Notes on Space- and Velocity-jump Models of Biological ... - CiteSeerX

Notes on Space- and Velocity-jump Models of Biological Movement Hans G. Othmer School of Mathematics University of Minnesota April 8, 2010

Contents 1 Background on continuum descriptions of motion

1

2 Elementary properties of random walks

3

3 Generalized random walks and their associated PDEs 8 3.1 Analysis of moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Diffusion limits for the exponential waiting time distribution . . . . . . . . . . . . . . . . . . . . . 12 4 Reinforced random walks

14

5 Velocity Jump Processes 5.1 The telegraph process in one space dimension . . . . . 5.2 The general velocity-jump process . . . . . . . . . . . . 5.3 The unbiased walk . . . . . . . . . . . . . . . . . . . . . 5.4 A biased walk in the presence of a chemotactic gradient 5.5 Inclusion of a resting phase . . . . . . . . . . . . . . . .

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17 18 20 21 24 26 29

Background on continuum descriptions of motion

In these notes we will describe deterministic continuum descriptions of the various stochastic processes used to describe movement of biological cells and organism1 . To understand the broad picture before delving into the details, let us first restrict attention to non-interacting particles. If the forces are deterministic and individuals are regarded as point masses their motion can always be described by Newton’s laws, and this leads to a classification of movement according to the properties of the forces involved. Initially the particles are regarded as structureless, but we admit the possibility that they can exert forces, and later we add internal states. Firstly, if the forces are smooth bounded functions the governing equations are smooth and the paths are smooth functions of time. In a phase space description in which the fundamental variables are position and 1 There

is a huge literature on this subject. A recent review that discusses some of the topics treated herein is given in Codling et al. (2008)

1

velocity, Newton’s equations are dx =v dt dv m = F. dt

(1) (2)

If we assume that the forces are independent of the velocity, then these are just the characteristic equations for the hyperbolic equation ∂ρ F + v · ∇x ρ + · ∇v ρ = 0. ∂t m

(3)

Here ρ is the density of individuals, so defined that ρ(x, v, t)dxdv is the number of individuals with position and velocity in the phase volume (dxdv) centered at (x, v). If we define the number density n and the average velocity u by ∫ n(x, t) = ρ(x, v, t)dv (4) ∫ n(x, t)u(x, t) = ρ(x, v, t)vdv, (5) then the evolution of these average quantities is governed by ∂n + ∇x · (nu) = 0. ∂t

(6)

If we admit impulsive (i.e., distributional) forces then we arrive at the second major type of movement, which is called a velocity jump process in Othmer et al. (1988). In this case the motion consists of a sequence of “runs” separated by re-orientations, during which a new velocity is chosen instantaneously. If we assume that the velocity changes are the result of a Poisson process of intensity λ, then in the absence of other forces we show later that we obtain the evolution equation ∫ ∂ρ + ∇x · vρ = −λρ + λ T (v, v0 )ρ(x, v0 , t) dv0 . (7) ∂t For most purposes one does not need the distribution ρ, but only its first few velocity moments. If we integrate this over v we again obtain (6). Similarly, multiplying (7) by v and integrating over v gives ∫ ∫ ∂(nu) + ∇ · ρvv dv = −λnu + λ T (v, v0 )vρ(x, v0 , t) dv0 dv. (8) ∂t Applications of this description will be given later and in subsequent lectures. The final description of motion, which in a sense is the roughest, is the familiar random walk, in which there are instantaneous changes in position at random times. These are called space-jump processes (Othmer et al. 1988), and later we show that the probability density for such a process satisfies the renewal equation ∫ t∫ φ(t − τ )T (x, y)P (y, τ |0) dy dτ.

ˆ P (x, t|0) = Φ(t)δ(x) + 0

(9)

Rn

Here P (x, t|0) is the conditional probability that a walker who begins at the origin at time zero is at x at time t, ˆ φ(t) is the density for the waiting time distribution, Φ(t) is the complementary cumulative distribution function associated with φ(t), and T (x, y) is the redistribution kernel for the jump process. If the initial distribution is given by F (x) then ∫ n(x, t) ≡ P (x, t|x0 )F (x0 ) dx0 Rn

2

can be regarded as the number density of identical non-interacting walkers at x at time t. Clearly n(x, t) satisfies ∫ t∫ ˆ n(x, t) = Φ(t)F (x) + φ(t − τ )T (x, y)n(y, τ ) dy dτ. (10) Rn

0

The final description used is one in which the changes of position or velocity are not generated by a jump process, but rather by the presence of small fluctuating components of velocity and/or position. This leads to the familiar stochastic differential equations dx = vdt + dX

(11)

mdv = Fdt + dV where X and V are random displacements and velocities, respectively. This approach leads to a Fokker-Planck equation under suitable conditions on the fluctuating forces (Kampen 1981).

2

Elementary properties of random walks

We begin with an unbiased random walk on a lattice in Z in which the walker takes a step of length ∆ at intervals τ to one of its nearest neighbors, each step having probability 1/22 . We then ask what is the probability that a walker beginning at the origin will be at site m∆ > 0 after N steps. We first note that every path of length N has the same probability, namely (1/2)N . Secondly, in order to be at m∆ after N steps the walker must have taken m more steps in the positive direction than in the negative direction, i.e. there must be (N + m)/2 steps to the right and (N − m)/2 steps to the left. (Note that m must be even or odd accordingly as N is even or odd.) Thus the probability p(m, N ) of being at m after N steps after whatever path is p(m, N ) =

( )N ( ) N 1 N +m 2 2

(12)

() where ·· is the binomial coefficient. If we assume that there are many steps but a small net displacement, then N is large and m  N , and we use Stirling’s approximation log n! = (n + 1/2) log n − n + 1/2 log 2π + O(1/n) for N → ∞, to obtain

[ ( ] N m) 1 1+ log p(m, N ) ∼ (N + 1/2) log N − (N + m + 1) log 2 2 N [ ( ] ) N m 1 −1/2(N − m + 1) log 1− − log 2π − N log 2. 2 N 2

Because m  N, we can expand the logarithm to obtain 1 1 log p(m, N ) ∼ − log N + log 2 − log 2π − m2 /2N 2 2 √

or p(m, N ) ∼

2 −m2 /2N e . πN

Now let x = m∆ and t = N τ , and define ( P (x, t)dx = p 2 See

Chandrasekhar (1943) for an ecomprehensive development.

3

x t , ∆ τ

)

dx 2∆

for then

2 2 1 P (x, t)dx = √ e−x /2∆ t/τ dx. 2 2π∆ t/τ

Thus far ((x, t) is only defined on a lattice, but if we let τ → 0 and ∆ → 0 while holding ∆2 = constant τ we obtain P (x, t) = √

= 2D

2 1 e−x /4Dt , 4πDt

(13)

which is now defined for (x, t) ∈ R×R+ . It is easy to verify that P (x, t) is a solution, in fact called the fundamental solution or Green’s function, for the parabolic initial-value problem ∂P ∂t

= D

P (x, 0)

∂2P ∂x2

x∈R

t ∈ R+ (14)

= δ(x)

where δ(·) is the Dirac distribution. It is easy to show that ∫ ∞ P (x, t)dx = −∞ ∫ ∞ xP (x, t)dx = −∞ ∫ ∞ x2 P (x, t)dx =

1 0

(15)

2Dt

−∞

Remark 1 An alternate approach is to begin with a continuus-time random walk on a lattice, in which one specifies a rate of jumping, rather than the interval between jumps. If the jumps are governed by a Poisson process of intensity λ and the jumps are restricted to nearest neighbors, then the probability P (n, t) of being at n at time t satisfies the Kolmogorov forward equation (or master equation) ∂P (n, t) ∂t

= λP (n − 1, t) − 2λP (n, t) + λP (n − 1, t)

P (n, 0) =

(16)

δ(n − n0 )

In this form one sees that the right-hand side can be viewed as a second-order finite difference approximation to the right-hand side of (14). One can also see how the diffusion coefficient should be defined in an appropriate limit. Later we will consider generalizations of this in which the intensities depend on another field. Next we introduce a barrier at y > 0. In the case of a reflecting barrier the method of images (cf. Figure 1) shows that 2 2 1 P (x, t) = √ {e−x /4Dt + e−(2y−x) /4Dt }, (17) 4πDt while for an absorbing barrier P (x, t) = √ In the former case it follows that

2 2 1 {e−x /4Dt − e−(2y−x) /4Dt }. 4πDt

∂P =0 ∂x x=y

whereas in the latter case P (y, t) = 0 4

(18)

t

x y

2y

Figure 1: A path and its image for the case of a reflecting barrier. thus confirming the physical meaning of these boundary conditions. The foregoing can be generalized to an arbitrary domain in the following sense. Let Ω = Rn or a bounded subset of Rn with a smooth boundary. Let x denote the coordinates of a point in Ω. Then the probability that a walker who begins at x0 ∈ Ω at t = 0 is at x ∈ Ω at t > t0 satisfies the diffusion equation. ∂P ∂t

= D∆P (19) δ(x − x0 ),

P (x, 0) =

plus boundary conditions if Ω is a bounded domain. Thus if one can find the Green’s function for (19) one can immediately determine the probability. In any case, if the domain is compact and has a smooth boundary then the solution of (19) has the eigenfunction expansion ∑ P (x, t) = an e−λn t ψn (x) (20) n

where the functions {ψn } satisfy D∆ψn = −λn ψn

(21)

with the appropriate boundary condition, and for a smooth domain these form a complete set. This eigenfunction expansion proves useful for many computations, for example, for funding the mean lifetime of a walker, or the time to first capture. Consider a bounded domain Ω with a smooth boundary. At t = 0 ∫ P (x, 0)dx = 1 Ω

but for t > 0 the integral may be less than one if the walker can escape, i.e., if either P = 0 on some portion of ∂Ω or a non-zero flux is specified on some portion of the boundary. In either case, if we integrate (19) over the domain and apply the divergence theorem then ∫ ∫ d P (x, t)dx = − n · j dS (22) dt Ω ∂Ω where n is the outward normal and j is the flux. For the diffusion equation we have j = −D∇x P 5

and therefore (22) may be written d dt

∫

∫ n · D∇x P dS.

P (x, t)dx = Ω

(23)

∂Ω

The right-hand side of this equation gives the probability per unit time of leaving the domain Ω, and so we define the waiting time distribution φ(t) as ∫ φ(t) = −

n · D∇x P dS.

(24)

∂Ω

This only makes sense if the integral is strictly non-positive, since otherwise the probability of escape may be negative. One may think of φ as the mean waiting time for crossing of the boundary in the following sense. If T is the first time the walker reaches the boundary, given that it begins in the interior of the domain, then φ(t)dt = P r{t ≤ T ≤ t + dt} From the definition of φ it follows that ∫ t∫ ∫ t Φ(t) ≡ φ(s)ds = − 0

0

n · D∇x P (x, s)dSds

(25)

∂Ω

is the probability that the walker will escape in the interval (0, t), i.e. that the walker lifetime in Ω is less than t. Similarly ∫ ∞∫ ˆ Φ(t) = − n · D∇x P (x, s)dSds = 1 − Φ(t) t

∂Ω

is the probability that the walker is still in Ω at t. The mean lifetime of a walker in Ω is ∫ ∞ λ−1 = sφ(s)ds 0

∫ =

−

(26)

∫

∞

n · D∇x P (x, s)dSds

s 0

∂Ω

and λ is the mean rate at which walkers leave Ω. If P (x, t) has the form given in (20) then ∫ ∑ −λn t φ(t) = − an e n · ∇x ψn dS ∂Ω

n

and therefore λ−1 =

] ∑ ∑ an [ ∫ bn − n · ∇x ψn dS ≡ λn λn ∂Ω n n

(27)

Here the an are determined by the initial data and (λn , ψn ) is an eigen-pair for the Laplacian on Ω. As an example, consider a one-dimensional domain [L1 , L2 ] with homogeneous conditions at the ends; Neumann at x = L1 and Dirichlet at x = L2 . We have to solve the problem ∂P ∂t P ∂P ∂x

∂2P x ∈ (L1 , L2 ) ∂x2 = 0 at x = L1 = D

(28) = 0

P (x, 0) =

at x = L2

δ(x − x0 )

Without loss of generality we let L1 = 0 and set L2 = L.

6

x0 ∈ (L1 , L2 )

One finds via separation of variables that λn

( =

n+ √

ψn and therefore

=

1 2

)2

π2 D L2 (29)

2 πx sin(n + 1/2) L L

( ) ( ) ∞ 2 ∑ −(n+ 12 )22 π2 Dt 1 πx 1 πx0 L P (x, t) = e sin n + sin n + . L n=0 2 L 2 L

Note that at t = 0 this reduces to

( ( ) ) 1 πx 1 πx0 2∑ sin n + sin n + L n 2 L 2 L

P (x, 0) =

= δ(x − x0 ). Thus

) −(n+ 1 )2 π2 Dt ( ) ∞ ( 2 2πD ∑ 1 1 πx0 2 L φ(t) = 2 n+ e sin n + L n=0 2 2 L

and

1 πx0 ∞ 2L2 ( x0 )3 ∑ sin(n + 2 ) L λ = tφ(t)dt = . 1 3 πx0 3 D L 0 ) n=0 (n + 2 ) ( L = 0, and if x0 = L one finds that −1

If x0 = 0 then clearly λ−1

∫

(30)

∞

λ−1 =

∞ 0.4204L2 2L2 ∑ sin(n + 21 )π ∼ . D n=0 [(n + 12 )π]3 D

(31)

(32)

If we define the average lifetime over all initial points as ∫ 1 L −1 λ−1 = λ dx L 0 then we find from (31) that τ1 ≡ λ−1

=

( )4 ∞ ∞ 2L2 ∑ 1 2 L2 ∑ 1 = 2 · 1 4 4 π D n=0 (n + 2 ) π D n=0 (2n + 1)4

=

L2 3D

where we have used the fact that

(33)

∞ ∑

1 π4 = 4 (2n + 1) 96 n=0 Note that the average lifetime is approximately 3/4 of the largest lifetime, and that τ1 → ∞ as either L → ∞ or D → ∞, as we should expect. It is also clear that if L1 > 0 then we can simply replace L by L2 − L1 in the above derivation. Remark 2 There are many ways to generalize this analysis. One is to consider the role of dimensionality in search problems. For example, consider the analog of the foregoing in on a square in 2D, with homogeneous Neumann data on three sides and homogoeneous Dirichlet on the fourth. Then it can be shown that the average lifetime is the same as in the 1D problem, despite the fact that there is a larger area to explore. A second generalization is to show that one doesn’t even have to solve for the Green’s function for the transient problem: λ−1 (x0 ) ≡ τ (x0 ) is the solution of the nonhomogeneous problem ∇2 τ = −1/D with appropriate boundary conditions. 7

3

Generalized random walks and their associated PDEs

In this section we show that the theory of random space jump processes can be generalized considerably3 . Consider a random jump process on Rn in which the walker executes a sequence of jumps of negligible duration, and suppose that the waiting times between successive jumps are independent and identically distributed. That is, if the jumps occur at T0 , T1 , . . . then the increments Ti − Ti−1 are identically and independently distributed, and therefore the jump process is a semi-Markov process (Feller 1968; Karlin and Taylor 1975). Let T be the waiting time between jumps and let φ(t) be the density for the waiting time distribution. T is experimentally observable, and in principle φ(t) can be determined from experimental observations. If a jump has occurred at t = 0 then φ(t) = P r{t < T ≤ t + dt}. The cumulative distribution function for the waiting times is ∫ t Φ(t) = φ(s) ds = P r{T ≤ t} 0

and the complementary cumulative distribution function is ∫ ∞ ˆ Φ(t) = φ(s) ds = 1 − Φ(t) = P r{T ≥ t} t

For example, if the jumps are governed by a Poisson process then Φ(t) = 1 − e−λt and φ(t) = λe−λt . This is the only smooth distribution for which the jump process is Markovian (Feller (1968), p. 458). Next we must specify how jumpers are redistributed in space, given that a jump occurs. For simplicity we shall assume that the spatial redistribution that occurs at jumps is independent of the waiting time distribution. Thus the probability of a transition from y to x at time t will simply be the product of Φ(t) times the function that gives the probability of the jump from y to x. This assumption of statistical independence between the event of deciding to jump and the event of deciding where to jump may clearly be too restrictive for some systems, for the direction and length of a jump may very well depend on the time elapsed since the last jump. Our formulation of the velocity jump process will incorporate some types of directional persistence, but for now we shall, in effect, assume that we have infinitely energetic jumpers that have no recollection of their previous location. Let T (x, y) be the probability density function for a jump from y to x. That is, if X(t) is a random variable giving the jumper’s position at time t, then given that a jump occurs at Ti , T (x, y) dx = P r{x ≤ X(Ti+ ) ≤ x + dx |X(Ti− ) = y},

(34)

where the superscripts ± denote limits from the right and left, respectively. This definition allows for the possibility that the underlying medium is spatially nonhomogeneous and nonisotropic, in which case the transition probability depends on x and y separately. In the case of a homogeneous and isotropic medium T (x, y) = T˜( x − y ), where T˜ gives the absolute (unconditioned) probability of a jump of length x − y . One of the purposes of the analysis is to show how the functions φ(t) and T (x, y) can be related to experimentally observable quantities. The statistics most accessible from observations are the various moments of the displacement and their dependence on t. To relate these to φ and T we must derive an evolution equation for the density function P (x, t|0), which is defined so that P (x, t|0)dx is the probability that the position of a jumper which begins at the origin at time t = 0 lies in the interval (x, x + dx) at time t. We shall derive this equation via equations for some auxiliary quantities. Let Qk (x, t) be the conditional probability that a jumper which begins at x = 0 at t = 0 takes its k th step at t− and lands in the interval (x, x + dx). Then it is clear that for x > 0, t > 0, Qk satisfies the first-order integro-difference equation 3 See

also Othmer et al. (1988) for the analysis in this section.

8

∫ t∫ φ(t − τ )T (x, y)Qk (y, τ ) dy dτ.

Qk+1 (x, t) = Rn

0

Summing this over k we obtain the density function for arriving in the interval (x, x + dx) at time t− after any number of steps. Thus we obtain the Volterra integral equation ∞ ∑

Q(x, t) =

∫ t∫ φ(t − τ )T (x, y)Q(y, τ ) dy dτ

Qk (x, t) = Q0 (x, t) +

(35)

Rn

0

k=0

and this must satisfy the initial condition Q(x, 0) = δ(x). Consequently (35) becomes ∫ t∫ φ(t − τ )T (x, y)Q(y, τ ) dy dτ.

Q(x, t) = δ(x)δ(t) + 0

Rn

The probability density function P (x, t|0) for the conditional probability that X(t) lies in (x, x + dx) at time t can be computed as the product of the probability of arriving in this interval at some time τ < t, multiplied by the probability that no transition occurs in the remaining time t − τ . Thus P (x, t|0) ∫ t ˆ − τ )Q(x, τ ) dτ = Φ(t 0 ∫ t ∫ τ∫ ˆ = Φ(t − τ ){δ(x)δ(τ ) + φ(τ − s)T (x, y)Q(y, s) dy ds} dτ 0 0 Rn (∫ ) ∫ t∫ t ˆ ˆ = Φ(t)δ(x) + Φ(t − τ )φ(τ − s) dτ T (x, y)Q(y, s)dy ds. 0

Rn

(36)

s

On the other hand, it follows from (36) that ∫ t∫ 0

φ(t − τ )T (x, y)P (y, τ |0) dy dτ ∫ t∫ ∫ τ ˆ − s)T (x, y)Q(y, s) dy ds dτ = φ(t − τ )Φ(τ 0 Rn 0 ) ∫ t ∫ (∫ t ˆ − s)φ(t − τ ) dτ T (x, y)Q(y, s) dy ds. = Φ(τ Rn

Rn

0

It is easy to show that

∫

s

∫

t

t

ˆ − τ )φ(τ − s) dτ = Φ(t s

ˆ − s) dτ φ(t − τ )Φ(τ s

by setting u = t − s, z = τ − s, and observing that the resulting integrals have the same Laplace transforms. Thus P (x, t|0) satisfies the renewal equation: ∫ t∫ φ(t − τ )T (x, y)P (y, τ |0) dy dτ.

ˆ P (x, t|0) = Φ(t)δ(x) + 0

If the initial distribution is given by F (x) then ∫ n(x, t) ≡

(37)

Rn

P (x, t|x0 )F (x0 ) dx0

Rn

can be regarded as the number density of identical non-interacting jumpers at x at time t. Clearly n(x, t) satisfies ∫ t∫ φ(t − τ )T (x, y)n(y, τ ) dy dτ.

ˆ n(x, t) = Φ(t)F (x) + 0

Rn

9

(38)

In order that the total number of jumpers be conserved in the jump process it is necessary that ∫ ∫ N (t) = n(x, t) dx = N0 ≡ F (x) dx Rn

i.e., that

∫

Rn

∫ t∫ φ(t − τ )T (x, y)n(y, τ ) dy dτ dx = N0

ˆ Φ(t)N 0+ Rn

Rn

0

We assume that T ∈ L1 (Rn × Rn ), and therefore the x and y integrations can be interchanged by Fubini’s theorem. It follows that the necessary and sufficient condition for conservation of jumpers is that ∫ T (x, y) dx = 1 Rn

Hereafter we assume that Φ and T have the proper normalizations and sufficient regularity that the indicated operations make sense. Special choices of φ and T lead to some of the standard random jump problems treated in the literature. For instance, if φ(t) = δ(t − t0 ) then Φ(t) = H(t0 − t), where H(·) is the Heaviside function, and (37) reduces to ∫ P (x, t|0) = H(t0 − t)δ(x) + [1 − H(t0 − t)] T (x, y)P (y, t − t0 |0) dy. Rn

This is the governing equation for a discrete time, continuous space process in which jumps occur at intervals of t0 . If in addition the support of T is concentrated on the points of a lattice Z n ⊂ Rn , then ∑ P (xi , t|0) = H(t0 − t)δi0 + [1 − H(t0 − t)] Tij P (xj , t − t0 |0) . j

where δi0 is the Kronecker delta, and xi is a lattice point. This can be written in the more conventional ChapmanKolmogorov form as follows. ∑ Pi0 (n + 1) = j Tij Pj0 (n) n≥1 Clearly the underlying process is Markovian for the above choice of φ. If the support of the kernel T (x, y) is a lattice and the waiting time distribution is exponential, as in a Poisson process, then one obtains the continuous time random walk ∑ ∂P (xi , t|0) = −λP (xi , t|0) + λ Tij P (xj , t|0). (39) ∂t j

3.1

Analysis of moments

As we remarked earlier, one of our purposes is to relate φ and T to the experimental observations. The statistics most accessible from observations are the various moments of the displacement, in particular their dependence on t. We shall compute these moments from (37), and for illustrative purposes we assume that the medium is one-dimensional and spatially homogeneous. Define ∫ +∞ hxn (t)i = xn P (x, t|0) dx ∫

−∞ +∞

∫ t∫

+∞

= −∞

0

−∞

Let

xn T˜(x − y)φ(t − τ )P (y, τ |0) dy dτ dx ∫

mk =

+∞

xk T˜(x) dx −∞

10

(40)

be the k-th moment of T˜ about zero. Then (40) can be written   ∫ t∑ n n   n−k hxn (t)i = (τ )i dτ.  mk φ(t − τ )hx  0 k=0 k

(41)

It follows that all the moments of x(t) can be gotten by solving a sequence of linear integral equations of convolution type. Let ∫ ∞ e−sτ hxk (τ )i dτ

Xk (s) = L{hxk (t)i} ≡

0

¯ = L{φ(t)}. Then one finds that be the Laplace transform of the k-th moment, and let φ(s) ¯ m1 φ(s) ¯ s 1 − φ(s) ( ¯ m2 ) φ(s) X2 (s) = 2m1 X1 (s) + ¯ s 1 − φ(s) X1 (s) =

(42)

If the first moment of T˜ vanishes then these simplify to X1 (s)

= 0 ¯ m2 φ(s) X2 (s) = ¯ . s 1 − φ(s)

(43)

The asymptotic behavior of the moments can be gotten by applying limit theorems for Laplace transforms (Widder 1946), but we shall merely illustrate the dependence of X2 on t for two particular choices of φ. Firstly, suppose that m1 = 0 and that φ(t) = λe−λt (44) ¯ = λ/(s + λ) and it follows which is the density function for an exponential waiting time distribution. Then φ(s) that ( ) ∫ t λ hx2 (t)i = m2 L−1 dτ = m2 λt. (45) s 0 Secondly, if we choose φ(t) = λ2 te−λt

(46)

which is the density function for a gamma waiting time distribution with parameters (2, λ), then ¯ = φ(s) One finds that

∫

t

hx2 (t)i = m2 0

L−1

(

λ2 s(s + 2λ)

λ2 . (s + λ)2 ) dτ =

m2 λ 2

{ } 1 t− (1 − e−2λt ) , 2λ

(47)

which is shown in Figure 2.7. It is clear from the analysis given earlier that (45) predicts the same mean squared displacement as a diffusion process with diffusion coefficient D = m2 λ/2. Similarly (46) leads to the same mean squared displacement as the telegraph process discussed later. Of course neither fact proves that the processes defined by (44) and (46) are diffusion and telegraph processes, respectively, but an experimentalist who can reliably measure only the first two moments of the displacement could not distinguish them from these processes. It is noteworthy that this conclusion holds under the reasonable hypothesis that the first two moments of T are finite, without any condition on the higher moments.

11

2





(a)

(b)

t

t

Figure 2: Theoretical curves of mean-squared displacement sketched for the space jump process with (a) exponential and (b) gamma waiting time distributions.

3.2

Diffusion limits for the exponential waiting time distribution

The results given by (45) and (47) raise the question as to whether, for some choice of T , the corresponding integral equations are equivalent to the diffusion and telegraph equations, respectively, in an appropriate limit. Consider first the choice φ(t) = λe−λt which leads to (45). After differentiating (37) and rearranging one finds that ∫ ∂P = −λP + λ T˜(x − y)P (y, t) dy (48) ∂t R where here and hereafter we suppress the conditioning argument in P . If 1 T˜(x − y) = [δ(x − y − ∆) + δ(x − y + ∆)] 2

(49)

then

∂P λ = [P (x + ∆, t) − 2P (x, t) + P (x − ∆, t)]. ∂t 2 The right-hand side can be written λ∆2 ∂ 2 P [ + O(∆2 )], 2 ∂t2 and therefore, in the diffusion limit (λ → ∞, ∆ → 0, λ∆2 = constant ) we obtain ∂2P ∂P =D 2 , ∂t ∂t

(50)

provided that the higher-order derivatives included in O(∆2 ) are bounded. In fact, a similar result holds in any dimension. Let δ( x − y − ∆) T˜(x − y) = ∆n−1 ωn where ωn is the surface measure of the unit sphere in Rn . For this choice of T˜ one finds that ∂P = λ[P¯ (x, ∆, t) − P (x, t)] ∂t where P¯ is the average of P over the surface of a sphere of radius ∆ centered at x. By expanding P about x and performing the indicated average one finds that in the diffusion limit ∂P = D∇2 P, ∂t

(51)

provided that P varies smoothly, i.e., provided that all higher-order derivatives are bounded. Here D ≡ λ∆2 /2n is the diffusion coefficient in n dimensions.

12

A more realistic choice for T˜(x − y) in one space dimension is a sum of Gaussians, one centered at +∆ and one centered at −∆. Thus suppose that For this kernel it is more convenient to work with the Fourier transform of (48). If P is absolutely integrable in x its Fourier transform is defined as ∫

+∞

Pˆ (k, t) =

eikx P (x, t) dx. −∞

Since

} 2 2 1 { i∆k−σ2 k2 /2 e + e−i∆k−σ k /2 Tˆ(k) = 2

it follows that Pˆ (k, t) satisfies the ordinary differential equation { } 2 2 dPˆ = λ −1 + (cos ∆k)e−σ k /2 Pˆ (k, t) dt Upon expanding the bracketed term and collecting like powers of k one finds that dPˆ λk 2 2 λk 4 =− [σ + ∆2 ]Pˆ + [3σ 4 + 6∆2 σ 2 + ∆4 ]Pˆ + O(k 6 ) dt 2 4! To obtain the Fourier transform of a second-order operator on the right-hand side we must let λ → ∞ and (σ, ∆) → (0, 0) in such a way that λ[σ 2 + ∆2 ] → constant

(52)

and λ[3σ 4 + 6∆2 σ 2 + ∆4 ] → 0 Thus it suffices to require that

λ

→ ∞

∆

→ 0

λ∆

2

→

constant (53)

In this case the diffusion coefficient is the limiting value of λ[σ 2 + ∆2 ]. If σ/∆ ∼ o(1) as ∆ → 0 then the term involving σ 2 in the diffusion coefficient vanishes. A similar conclusion holds for much more general kernels T˜. Suppose that T˜ has the form 1 x−y T˜(x − y) = T0 ( , ∆). ∆ ∆ Then

( ∫ ) ( 2∫ ) 2 ∂P ∆ ∂ P ∂P 2 =λ ∆ T0 (r, ∆)rdr +λ T0 (r, ∆)r dr + O(∆3 ). 2 ∂t ∂x 2 ∂t R R

(54)

It follows that if the first moment of T0 is O(∆) for ∆ → 0, if the second moment of T0 tends to a constant, and if all higher moments are bounded, then in the diffusion limit (λ → ∞, ∆ → 0, λ∆2 = constant ) we obtain a diffusion equation with drift. The diffusion coefficient is given by ∫ ∆2 D=λ T0 (r, ∆)r2 dr (55) lim 2 ∆→0 R and the drift coefficient is given by β=λ

∆2 lim 2 ∆→0

∫ R

13

T0 (r, ∆) rdr. ∆

(56)

If the kernel is symmetric then the drift coefficient vanishes. The reader can check that the foregoing conditions are satisfied for the kernel { } 2 2 2 2 1 e−(x−y−∆) /2σ + e−(x−y+∆) /2σ . T˜(x − y) = √ 2 2πσ 2 provided that σ/∆ ∼ O(1) as

4

∆ → 0.

Reinforced random walks

The rigrorous analysis of random walks is more complicated when particle interactions, either direct or indirect, are taken into account (cf. Spohn (1991); Oelschl¨ager (1987)). Thus for instance in the case of myxobacteria, a bacterium gliding on a slime trail reacts to its own contribution to these trails and to the contributions of the other bacteria. Similarly bacteria that release an attractant and react to that released by others interact indirectly via the attractant. There is a growing mathematical literature on what are called reinforced random walks that began with the work of Davis (1990); a recent review can be found in Pemantle (2007). Here we sketch the approach developed in Othmer and Stevens (1997), where the particle motion is governed by a jump process and the walkers modify the transition probabilities on intervals for subsequent transitions of an interval. Davis (1990) considered a reinforced random walk for a single particle in one dimension. Initially there is a weight wni on each interval (i, i + 1), i ∈ ZZ which is equal to wn0 . If at time n an interval has been crossed by the particle exactly k times, its weight will be wni = wn0 +

k ∑

aj ,

j=1

where aj ≥ 0 , j = 1, ..., k. Furthermore, the transition probablilites are given by P (xi+1 = n + 1|xi = n) =

wni

wni i + wn−1

Davis’ main theorem asserts that localization of the particle will occur if the weight on the intervals grows quickly enough with each crossing, as summarized in the following. Let ( )−1 ∞ n ∑ ∑ X ≡ {xi , i ≥ 0} and φ(a) ≡ 1+ ai n=1

i=1

Theorem Suppose that wn0 = 1. Then (i) If φ(a) = ∞ then X is recurrent (ii) If φ(a) < ∞ then X has finite range and there are random integers n and I such that xi ∈ (n, n+1) if i > I Here reccurent means that every integer is visited infinitiely often a.s., i.e.,the walker does not become trapped. From this it follows that if aj = constant, for instance, which corresponds to linear growth of the weight, then X is recurrent almost surely, whereas if the growth is superlinear then the particle oscillates between two random integers almost surely after some random elapsed time. Since the result deals with a single particle it does not directly address the aggregation of particles, but it does at least suggest that if the particles interact only through the modification of the transition probability there may be aggregation if this modification is strong enough. This theorem motivated the following development, taken from Othmer and Stevens (1997), in which we begin with a master equation for a continuous-time, discrete-space random walk. and we postulate a generalized form of (39) in which the transition rates depend on the density of a control or modulator species that modulates the 14

transition rates. We restrict attention to one-step jumps, although it is easy, using the framework given earlier, to apply this to general graphs, but one usually does not obtain diffusion equations in the continuum limit. Suppose that the conditional probability pn (t) that a walker is at n ∈ ZZ at time t, conditioned on the fact that it begins at n = 0 at t = 0, evolves according to the continuous time master equation ∂pn + − = Tˆn−1 (W ) pn−1 + Tˆn+1 (W ) pn+1 − (Tˆn+ (W ) + Tˆn− (W )) pn . ∂t

(57)

Here Tˆn± (·) are the transition probabilities per unit time for a one-step jump to n ± 1, and (Tˆn+ (W ) + Tˆn− (W ))−1 is the mean waiting time at the nth site. We assume throughout that these are nonnegative and suitably smooth functions of their arguments. The vector W is given by W

=

(· · · , w−n−1/2 , w−n , w−n+1/2 , · · · , wo , w1/2 , · · · ).

(58)

Note that the density of the control species w is defined on the embedded lattice of half the step size. The evolution of w will be considered later; for now we assume that the distribution of w is given. Clearly a timeand p-independent spatial distribution of w can model sa heterogeneous environment, but this static situation is not treated here. As (57) is written, the transition probabilities can depend on the entire state and on the entire distribution of the control species. Since there is no explicit dependence on the previous state the jump process may appear to be Markovian, but if the evolution of wn depends on pn , then there is an implicit history dependence, and the space jump process by itself is not Markovian. However, if one enlarges the state space by appending the w one gets a Markov process in this new state space. Three distinct types of models are developed and analyzed in Othmer and Stevens (1997), which differ in the dependence of the transition rates on w; (i) strictly local models, (ii) barrier models, and (iii) gradient models. In the first of these the transition rates based on local information, so that Tˆn± = Tˆ (wn ), and to ssimplify the analysis we assume that the jumps are symmetric, so that Tˆ + = Tˆ − ≡ Tˆ . In this case (57) reduces to ∂pn = Tˆ (pn−1 , wn−1 )pn−1 + Tˆ (pn+1 , wn+1 )pn+1 − 2Tˆ (pn , wn )pn , ∂t and in the formal diffusion limt lim λh2 = constant ≡ D

h→0 λ→∞

we obtain the nonlinear diffusion equation ∂p ∂2 = D 2 (T (w)p). ∂t ∂x

(59)

wherein the flux is defined as

∂ (T (w)p). ∂x The second type is one called a barrier model, for which there are two sub-cases, dending on whether or not the transition rates are renormalized. In the first case we assume that j = −D

Tˆn± (W ) = Tˆ (wn±1/2 ) which leads to ∂p =D ∂τ

{

∂2 ∂ (T (p, w)p) − 2 ∂x ∂x

( pTw (p, w)

∂w ∂x

in one space dimension, and to ∂p = D∆(T (w)p) − D∇ · (pTw ∇w). ∂t in general. The flux is now given as j = −D∇(T p) + DpTw ∇w = −DT ∇p, 15

)} ,

Clearly the general equation can also be written as ∂p = D∇ · (T ∇p). ∂t One may also renormalize the transition rates so that λ(Tˆn+ (W ) + Tˆn− (W )) = constant ≡ λ and then define Nn± (W ) =

T (wn±1/2 ) T (wn+1/2) + T (wn−1/2 )

+ Nn−1 (wn−1/2 , wn−3/2 ) =

T (wn−1/2 ) T (wn−3/2) + T (wn−1/2 )

− Nn+1 (wn+1/2 , wn+3/2 ) =

T (wn+1/2 ) T (wn+3/2) + T (wn+1/2 )

(60)

The master equation then reads 1 ∂pn λ ∂t

=

N + (wn−1/2 , wn−3/2 )pn−1 + N − (wn+1/2 , wn+3/2 )pn+1 ( ) − N + (wn+1/2 , wn−1/2 ) + N − (wn−1/2 , wn+1/2 ) pn

and in the diffusion limit this leads to ∂p ∂t

= D

∂ ∂x

( ) ∂ ( p) p ln ∂x T

(61)

For later use we define the chemotactic velocity and sensitivty as χ = D (lnT )w

u = −D

∂ 0 ∂w ln p + D (lnT (w)) . ∂x ∂x

Thus the taxis is positive if T 0 (w) > 0. If we set T (w) = α + βw, (61) reduces to ( ) ∂p ∂ ∂p β ∂w =D −p ∂t ∂x ∂x α + βw ∂x

(62)

and we use this form later in examples. + The last type of model is the gradient-based, or look-ahead model, for which Tn−1 = α + β(τ (wn ) − τ (wn−1 )) − and Tn+1 = α + β(τ (wn ) − τ (wn+1 )), α ≥ 0, which leads to [ ( )] ∇p β ∂p = Dα∇ · p − 2 τw ∇w ∂t p α if the rates are not renormalized, and if they are, leads to { } ∂p 1 βτw = D∇ · ∇p − p ∇w ∂t 2 α The results for the different types of models is summarized in Table 1. Of course we also have to specify the local dynamics for the evolution of w, and here we use the general form ∂w ∂t

=

pw p + γr − µw ≡ R(p, w) 1 + λw K +p

in the examples shown if Figure 3. For all cases we set D = 0.36, and in the first panel we show the solution of (62) and (4) for α = γr = µ = 0 and β = 1, λ = 1. × 10−5 . The second panel is as in the first, but with λ = 0, 16

Table 1: Dependence of the response on the sensing mechanism

1.

Type of

Taxis

Chemotactic

Type of

Sensing

Velocity

Sensitivity

Taxis

Local

-D∇T

−DT 0 (w)

0

0

D∇lnT

D (lnT (w))0

2D∇lnT

2D (lnT (w))0

2Dβ∇τ

2Dβτ 0 (w)

β Dα ∇τ

β 0 Dα τ (w)

Negative if T 0 (w) > 0

Barrier without 2.

re-normalization Barrier with

3.

re-normalization Nearest neighbor with

4.

re-normalization Gradient without

5.

re-normalization Gradient with

6.

re-normalization

None Positive if T 0 (w) > 0 Positive if T 0 (w) > 0 Positive if βτ 0 (w) > 0 Positive if βτ 0 (w) > 0

and in the third panel a more complicated transition rate is used (cf. Othmer and Stevens (1997)). One sees in that figure that both the dependence of the transition rates on the local modulator w, and the dynamics of w itself, play an important role in the dynamics of the system. In the first panel the solution stabilizes at some smooth distribution, in the second panel th esolution blows up in finite time (around t - 9.3 – this assertion is supported by analysis of the Foruier components) and in the thord panel the solution ulitmately collapses, in a very intersting step-wise fashion that is not understood at present.

(a)

(b)

(c) Density

Density

Density

100

25

0 Ti m

e

9.3

0

0

e

Tim

x

x 1

1

x

e

Tim

1

Figure 3: Three examples of the dynamics. See Othmer and Stevens (1997) for details.

5

Velocity Jump Processes

The prototypal organisms whose motion can be described as a velocity jump process are the flagellated bacteria, the best studied of which is E. coli. To search for food or escape an unfavorable environment, E. coli alternates two basic behavioral modes, a more or less linear motion called a run, and a highly erratic motion called tumbling, the purpose of which is to reorient the cell (cf. Figure Fig-02). Run times are typically much longer than the

17

The unbiased process

The biased process 111111111111111111111111 000000000000000000000000 000000000000000000000000 111111111111111111111111

11 00 00 11

1 0 1 0

1 0 0 1

1 0 0 1

Figure 4: The movement of a particle executing a velocity-jump. time spent tumbling, and when bacteria move in a favorable direction (i.e., either in the direction of foodstuffs or away from harmful substances) the run times are increased further. During a run the bacteria move at approximately constant speed in the most recently chosen direction. New directions are generated during tumbles, and when bacteria move in an unfavorable direction the run length decreases and the relative frequency of tumbling increases. The distribution of new directions is not uniform on the unit sphere, but has a bias in the direction of the preceding run. The effect of alternating these two modes of behavior, and in particular, of increasing the run length when moving in a favorable direction, is that a bacterium executes a three-dimensional random walk with drift in a favorable direction when observed on a sufficiently long time scale (Koshland, 1980; Berg, 1983). We begin with a simple example that illustrates the main points. We assume as before that there is no onteraction between walkers, and therefore can consider either the probability of a single walker being at a given position with a given velocity at time t, or the density of walkers. Here we choose the latter.

5.1

The telegraph process in one space dimension

Supose that the walkers are confined to the interval [0, 1] with homogeneous Neumann data at the ends, that the speeds s± to the right and left are constant, and that direction is reversed at random instants governed by Poisson processes of intensity λ± . Let p± denote the density of walkers moving to the right and left, respectively. Then the conservation equations for these densities are ∂p+ ∂(s+ p+ ) + ∂t ∂x

= −λ+ p+ + λ− p−

∂p− ∂(s− p− ) − ∂t ∂x

= λ p −λ p .

(63) + +

− −

Define p ≡ p+ + p− and j ≡ (s+ p+ − s− p− ); then these can be written in the alternative form ∂j ∂p + ∂t ∂x

=

∂j + 2λj ∂t

∂ + + ∂ = −s (s p ) − s− (s− p− ) + λ(s+ p− − s− p+ ) ∂x ∂x

0 (64)

+

To illustrate the essence of chemotaxis in this simple context, we ask how the walkers should modify their behavior so as to produce a nonuniform distribution on the interval. We consider three possible cases. Case I: Constant and equal speeds and turning rate – λ+ = λ− and s+ = s− . Combining the two equations in (64) leads to the classical telegrapher’s equation ∂p ∂2p ∂2p + 2λ = s2 2 . 2 ∂t ∂t ∂x

18

(65)

The diffusion equation results by formally taking the limit λ → ∞, s → ∞ with s2 /λ ≡ 2D constant in (65), but this can be made more precise because the equation can be solved explicitly. The solution when the spatial domain is the entire line is [ ])  −λt ( e λ λt   δ(x − st) + δ(x + st) + I0 (Λ) + I1 (Λ) |x| < st 2 s Λ p(x, t) =   0 |x| > st Here I0 and I1 are modified Bessel functions of the first kind. If we make use of the asymptotic expansions ( ) ( ) ez ez 1 1 I0 (z) = √ +O I1 (z) = √ +O as z → ∞ z z 2πz 2πz we see that x2 − 1 p(x, t) = √ e 4Dt + e−λt O(ξ 2 ) 4πDt

ξ 2 ≡ (x/st)2

and thus the telegraph process reduces to a diffusion process on short space scales and long time scales. This fact was known to Einstein and this process has since been studied by many (Taylor 1920; F¨ urth 1920; Goldstein 1951; Kac 1956; Othmer et al. 1988). If we define τ = 2 t and ξ = x, where  is a small parameter, then (65) reduces to 2

2 ∂2n ∂n 2∂ n + 2λ = s . ∂τ 2 ∂τ ∂ξ 2

(66)

The diffusion regime defined by the exact solution now becomes x ξ = st sτ and this requires only that ξ/(sτ ) ≤ O(1). In the limit  → 0 the exact solution can be used to show that (66) again reduces to the diffusion equation, both formally and rigorously (for t bounded away from zero). However this shows that the approximation of the telegraph process by a diffusion process hinges on the appropriate relation between the space and time scales, not necessarily on the limit of speed and turning rate tending to infinity In any case, it is clear that the spatial distribution of p is asymptotically constant, and thus there is no localization of walkers in this case. Imposing no-flux boundary conditions on a finite interval does not change this conclusion. Case II: λ constant, speed depending on direction Here one finds that the time-independent solutions are given by ] s (0)p (0) λ + e p (x) = s+ (x) [

[

+

] s (0)p (0) λ p− (x) = e s− (x) +

∫

x

+

0

∫

+

0

x

s+ − s− dξ s+ s−

≡ p+ (0)F + (x),

s+ − s− dξ s+ s−

≡ p+ (0)F − (x).

where the constant p+ (0) is determined by the conservation of walkers. Clearly the flux vanishes pointwise, as it must at steady state. It is also clear that these distributions differ if s+ (x) 6= s− (x). Case III: λ+ 6= λ− , constant and equal speeds

19

∂p+ ∂p+ +s ∂t ∂x

=

−λ+ p+ + λ− p−

∂p− ∂p− −s ∂t ∂x

=

λ+ p+ − λ− p−

We write

λ+ + λ− λ+ − λ− ± ≡ λ0 ± λ1 2 2 and then the density-flux form of the sytem is λ± =

∂j ∂p + ∂t ∂x ∂j + 2λ0 j ∂t

= 0 = −s2

∂p − 2sλ1 p. ∂x

One finds that the steady-state solution is N0 e− s

2

p(x) = ∫ 1 0

− 2s

e

Rx 0

Rx 0

λ1 (ξ)dξ

λ1 (ξ)dξ

. dx

and again there may be a non-constant solution; now the difference in turning leads to this, and one can see that the chemotactic velocity should be defined as sλ1 uc = − λ0 .

5.2

The general velocity-jump process

Our approach to velocity jump processes will be a direct generalization of the earlier derivation of the telegrapher’s equation. Thus we shall work directly with the differential equation form of the conservation equation for a phase space density function that depends only on the position, velocity and time. The development is similar to that which leads to the Boltzmann equation and its related moment equations in the kinetic theory of gases (cf. Resibois and DeLeener (1977). Here we deal with the case of no internal variables; that case will be dealt with in subsequent lectures. Let p(x, v, t) be the density function for individuals in a 2n-dimensional phase space with coordinates (x, v), where x ∈ Rn is the position of a individual, and v ∈ Rn is its velocity. Then p(x, v, t) dx dv is the number density of individuals with position between x and x + dx and velocity between v and v + dv, and ∫ n(x, t) = p(x, v, t) dv is the number density of individuals at x, whatever their velocity. The evolution of p is governed by the partial differential equation ∂p + ∇x · vp + ∇v · Fp = R, (67) ∂t where F denotes the external force acting on the individuals and R is the rate of change of p due to reaction, random choice of velocity, etc. For the present we assume that F ≡ 0 and that only two processes contribute to the changes on the right-hand side of (67), namely, a birth/death process and a process that generates random velocity changes. We assume that the former is independent of the velocity and that it can be written (

∂p ) = kr(n)p ∂t bd 20

(68)

where k is a constant. We suppose that the random velocity changes are the result of a Poisson process of intensity λ, where λ may depend upon other variables. Thus λ−1 is a mean run length time between the random choices of direction. The net rate at which individuals enter the phase-space volume at (x, v) is given by ∫ ∂p ( )sp = −λp + λ T (v, v0 )p(x, v0 , t) dv0 (69) ∂t where ‘sp’ denotes the change due to the stochastic process. Clearly this equation is the velocity-space analog of the master equation for a space-jump process. The kernel T (v, v0 ) gives the probability of a change in velocity from v0 to v, given that a reorientation occurs, and therefore T (v, v0 ) is non-negative and normalized so that ∫ T (v, v0 ) dv = 1. This normalization condition merely expresses the fact that no individuals are lost during the process of changing velocity. At present we assume that T (v, v0 ) is independent of the time between jumps. In light of the foregoing assumptions, (7) becomes ∫ ∂p + ∇x · vp = −λp + λ T (v, v0 )p(x, v0 , t) dv0 + kr(n)p. (70) ∂t For most purposes one does not need the distribution p, but only its first few velocity moments. The first two are the number density n(x, t) introduced previously, and the average velocity u(x, t), which is defined by ∫ n(x, t)u(x, t) ≡ p(x, v, t)v dv. (71) If we integrate (70) over v we find that ∂n + ∇x · nu = R(n) ∂t where R(n) ≡ knr(n). Similarly, multiplying by v and integrating over v gives ∫ ∫ ∂(nu) + ∇ · pvv dv = λ T (v, v0 )vp(x, v0 , t) dv0 dv − λnu + knur(n). ∂t

(72)

(73)

External signals enter either through a direct effect on the turning rate λ and the turning kernel T , or indirectly via internal variables that reflect the external signal and in turn influence λ and/or T . The first case arises when experimental results are used to directly estimate parameters in the equation (Ford and Lauffenburger 1992), but the latter approach is more fundamental. The reduction of (67) to the macroscopic chemotaxis equations for the first case is done in (Hillen and Othmer 2000; Othmer and Hillen 2002), and this will be discussed in a subsequent lecture.

5.3

The unbiased walk

When the underlying space is one-dimensional we define T (v, v0 ) = δ(v + v0 ) and thus demand that individuals change direction each time a choice is made. This is consistent with the scheme that led to the telegrapher’s equation earlier, but not for instance, with the random choice of direction made at each tumble in bacterial motion. (How would one define T in the latter case?) When the speed is constant v = ±s and nu = s(p+ − p− ), where p± ≡ p(x, ±s, t). Furthermore ∫ ∂ ∂n = s2 (p+ + p− ). ∇ · pvv dv = s2 ∂x ∂x

21

For the foregoing choice of T the integral term in (73) reduces to −λs(p+ − p− ), and thus in the absence of reaction (72) and (73) reduce to ∂ + ∂ (p + p− ) + s (p+ − p− ) = 0 ∂t ∂x ∂ ∂ s (p+ − p− ) + s2 (p+ + p− ) = −2λs(p+ − p− ), ∂t ∂x These are just the equations given at (64), written in a slightly different form. In higher space dimensions equations (72) and (73) do not specify n and u as they stand, for they involve the second vmoment of p and the as yet unspecified kernel T (v, v0 ). Some further simplifying assumptions are necessary, and to describe some that are biologically meaningful we shall first introduce the notion of persistence. Let v = sξ where s = k v k is the speed (the Euclidean norm of v) and ξ = v/ k v k is the direction of v. For a fixed v0 , the average velocity v after reorientation is defined by ∫ ∫ 0 v = T (v, v )v dv = T (v, v0 )ξsn ds dωn where dωn is the surface measure on the unit sphere S0n−1 centered at the origin in Rn . While the average speed ∫ ∫ 0 s = T (v, v ) k v k dv = T (v, v0 )sn ds dωn is always positive (since T ≥ 0 and T is not concentrated at v = 0), the average velocity vector may vanish, and k v k≤ s, see Figure 2. The angle between v/s and ξ 0 = v0 /s0 , provides a measure of the tendency of the motion to persist in any given direction ξ 0 . Therefore we define the index of directional persistence as ψd ≡

v · v0 ss0

(74)

where ψd ∈ [−1, +1]. Of particular interest is the case in which the speed does not change with reorientation and the turning probability depends only on the cone angle θ between v0 and v, which is given by θ(v, v0 ) ≡ arccos

v · v0 , ss0

where θ ∈ [0, π]. Then T (v, v0 ) has the form δ(s − s0 ) h (θ(v, v0 )) sn−1

T (v, v0 ) =

(75)

for any n ≥ 2. The distribution h is normalized so that ∫ π 2 h(θ) dθ = 1 0

for n = 2 and ∫ 2π

π

h(θ) sin θ dθ = 1 0

for n = 3. Given a velocity v0 , the average velocity after reorientation can be resolved into a component along v0 and 0 a component v⊥ orthogonal to v0 . Since the probability of choosing a given direction depends only on θ for the 0 foregoing T , it follows that v⊥ = 0. Furthermore, in this case ψd in (74) is independent of v0 and v = ψd v 0 ,

22

(76)

where the persistence index or mean cosine is given by  ∫   2 π h(θ) cos θ dθ 0 ψd = ∫   2π π h(θ) cos θ sin θ dθ 0

for n = 2

(77)

for n = 3

(cf. Patlak (1953)). Some specific examples of interest will help to illustrate this. For the simple case of uniform random selection of direction on the unit circle, h(θ) = 1/(2π) and ψd = 0. For the the circular normal distribution (Johnson and Kotz 1970) with pole θ0 = 0, we have h(θ) = [2πI0 (k)]−1 exp(k cos θ), where I0 is the Bessel function of order zero of imaginary argument. For this distribution one finds that ψd = I1 (k)/I0 (k) ((Abramowitz and Stegun 1965)), equation 9.6.19). For k = 0 we have uniform random selection of direction, while as k → ∞ the new direction of motion tends to be the same as the previous direction, and ψd → 1. From observations of the two-dimensional locomotion of Dictyostelium amoeba, the data from Hall (1977) yield ψd ≈ 0.7 whereas the three-dimensional bacterial random walk data in Berg and Brown (1972)) show ψd ≈ 0.33 ( cf. Berg (1983)). It is also possible to derive simple equations for the mean squared displacement of individuals which begin at the origin at t = 0. Let ∫ ∫ D2 (t) ≡ hk x(t) k2 i ≡ k x k2 p(x, v, t) dx dv/ p(x, v, t) dx dv (78) be the mean squared displacement, and let S (t) ≡ hs i ≡ m

m

∫

∫ m

s p(x, v, t) dx dv/

p(x, v, t) dx dv

be the m-th moment of the speed distribution. If N0 individuals are released at x = 0 at t = 0 then n(x, 0) = N0 δ(x) and (nu)(x, 0) = 0. We shall assume that there is no birth/death term in (70),(72) and (73) until stated otherwise, and as a result ∫ p(x, v, t) dx dv ≡ N0 . To obtain a differential equation for D2 (t), multiply (70) by k k x k k2 and integrate over x and v. Under the ∫∫ 2 assumption that terms of the form xi vj p dx dv vanish at infinity we obtain the equation ∫ d 2 2 D (t) = (x · v)p(x, v, t) dx dv ≡ 2B(t). (79) dt N0 By multiplying (70) by x · v and integrating we obtain the following equation for B(t): ∫ d −1 B(t) = (x · v)∇x · (vp) dx dv dt N0 ∫ λ (x · v)p(x, v0 , t) dv0 dx −λB(t) + N0

(80)

In cases where the relation (76) holds, the last term is simply λψd B(t). Suppose that this is the case, and that terms of the form ∫ ∫ (xi vj vk )p(x, v, t) dx dv vanish at infinity; then (80) reduces to dB + λ(1 − ψd )B = S 2 (81) dt where S 2 , the second moment of the speed distribution, is a constant. Therefore integration of (81), subject to B(0) = 0, and of (79) subject to D2 (0) = 0, yields  S2   [1 − e−λ(1−ψd )t ] for ψd 6= 1 λ(1 − ψ ) d (82) B(t) =   S 2t for ψd = 1 23

and D2 (t) =

  

2S 2 1 [t − (1 − e−λ(1−ψd )t )] λ(1 − ψd ) λ(1 − ψd )

  S 2 t2

for ψd 6= 1

(83)

for ψd = 1.

The quantity λ0 = λ(1 − ψd ) is a modified turning frequency associated with the reorientation kernel T (v, v0 ), and the inverse P = 1/λ0 is a characteristic run time that incorporates the effect of persistence. This is called a “persistence time” by Dunn (1983). The “motility” or diffusion coefficient is defined as D = S 2 P/n in a space of dimension n. In terms of D the first equation in (83) reads D2 (t) = 2nD[t −

1 (1 − e−λ0 t )] λ0

(84)

To reduce this to the result obtained earlier, note that when an individual reverses direction at every step ψd = −1, and therefore λ0 = 2λ. Consequently (84) is equivalent to (47) in the one-dimensional case.

Figure 5: A sketch of the the theoretical values of the mean squared displacement (a) versus time t, according to equation (50) and (b) versus the number of consecutive moves m. See also Hall (1977), Figure 7.

5.4

A biased walk in the presence of a chemotactic gradient

Some statistics of the density distribution in the first case, wherein the external field modifies the turning kernel or turning rate directly, can easily be derived and used to interpret experimental data (Erban and Othmer 2007). To outline the procedure, we consider two-dimensional motion of amoeboid cells in a constant chemotactic gradient directed along the positive x1 axis of the plane, i.e. ∇S = k∇Sk e1 ,

where we denoted

e1 = [1, 0].

(85)

Moreover, we assume that the gradient only influences the turn angle distribution T ; details of the procedure are given in (Othmer et al. 1988). We assume for simplicity that the individuals move with a constant speed s. i.e. a velocity of an individual can be expressed as v(φ) ≡ s[cos(φ), sin(φ)] where φ ∈ [0, 2π). We assume that T (v, v0 ) ≡ T (φ, φ0 ) is the sum of a symmetric probability distribution h(φ, φ0 ) ≡ h(φ − φ0 ) = h(|φ − φ0 |) and a bias term k(φ) that results from the gradient of the chemotactic substance. Since the gradient is directed along the positive x1 axis, we assume that the bias is symmetric about φ = 0 and takes its maximum there. Thus we write T (φ, φ0 ) = h(φ − φ0 ) + k(φ) where h and k are normalized as follows. ∫ 2π ∫ 2π h(φ)dφ = 1 k(φ)dφ = 0 (86) 0

0

24

Let p(x, φ, t) be the density of cells at position x ∈ R2 , moving with velocity v(φ) ≡ s[cos(φ), sin(φ)], φ ∈ [0, 2π), at time t ≥ 0. The statistics of interest are the mean location of cells X(t), their mean squared displacement D2 (t), and their mean velocity V(t), which were defined earlier. Two further quantities that arise naturally are the taxis coefficient χ, which is analogous to the chemotactic sensitivity defined earlier because it measures the response to a directional signal, and the persistence index ψd . These are defined as ∫ π ∫ 2π k(φ) cos φ dφ and ψd = 2 h(φ) cos φ dφ. (87) χ≡ 0

0

The persistence index measures the tendency of a cell to continue in the current direction. Since we have assumed that the speed is constant, we must also assume that χ and ψd satisfy the relation χ < 1 − ψd , for otherwise the former assumption is violated (cf. (90)). One can now show, by taking moments of (67), using (86) and symmetries of h and k, that the moments satisfy the following evolution equations (Othmer et al. 1988). dX dV =V = −λ0 V + λχse1 dt dt 2 dB dD = 2B = s2 − λ0 B + λχsX1 dt dt where λ0 ≡ λ(1 − ψd ). The solution of (88) subject to zero initial data is ( ) 1 −λ0 t X(t) = sCI t − (1 − e ) e1 , V(t) = sCI (1 − e−λ0 t ) e1 λ0

(88) (89)

(90)

where CI ≡ χ/(1 − ψd ) is sometimes called the chemotropism index. Thus the mean velocity of cell movement is parallel to the direction of the chemotactic gradient and approaches V∞ = s CI e1 as t → ∞. Thus the asymptotic mean speed is the cell speed decreased by the factor CI . A measure of the fluctuations of the cell path around the expected value is provided by the mean square deviation, which is defined as ∫ ∫ 2π 1 σ 2 (t) = kx − X(t)k2 p(x, φ, t) dφdx = D2 (t)− kX(t)k2 . (91) N 0 R2 0 Using (88) – (89), one also finds a differential equation for σ 2 . Solving this equation, we find )} { ( 1 5 2 2s2 2 2 as t→∞ σ ∼ (1 − CI )t + C −1 λ0 λ0 2 I and from this one can extract the diffusion coefficient as 2s2 D= (1 − CI2 ). λ0 Therefore if the effect of an external gradient can be quantified experimentally and represented as the distribution k(φ), the macroscopic diffusion coefficient, the persistence index, and the chemotactic sensitivity can be computed from measurements of the mean displacement, the asymptotic speed and the mean-squared displacement. However, it is not as straightforward to derive directly the macroscopic evolution equations based on detailed models of signal transduction and response. Suppose that the internal dynamics that describe signal detection, transduction, processing and response are described by the system dy = f (y, S) (92) dt where y ∈ Rm is the vector of internal variables and S is the chemotactic substance (S is extracellular cAMP for Dd aggregation – this will be discussed in the third lecture). Models that describe the cAMP transduction pathway exist (Martiel and Goldbeter 1987; Tang and Othmer 1994; Tang and Othmer 1995), but for describing chemotaxis one would have to formulate a more detailed model. The form of this system can be very general but it should always have the “adaptive” property that the steady-state value (corresponding to the constant stimulus) of the appropriate internal variable (the “response regulator”) is independent of the absolute value of the stimulus, and that the steady state is globally attracting with respect to the positive cone of Rm . 25

5.5

Inclusion of a resting phase

As is suggested by the notation used in (69), there is no necessity that the random process generating the ∂p velocity changes be a Poisson process. Whatever the underlying process, one simply has to compute ( )sp ∂t for that process, but of course if it is not Markovian the right-hand side of (69) will involve an integral over time. Secondly, we can include a resting time in the reorientation or tumbling phase, in order to more accurately describe bacterial motion discussed in the following lecture. When a resting phase is incorporated, the total population is divided into two subpopulations, one consisting of the moving bacteria and the other comprising the resting bacteria. As before, let p = p(x, v, t) be the density of bacteria at (x, v), and let r = r(x, v, τ, t) be the density of bacteria in the resting phase, defined so that r = r(x, v, τ, t) dx dv dτ is the number of bacteria with position between x and x + dx, whose most recent nonzero velocity lies between v and v + dv, and whose rest time lies between τ and τ + dτ . We assume as before that there are no external forces on the bacteria, and that the loss of bacteria from a given (x, v) point in positionvelocity space is governed by a Poisson process of intensity λ. Now however the change is not to a non-zero velocity, but rather into the resting phase. Bacteria leave the resting phase at random times and choose a new velocity. The random exits from the resting phase are supposed to be governed by a Poisson process of intensity µ, and the new choice of velocity depends on the time spent in the resting phase as follows: T(v, v0 , τ ) = e−γτ T (v, v0 ) + (1 − e−γτ )g(k v k).

(93)

Here T (v, v0 ) is a velocity kernel of the type given at (75), and the speed distribution g(s) is such that g(0) = 0 and ∫ ∞ ωn g(s)sn−1 ds = 1. (94) 0 n/2

The factor ωn = 2π /Γ(n/2) is the surface area of the unit sphere in Rn . Thus the probability of choosing a random direction with speed g(s) increases with the resting time, and any directional persistence embodied in the kernel T(v, v0 ) is exponentially fading in the resting time. In the absence of birth/death terms, the governing equations for p and r are ∫ ∫ ∞ ∂p + v · ∇x p = −λp + µ T(v, v0 , τ )r(x, v0 , τ, t) dτ dv0 (95) ∂t Rn 0 and

∂r ∂r + = −µr ∂t ∂τ with the initial condition on r having the renewal form r(x, v, 0, t) = λp(x, v, t). ∫

If we define

p(x, v, t) dv dx Rn

∫ Nr (t) ≡

∞

(97)

∫

Np (t) ≡ and

(96)

∫

Rn

∫ r(x, v, τ, t) dv dx dτ,

0

Rn

Rn

then, since the total number of particles are conserved in the absence of a birth/death process, Np (t) and Nr (t) must satisfy Np (t) + Nr (t) = N0 . It is easy to see that the solution of (96) and (97) is given by r(x, v, τ, t) = λe−µτ p(x, v, t − τ ), and it is convenient to introduce the following notation for the two moments ∫ ∞ r0 (x, v, t) ≡ r(x, v, τ, t) dτ 0

26

∫

and r1 (x, v, t) ≡

∞

e−γτ r(x, v, τ, t) dτ.

0

The governing equation for p can now be written ∫

∂p + v · ∇x p = ∂t

T (v, v0 )r1 (x, v0 , t) dv0

−λp + µ ∫

Rn

(r0 (x, v0 , t) − r1 (x, v0 , t)) dv0

µg(k v k)

+

(98)

Rn

As before, we define the mean squared displacement in x of moving bacteria as ∫ ∫ k x k2 p(x, v, t) dv dx /Np (t), Dp2 = Rn

and of resting bacteria as

∫

∫ Dr2 =

Rn

k x k2 r0 (x, v, t) dv dx /Nr (t). Rn

Rn

Furthermore, we define the corresponding second-order moments ∫ ∫ Bp = (x · v)p(x, v, t) dv dx /Np (t) Rn

∫

and

Rn

∫

Br =

(x · v)r1 (x, v, t) dv dx /Nr (t). Rn

Rn

These satisfy the following system of ordinary differential equations. dDp2 Np = 2Bp Np − λDp2 Np + µDr2 Nr dt dDr2 Nr = λDp2 Np − µDr2 Nr dt dBp Np = Sp2 Np − λBp Np + µψd Br Nr dt dBr Nr = λBp Np − (µ + γ)Br Nr dt

(99) (100) (101) (102)

This system is not closed, for the second moments Sp2 and Sr2 of the speed distribution, which are defined ∫ ∫ s2 p(x, v, t) dv dx /Np (t) Sp2 = Rn

∫

and Sr2 =

Rn

∫ s2 r1 (x, v, t) dv dx /Nr (t),

Rn

Rn

are time-dependent, in contrast to the case analyzed in Section 3. One finds that dSp2 Np dt dSr2 Nr dt dNr dt dR1 dt

= −λSp2 Np + µSr2 Nr + µs20 (Nr − R1 ) = λSp2 Np − (µ + γ)Sr2 Nr = λNp − µNr = λNp − (µ + γ)R1

27

(103)

∫

where

∞

s20 ≡ ωn

g(s)sn+1 ds.

0

∫

∫

and R1 ≡

r1 (x, v, t) dv dx. Rn

Rn

Since these equations are linear in Sp2 Np , etc., they can be solved explicitly and the results can be used in (99-102). However, if λ and µ are large the solution quickly relaxes to the steady-state solution, which is given by λN0 λ+µ µN0 λ+µ λµN0 (µ + λ)(µ + γ)

Nr

=

Np

=

R1

=

Sp2

= s20

Sr2

=

µ 2 s µ+γ 0

(104)

Moreover, if we assume that initially the cells are released at the origin x = 0, then Dp2 (0) = Dr2 (0) = 0, and if they have no preferential direction of motion then Bp (0) = Br (0) = 0. If in addition the initial distribution between moving and nonmoving cells is the steady-state distribution given by (104), then (104) holds for all time and the mean squared speed Sp2 = s20 (105) is a constant. With these assumptions we obtain from (99) and (100) the usual formula dD2 = 2bNp /N0 dt

(106)

for the weighted mean squared displacement D2 (t) =

Dp2 (t)Np + Dr2 (t)Nr N0

The quantity B ≡ Bp satisfies the following second order equation, which is derived from (101), (102) dB d2 B + (λ + µ + γ) + λ(µ(1 − ψd ) + γ)B = (µ + γ)s20 . 2 dt dt

(107)

It should be noted that in the limit µ → ∞, in which case the mean resting time 1/µ tends to zero, equation (107) formally reduces to equation (81) with S 2 = s20 . The solution of (107) and the solution of the reduced equation agree to within terms of O(1/µ), except in a neighborhood of t = 0. This suggests the following definition of a modified turning frequency λ0 = λ

µ(1 − ψd ) + γ , µ+γ

and if we solve (107) subject to Br (0) = Bp (0) = 0 we obtain { } s2 λ+ − λ0 −λ− t λ− − λ0 −λ+ t B(t) = 0 1 − e + e , λ0 λ+ − λ − λ+ − λ − where λ± are given by

√ [ ] λ+µ+γ (1 − ψd )µ + γ λ± = 1 ± 1 − 4λ . 2 (λ + µ + γ)2 28

(108)

(109)

(110)

Note that λ+ ∼ µ and λ− → λ0 in the limit µ → ∞. The solution of (106) subject to the initial condition D2 (0) = 0 gives a relation for the mean squared displacement, namely { } 2s20 µ 1 λ+ − λ0 −λ− t 1 λ− − λ0 −λ+ t 2 D (t) = t+ (e − 1) − (e − 1) . (111) λ0 λ + µ λ− λ+ − λ − λ+ λ+ − λ− As we saw in earlier, the first term in (111) arises in a diffusion process. It can be shown that λ± are both real and therefore the foregoing generalization deviates from this by two exponentially decreasing terms with the relaxation times 1 P± = . λ± A plot of the relation in (111) similar to Figure 5 has an asymptote whose intercept with the t-axis is the persistence time λ+ + λ− − λ0 . P = λ+ λ−

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