{A}(t)\right) a_n(t)$,  
	that are soluble and lead, after some spatial averaging, to the 
	conclusion that
	\begin{displaymath}
		\otherblue
		\kappa(t) = k + \left( 
			\frac{\epsilon^2 k }{\overline{A}^2(0) 
				\left(1 + \overline{A}(0) kt \right)^2 
			}
			\right) \sum_{n=1}^\infty
			e^{-2 \lambda_n Dt} a_n^2(0) \; + O(\epsilon^3).
	\end{displaymath}
	\medskip
	
	The underlying fractal geometry influences this sum through the
	spectrum of the Laplacian and enforces a constraint on the way 
	$\kappa(t)$ relaxes toward $k$.
	
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{wideslide}[toc=]{Bounds on $\kappa(t) - k$} % 
	\vspace*{-0.5cm}
	If we assume the initial perturbation $\alpha(x, 0)$
	is such that there are positive constants $\gamma$ and $a_\pm$ such that
	\begin{displaymath}
		a_- e^{-\gamma \lambda_n} \leq \epsilon a_n(0) \leq a_+ e^{-\gamma \lambda_n}
	\end{displaymath}
	then it is possible to prove that there are constants $K_\pm$ such that
	$K_- F(t) &lt; (\kappa(t) - k) &lt; K_+ F(t)$ where
	\begin{displaymath}
		F(t) = \frac{\Gamma\left(1 + d_s/2, \, 2 \lambda_1 (\gamma + Dt)\right)}{
					\left[1 +  \overline{A}(0) kt \right]^2 \left[ 2(\gamma + Dt)\right]^{d_s/2}
				}
	\end{displaymath}
	where $\Gamma(., \, .)$ is the incomplete gamma function. 
	\medskip
	
	For suitable
	choices of initial data ($\gamma$, really) and times $t &gt; \gamma/D$, this
	function can appear to exhibit power-law behaviour, though in the very long term
	the relaxation of $\kappa(t)$ to $k$ is controlled by $\lambda_1$, the
	eigenvalue nearest zero.
\end{wideslide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{wideslide}[toc=Two species]{Two reacting and diffusing species} % 
	\vspace*{-0.5cm}
	To conclude, I'll consider briefly the reaction-diffusion system in which
	two distinct chemical species react: $A + B \rightarrow \mbox{\sf product}.$
	If we define the averages $\overline{A}$ and $\overline{B}$ as
	we did earlier, a very similar argument leads to the conclusion 
	\begin{displaymath}
		\kappa(t) = k \left( \frac{\overline{A B}}{\overline{A}\; \overline{B}} \right)
	\end{displaymath}
	\medskip
	
	\begin{description}
		\item[{\darkred$\bullet$}] The quantity  {\otherblue$\overline{A
		B}$} measures the correlation between the spatial distribution of
		$A$ and $B$ 
		
		\item[{\darkred$\bullet$}] If {\otherblue $A$} and
		{\otherblue $B$} are uncorrelated, {\otherblue  $\overline{A B} =
		\overline{A}\; \overline{B}$} and hence {\otherblue $\kappa = k$}
		
%		\item[{\darkred$\bullet$}] Initially, {\otherblue $A$} and
%		{\otherblue $B$} could well be uncorrelated ${\otherblue
%		\Rightarrow \kappa(0)=k}$.
		
		\item[{\darkred$\bullet$}] If the kinetics dominate---if the reaction
		proceeds rapidly, consuming molecules of $A$ and $B$ faster than
		diffusion can resupply them---then {\otherblue $A$} and 
		{\otherblue $B$} become \emph{anticorrelated}: 
		\begin{displaymath}
		{ \otherblue
			\overline{A B}  &lt; \overline{A}\; 
			\overline{B} \quad \Rightarrow \quad \kappa(t)&lt; k
		}
		\end{displaymath}
	\end{description}


\end{wideslide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{wideslide}[toc=Two species: numerics]{Two species: numerics} % 
	\vspace*{-0.5cm}
	\dualslide{lcolwidth=0.6\linewidth,rcolwidth=0.4\linewidth}{%
		We solve the system of PDEs (with $k=0.1$ and $D=0.01$):
		\begin{eqnarray*}  
			\frac{\partial A}{\partial t} &amp; = &amp;  D \nabla^2 A - k A B  \\
			\frac{\partial B}{\partial t} &amp; = &amp;  D \nabla^2 B - k A B   
		\end{eqnarray*}
		\medskip
		
		The colour scheme is:
		\begin{eqnarray*}
			&amp; \mbox{\red$\bullet$} &amp; {\red (A&gt;\overline{A}) \wedge (B&lt;\overline{B}) } \\
			&amp; \mbox{\green$\bullet$} &amp; {\green (A&lt;\overline{A}) \wedge (B&gt;\overline{B})} \\
			&amp; \mbox{\cyan$\bullet$} &amp; {\cyan ((A&gt;\overline{A}) \wedge (B&gt;\overline{B}))\vee ((A&lt;\overline{A}) \wedge (B&lt;\overline{B}))}
		\end{eqnarray*}
	}{%
		\begin{center}
			\includegraphics[scale=0.3]{figures/ABConcentrations1.eps}
			\includegraphics[scale=0.3]{figures/ABConcentrations2.eps}
		\end{center}
	}
\end{wideslide}
%------------------------------------------------------------------------------


\overlays{2}{%
%---------------------------------------------------------------------- SLIDE -
\begin{wideslide}{On segregation}
\fromSlide*{1}{
	\vspace{-0.7cm}
	\begin{description}
		\item[{\otherblue$\bullet$}] This anti-correlation effect
		(segregation) was predicted by Zeldovich \emph{et al.} in 1978.

%		\item[{\otherblue$\bullet$}] The possibility of 
%		satisfying the condition that chemistry
%		must act faster than diffusion depeds on dimension:
%		segregation like this is not found in steady state in
%		$\RR^2$ or $\RR^3$ (Kopelman)

		\item[{\otherblue$\bullet$}] Segregation can be observed in 
		dimensions $&lt; 2$: Kopelman (1989) demonstrated it in a lattice gas 
		on the Sierpinski Gasket. His observations are
		consistent with our analysis.
		
	    \item[{\otherblue$\bullet$}] Still not a power law, nor Zipf-Mandlebrot 
	    (Schnell and Turner: lattice gas model of Michaelis-Mentin) 
	    {\otherblue  $\kappa(t)= k \tau/ (\tau + t)^{h}$}
	    
%	    \item[{\otherblue$\bullet$}] Our numerics suggest 
%	    {\otherblue $\kappa(t) \sim t$} for $t$ large.
	\end{description}	 
}
\onlySlide*{1}{
	\begin{center}
		\includegraphics[scale=0.2]{Figures/KopFracAB.eps}
		\includegraphics[scale=0.25]{Figures/ABreactionK.eps}
	\end{center}
	
	{\tiny R. Kopelman (1988) \emph{Science} \textbf{241} 1620--1626. \hfil}
}
\fromSlide*{2}{
	\begin{center}
		\includegraphics[scale=0.2]{Figures/KopFracAB.eps}
		\includegraphics[scale=0.25]{Figures/ABRate2.eps}	
	\end{center}
	
	{\tiny R. Kopelman (1988) \emph{Science} \textbf{241} 1620--1626. \hfil}
}
\end{wideslide}
%------------------------------------------------------------------------------
}%end of overlays

\begin{slide}{Concluding Remarks}
\vspace{-0.5cm}
\begin{description}
	\item[{\otherblue$\bullet$}] Continuous dynamics on fractal 
	sets differs in a qualitative way from dynamics involving discrete reactants. 
	\item[{\otherblue$\bullet$}] Lattice gas type modelling and 
	reaction-diffusion seem complimentary
	\item[{\otherblue$\bullet$}] We can define and analyse 
	reaction-diffusion models in a class of fractal sets: 
	\begin{description}
		\item[{\darkred$\bullet$}] Analogous sets based on a tetrahedron\ldots generally on an $n$-simplex.
		\item[{\darkred$\bullet$}] Post-critically finite sets (Kigami)
		\item[{\darkred$\bullet$}] Certain random fractals
	\end{description}
	
	\item[{\otherblue$\bullet$}] The behaviour is dependent on the 
	topology of the set rather than a particular embedding and so 
	continuous images of these sets could be used to model spatial detail.
	
	\item[{\otherblue$\bullet$}] Hybrid models which incorporate 
	both discrete and continuous approaches?

\end{description}
\end{slide}

\end{document}
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