Mathematical Biology - Systems Theory for Mathematical ... · Geometric control and quantitative...

ControllabilityPositive state control

Geometric control and quantitative genetics

Mathematical Biology - Systems Theory forMathematical Biology: Further and future topics.

Stuart Townley

University of Exeter, UK

March 21, 2014

Stuart Townley Math Biol – Future topics 1/ 46

Overview of week

X Lecture 1: Mathematical biology for one and two dimensionalmodels

X Lecture 2: Population Projection Models - A Lecture by“Crowd Sourcing” Built into this topic is a group researchproject - Presentations on Friday

X Lecture 3: A Feedback Control Approach to NonlinearPopulation Dynamics

X Lecture 4: Diffusion Driven Instability and links to SwitchedSystems

Lecture 5: Further and future topics in “Systems Theory forMathematical Biology”

RecapPositive input control

x(t+ 1) = Ax(t) +Bu(t) , x(0) = x0 , t ∈ N0 . (∗)

Controllability is one of the most fundamental properties incontrol theory.

For (any of) those not familiar with the concept, looselyspeaking, controllability asks to what states can the state xwith dynamics given by (∗) be steered to via a control signalu.

Three common concepts are that of

Reachability: which states can be reached from the zero state?Null–controllability: which states can be steered to the zerostate?Controllability: which initial states can be steered to whichtarget states?

x(t+ 1) = Ax(t) +Bu(t) , x(0) = x0 , t ∈ N0 . (∗)

Reachability: which states can be reached from the zero state?

Null–controllability: which states can be steered to the zerostate?Controllability: which initial states can be steered to whichtarget states?

x(t+ 1) = Ax(t) +Bu(t) , x(0) = x0 , t ∈ N0 . (∗)

Reachability: which states can be reached from the zero state?Null–controllability: which states can be steered to the zerostate?

Controllability: which initial states can be steered to whichtarget states?

x(t+ 1) = Ax(t) +Bu(t) , x(0) = x0 , t ∈ N0 . (∗)

The formulation of controllability most commonly used todaydates back to the work of Kalman.

Controllability (at least for continuous time systems) is an apriori analytic concept– it regards the existence of continuousor piecewise continuous functions and the solutions of ODEs.

Part of the appeal of the Kalman formulation (and itssolution) is that controllability is in fact equivalent to analgebraic property, which can easily be implemented andtested numerically.

In fact, for linear, time–invariant, continuous–time,finite–dimensional systems reachability, null–controllabilityand controllability are all equivalent.

If A−1 exists then the same is true for time–invariant,discrete–time, finite–dimensional systems.

Note that this equivalence does not hold for more complexsystems (non–linear, time–varying and so on).

x(t+ 1) = Ax(t) +Bu(t) , x(0) = x0 , t ∈ N0 . (∗)

For each t ∈ N the solution of (∗) is given by

x(t) = Atx0 +t−1∑j=0

At−1−jBu(j) .

For reachability, with x0 = 0 this gives

x(t) =t−1∑j=0

At−1−jBu(j) =[B AB . . . At−1B

] u(t)u(t−1)

...u(0)

. (∗∗)

x(t+ 1) = Ax(t) +Bu(t) , x(0) = x0 , t ∈ N0 . (∗)

x(t) = Atx0 +

t−1∑j=0

At−1−jBu(j) .

x(t) =t−1∑j=0

] u(t)u(t−1)

...u(0)

. (∗∗)

x(t+ 1) = Ax(t) +Bu(t) , x(0) = x0 , t ∈ N0 . (∗)

x(t) = Atx0 +

t−1∑j=0

At−1−jBu(j) .

x(t) =t−1∑j=0

] u(t)u(t−1)

...u(0)

. (∗∗)

x(t) =

t−1∑j=0

] u(t)u(t−1)

...u(0)

. (∗∗)

Clearly from (∗∗), a state x ∈ Rn is reachable if, and only if,x ∈ im

[B AB . . . At−1B

By the Cayley–Hamilton Theorem, the image (and thus therank) of the nested sequence of operators[

B AB A2B . . .],

terminates at t = n (if not before).

So if a state is reachable, it is reachable in at most n steps.

x(t) =

t−1∑j=0

] u(t)u(t−1)

...u(0)

. (∗∗)

[B AB . . . At−1B

B AB A2B . . .],

x(t) =

t−1∑j=0

] u(t)u(t−1)

...u(0)

. (∗∗)

[B AB . . . At−1B

B AB A2B . . .],

x(t) =

t−1∑j=0

] u(t)u(t−1)

...u(0)

. (∗∗)

[B AB . . . At−1B

B AB A2B . . .],

The object

R(A,B) =[B AB . . . An−1B

is called the reachability matrix of the pair (A,B), and thelinear system (∗) is said to be reachable if rankR(A,B) = n(maximal).

More generally, the reachable space is the set of all states thatcan be steered to and is equal to imR(A,B).

As mentioned earlier, for linear, time–invariant,finite–dimensional systems reachability, null–controllabilityand controllability are all equivalent (when A−1 exists indiscrete–time).

And thus are all equivalent to R(A,B) having full rank.

We note that there are many other equivalent conditions forcontrollability (Hautus criterion)!

The object

R(A,B) =[B AB . . . An−1B

The object

R(A,B) =[B AB . . . An−1B

The object

R(A,B) =[B AB . . . An−1B

The object

R(A,B) =[B AB . . . An−1B

x(t+ 1) = Ax(t) +Bu(t) , x(0) = x0 , t ∈ N0 . (∗)

This week we have been considering population models of theform (∗), where A ∈ Rn×n+ and for meaningful models werequire that the state x(t) ∈ Rn+.

Controllability for population models is certainly relevant forpopulation managers, as it has a natural interpretation interms of conservation and management.

However, controllability as presented thus far does not respectthe above componentwise nonnegativity.

x(t+ 1) = Ax(t) +Bu(t) , x(0) = x0 , t ∈ N0 . (∗)

There are many (physically motivated) applications where it isnatural that both the state and control variables takenonnegative values (such an input–output economic models,or drug ingestion rates, or flow in pipes).

These are examples of so–called positive linear systems (orpositive input systems).

Positive linear systems is now a well–established theory, withtextbooks by Krasnosel′skij et al. (1989) and Farina andRinaldi (2000) (for example).

It is worth noting that positive systems do not evolve on linear(or vector) spaces, where there is a well–defined notion ofsubtraction via the additive inverse. Instead, positive systemsevolve on cones, which are sets K such that αK ⊆ K for allα ≥ 0.

As such, positive input controllability is not (quite) asstraightforward as the usual case.

For example, the positive input reachable set in finite time ofthe pair (A,B) ∈ Rn×n+ × Rn×m+ is equal to⋃

N∈N〈B ,AB , . . . , AN−1B〉+ ,

where for X ∈ Rn1×n2+ , 〈X〉+ denotes all nonnegative linear

combinations of the nonnegative columns of X.

The positive input reachable set in finite time may beincreasing with increasing N , so that there are states that arereachable only in infinite–time.

N∈N〈B ,AB , . . . , AN−1B〉+ ,

The notions of positive input reachability, positive inputnull–controllability and positive input controllability are notequivalent.

Indeed, null–controllability is impractical as addingnonnegative inputs only makes the state larger. Essentially,null–controllability is possible in finite–time if, and only if, Ais nilpotent. In infinite–time it is required that A is Schur(that is r(A) < 1).

The following characterisation of positive input reachability isproven by Coxson and Shapiro1.

Corollary

A pair (A,B) ∈ Rn×n+ × Rn×m+ is positive input reachable in finitetime if, and only if, for some N ∈ N the matrix[B AB . . . AN−1B

]contains an n× n monomial submatrix.

1P. G. Coxson and H. Shapiro, “Positive input reachability and controllabilityof positive systems,” Linear Algebra Appl., vol. 94, pp. 35–53, 1987.

Corollary

MotivationTheoryExamples

x(t+ 1) = Ax(t) +Bu(t) , x(0) = x0 , t ∈ N0 . (∗)

In some applications (such as population ecology) therequirement that u(t) ≥ 0 is unnecessarily restrictive.

It is reasonable to desire that individuals can be removed froma population, so that u(t) ≤ 0 (or at least has negativecomponents), provided that x(t) ∈ Rn+ is not violated.

Such a framework is not addressed by the existing positive(input) systems theory.

x(t+ 1) = Ax(t) +Bu(t) , x(0) = x0 , t ∈ N0 . (∗)

We recall that an n× n Leslie2 matrix has the followingstructure

f1 f2 ... ... fns1 0 ... 00 s2 0...

. . .. . .

0 ... 0 sn−1 0

which models a population partitioned into discrete,increasing age–stages.

Thus, fi ≥ 0 denote reproductive rates and si ≥ 0 denotesurvival rates.

For ecologically meaningful models3, we will always assumethat s1, . . . , sn−1 > 0, f1, . . . , fn ≥ 0 and there exists at leastone i ∈ {1, 2, . . . , n} such that fi > 0.

2P. Leslie, “On the use of matrices in certain population mathematics,”Biometrika, vol. 33, no. 3, pp. 183–212, 1945.

3I. Stott, S. Townley, D. Carslake, and D. Hodgson, “On reducibility andergodicity of population projection matrix models,” Methods in Ecology andEvolution, vol. 1, no. 3, pp. 242–252, 2010.

f1 f2 ... ... fns1 0 ... 00 s2 0...

. . .. . .

0 ... 0 sn−1 0

f1 f2 ... ... fns1 0 ... 00 s2 0...

. . .. . .

0 ... 0 sn−1 0

Noting that the n× n positive diagonal matrix

T = diag

s1s2, . . . ,

s1 . . . sn−1

), has T−1 ≥ 0,

for positive state controllability it is sufficient to consider thesimilarity transformed pair (T−1AT, T−1B), and note that

T−1AT =

f1 f2 ... ... fn1 0 ... 00 1 0...

. . .. . .

0 ... 0 1 0

When B = ei, (ith standard basis vector in Rn), thenR(A,B) is square and invertible (easily checked), so thesystem is controllable and moreover the control steeringx(0) = 0 to x(n) = xT ∈ Rn+ (in n steps) is unique.

What is this control and what does it do to the state?

T = diag

s1s2, . . . ,

s1 . . . sn−1

), has T−1 ≥ 0,

T−1AT =

f1 f2 ... ... fn1 0 ... 00 1 0...

. . .. . .

0 ... 0 1 0

,When B = ei, (ith standard basis vector in Rn), thenR(A,B) is square and invertible (easily checked), so thesystem is controllable and moreover the control steeringx(0) = 0 to x(n) = xT ∈ Rn+ (in n steps) is unique.

T = diag

s1s2, . . . ,

s1 . . . sn−1

), has T−1 ≥ 0,

T−1AT =

f1 f2 ... ... fn1 0 ... 00 1 0...

. . .. . .

0 ... 0 1 0

,When B = ei, (ith standard basis vector in Rn), thenR(A,B) is square and invertible (easily checked), so thesystem is controllable and moreover the control steeringx(0) = 0 to x(n) = xT ∈ Rn+ (in n steps) is unique.

Writing down the control exactly is cumbersome, let’s look atan example. Consider

Which gives rise to

Writing down the control exactly is cumbersome, let’s look atan example. Consider

Which gives rise to

Here the unique control u takes negative values, so is notpermitted from a positive systems point of view, butmaintains nonnegativity of the state x.

We desire a notion of positive state reachability, which isweaker than positive input reachability.

How to formalise this?

It turns out that there is a complete characterisation ofpositive state control in terms of positive input control of arelated system under the following assumption:

(A) Given the pair (A,B) ∈ Rn×n+ ×Rn×m+ there exists F ∈ Rm×n

such that with A := A−BF both A ∈ Rn×n+ and if

v ∈ Rn+, w ∈ Rm satisfy Av +Bw ≥ 0 then w ≥ 0.

The idea of assumption (A) is that by decomposing A intoA+BF , then negative controls u in Ax+Bu can be can beabsorbed as Ax+B(Fx+ u).

Assumption (A) always holds in the SISO case B = ei, or theMIMO case B =

[ci1ei1 . . . cikeik

], for cij > 0 by taking

fT or F as the corresponding rows of A.

More generally, assumption (A) can be tested for by

Assumption (A) holds for (A,B) ∈ Rn×n+ × Rn×m+ if, and only if,there exist m rows of B such that the m×m submatrix, denotedB, formed by taking these m rows from B is a positive monomialmatrix and

A−BB−1A ≥ 0 .

Here A is formed of the m rows of A that appear in B.

A−BB−1A ≥ 0 .

Assumption (A) is useful because of the followingcharacterisation:

Theorem

Let the pair (A,B) satisfy (A) and denote A := A−BF . Thenthe state trajectories of (A,B) from initial state x0 ∈ Rn+ with

nonnegative state are precisely the state trajectories of (A, B) frominitial state x0 ∈ Rn+ with nonnegative control.

Positive state results for the pair (A,B) then follow ascorollaries from classical positive input results for the pair(A, B).

When assumption (A) does not hold, then some partialconverse results are available (we can describe part of thereachable space with nonnegative state by considering the pair(A, B) with positive input, but might not reach everything).

Theorem

Consider the pair A = [ 2 21 0 ], b = e1.

Trivially, e1 is reachable from zero with positive state in onestep and the control u(0) = 1, u(1) = −2 steers the statefrom zero to e2 in two steps.

By taking suitable linear combinations of these inputs all ofR2+ can be reached with nonnegative state.

If we restrict attention to (A, b) with only positive inputs,then the positive input reachable space is spanned by b = e1and Ab = 2e1 + e2, and is depicted in Figure 2 (b): note thatnot all of R2

+ is reachable.

Even in this very simple example there is a difference betweenpositive state controllability and classical positive inputcontrollability.

+ is reachable.

Figure: (a) Positive state trajectories steering the state of (A, b) to e1and e2 respectively. (b) The positive input reachable space of thepositive system (A, b) is the hatched area between b and Ab = 2e1 + e2.The line w is parallel to limk→∞Akb and so here the positive inputreachable set is not all of R2

+ but is attained in finite time.

The invasive weed Cirsium vulgare (spear thistle) in Nebraska,USA, is modelled by matrix PPM4 with A given by

[0 0 f1 f2s1 0 f3 f40 s2 s3 00 s4 s5 s6

In (∗) time–steps correspond to years and the four statesdenote the seed bank, small plants, medium plants and largeplants respectively.

The si ∈ [0, 1] denote survival and growth parameters and thefi ≥ 0 are reproductive values.

The meaning of s4 > 0 is that small plants can grow intolarge plants in one year. Furthermore, f3, f4 > 0 means thatboth medium and large plants can produce seeds thatgerminate and grow into small plants.

4B. Tenhumberg, S. M. Louda, J. O. Eckberg, and M. Takahashi, “Montecarlo analysis of parameter uncertainty in matrix models for the weed cirsiumvulgare,” Journal of Applied Ecology, vol. 45, no. 2, pp. 438–447, 2008.

[0 0 f1 f2s1 0 f3 f40 s2 s3 00 s4 s5 s6

For the parameter values provided, the uncontrolledpopulation is unstable as r(A) = 1.57 > 1.

We seek to reduce the weed population by using astate–feeback acting on large plants so that that B = b = e4.

When this action is performed (shortly) before the census ormeasurement then the resulting model is well described by (∗).

Here assumption (A) holds with F = fT the fourth row of Aso that A := A− bfT is given by

0 0 f1 f2s1 0 f3 f40 s2 s3 00 0 0 0

As Ak has no zero columns for any k ∈ N we see that nostate is positive state null controllable in finite–time.

Here r(A) = 1.0024 > 1 and although A is not primitive (oreven irreducible), r(A) is a simple eigenvalue and the limit

limk→∞

(r(A))kx0 =

vTx0vTw

w holds, (§)

where vT and w are (positive, once positively scaled) left andright eigenvectors of A corresponding to r(A) respectively.

0 0 f1 f2s1 0 f3 f40 s2 s3 00 0 0 0

.As Ak has no zero columns for any k ∈ N we see that nostate is positive state null controllable in finite–time.

limk→∞

(r(A))kx0 =

vTx0vTw

w holds, (§)

0 0 f1 f2s1 0 f3 f40 s2 s3 00 0 0 0

.As Ak has no zero columns for any k ∈ N we see that nostate is positive state null controllable in finite–time.

limk→∞

(r(A))kx0 =

vTx0vTw

w holds, (§)

Recall

limk→∞

(r(A))kx0 =

vTx0vTw

w . (§)

When x0 ≥ 0 and x0 6= 0 the right hand side of (§) is positiveand hence there are no non–trivial states that are positivestate null controllable in infinite–time.

Equivalently, the negative control u(t) = −fTx(t) does notstabilise any nonzero initial population.

Furthermore, the characterisation from the earlier theoremshows that this system cannot be stabilised by positive statecontrol.

Recall

limk→∞

(r(A))kx0 =

vTx0vTw

w . (§)

Recall

limk→∞

(r(A))kx0 =

vTx0vTw

w . (§)

Recall

limk→∞

(r(A))kx0 =

vTx0vTw

w . (§)

[0 0 f1 f2s1 0 f3 f40 s2 s3 00 s4 s5 s6

If instead the control action is in fact performed (shortly) afterthe census or measurement, then a more accurate model is

x(t+ 1) = A(x(t) + bu(t)) = Ax(t) +Abu(t), t ∈ N0 ,

and so we replace b = e4 by Ab =[f2 f4 0 s6

For the given parameters, (A) holds for (A,Ab) withF = fT = [ 0 0 f1/f2 1 ] and then

A := A−AbfT =

0 0 0 0

s1 0 f3 − f1f4f2

0 s2 s3 0

0 s4 s5 − f1s6f2

≥ 0 ,

[0 0 f1 f2s1 0 f3 f40 s2 s3 00 s4 s5 s6

A := A−AbfT =

0 0 0 0

s1 0 f3 − f1f4f2

0 s2 s3 0

0 s4 s5 − f1s6f2

≥ 0 ,

[0 0 f1 f2s1 0 f3 f40 s2 s3 00 s4 s5 s6

A := A−AbfT =

0 0 0 0

s1 0 f3 − f1f4f2

0 s2 s3 0

0 s4 s5 − f1s6f2

≥ 0 ,

A := A−AbfT =

0 0 0 0

s1 0 f3 − f1f4f2

0 s2 s3 0

0 s4 s5 − f1s6f2

≥ 0 ,

As the fourth column of A is zero, we have that x = e4 is nullcontrollable (in finite–time).

Furthermore, r(A) = 0.1153 < 1, every state is positive statenull controllable in infinite–time.

The above observations suggest that when control actions acton large weeds, organising these actions to take place postcensus is preferable to pre census. This is not biologicallysurprising because, loosely speaking, the fourth stage class isthe most reproductive and the post census control strategylimits to a greater extent reproduction in this stage class.

A := A−AbfT =

0 0 0 0

s1 0 f3 − f1f4f2

0 s2 s3 0

0 s4 s5 − f1s6f2

≥ 0 ,

A := A−AbfT =

0 0 0 0

s1 0 f3 − f1f4f2

0 s2 s3 0

0 s4 s5 − f1s6f2

≥ 0 ,

A := A−AbfT =

0 0 0 0

s1 0 f3 − f1f4f2

0 s2 s3 0

0 s4 s5 − f1s6f2

≥ 0 ,

What is quantitative geneticsLande & Kirkpatrick Maternal Effects ModelGeometric Control Theory

As Monty Python would say

And now for something completelydifferent!

Quantitative genetics is an approach used by evolutionarybiologists to understand how phenotypic traits evolve fromgeneration to generation.

In any generation, a given trait has distributed values acrossthe population. Any single trait distribution has mean andvariance (and possibly covariance with other traits).

Modification of traits is split into random components, fitnesseffects and environmental/selected influences.

Using simplifications that the trait distribution isapproximately normal, then the evolution of the mean of asingle trait z(t) is described by the Breeder’s equation:

z(t+ 1) = z(t) + gβ(t) ,

where β is like a control action which might be selected on orit might be an external environmental effect.

z(t+ 1) = z(t) + gβ(t) ,

The G Matrix Approach 5

∆z(t) = Gβ(t), t ∈ N . Multi-variable Breeders Equation

∆z(t) = z(t+ 1)− z(t); change in the vector of meanphenotypes, β the selection gradient.

The G matrix of genetic variance/covariances - studiedintensively6 - gives clouds of phenotype.

Often these clouds are “cigar-like” and evolutionary change isheavily constrained7 in a principal direction determined bythe leading eigenvector gmax of G

There are, however, many inputs to the phenotype thatcircumvent constraints imposed by genetic architecture.

5Lande. Quantitative genetical analysis of multivariate evolution applied tobrain:body size allometry. Evolution 33, (1979).

6Jones, Arnold & Burger. Evolution and stability of the g-matrix on alandscape with a moving optimum. Evolution 58, (2004).

7Arnold. Constraints on phenotypic evolution. Am. Nat. 140, (1992).Stuart Townley Math Biol – Future topics 29/ 46

Parental effects8 “trans–generationally” enhance theevolutionary importance of environmentally–induced changes– selection pressures on previous generations matter

Parental, especially non–Mendelian maternal, inheritance iscaptured in:

∆z(t) = M∆z(t−1)+(Caz+MP )β(t)−MPβ(t−1), t ∈ N .

8Kirkpatrick & Lande. The evolution of maternal characters. Evolution 43,(1989).

Parental effects8 “trans–generationally” enhance theevolutionary importance of environmentally–induced changes– selection pressures on previous generations matter

Parental, especially non–Mendelian maternal, inheritance iscaptured in:

∆z(t) = M∆z(t−1)+(Caz+MP )β(t)−MPβ(t−1), t ∈ N .

8Kirkpatrick & Lande. The evolution of maternal characters. Evolution 43,(1989).

∆z(t) = M∆z(t− 1) + (Caz + MP )β(t)−MPβ(t− 1), t ∈ N .

M is the matrix of maternal influence that models thenon–Mendelian transmission of phenotypes directly frommother to offspring.

Caz is the change in genetic component (≈ (G(I − 12M

T )−1).

E represents the environmental influences on the phenotype.

P is the phenotypic variance-covariance matrix analogous toG and determined by solving the matrix Lyapunov equation

MPMT −P + G + E + 12(MCT

az + CazMT ) = 0 .

T )−1).

az + CazMT ) = 0 .

T )−1).

az + CazMT ) = 0 .

T )−1).

az + CazMT ) = 0 .

T )−1).

az + CazMT ) = 0 .

Maternal inheritance causes evolutionary momentum9

0 2 4 6 81.5

Generation

0 2 4 6 81.5

Generation

0 2 4 6 81.5

Generation

0 2 4 6 81.5

Generation

Trait 1Trait 2Trait 3

Figure: G = 0.25I, E = I, M = s

0 0.5 0.250.5 0 00.25 0 0

s = 0, 1/3, 2/3, 1. β(t) selects only on trait 1 only for generationt = 1, 2, 3, 4 and is zero otherwise. Increasing M causes increasingmomentum/transients in the change in phenotype.

9Lande & Kirkpatrick. Selection response in traits with maternal inheritance.Genet. Res. 55, (1990).

Kirkpatrick and Lande’s model showed how biologicalmechanisms that bypass genetic constraints can divert thepath of evolution away from that under strict geneticinheritance.

Empirical and theoretical evidence for such epigeneticinfluences grows:Without maternal effects the range of phenotypic evolution ispredicted using the principal components of G;With additional biological inheritance systems predictions onthe response to selection based solely on G will necessarily beincomplete, ignoring, for example, the amplification(attenuation) from transient fluctuations that affect long-termadaptationWhat is unclear is how this attenuation alters the range ofphenotypes by bypassing gmax and whether the eccentricity ofthe phenotypic ellipses changes as the variation around themean evolutionary path is diverted by maternal inheritance.

Kirkpatrick and Lande’s model showed how biologicalmechanisms that bypass genetic constraints can divert thepath of evolution away from that under strict geneticinheritance.Empirical and theoretical evidence for such epigeneticinfluences grows:

Without maternal effects the range of phenotypic evolution ispredicted using the principal components of G;With additional biological inheritance systems predictions onthe response to selection based solely on G will necessarily beincomplete, ignoring, for example, the amplification(attenuation) from transient fluctuations that affect long-termadaptationWhat is unclear is how this attenuation alters the range ofphenotypes by bypassing gmax and whether the eccentricity ofthe phenotypic ellipses changes as the variation around themean evolutionary path is diverted by maternal inheritance.

Kirkpatrick and Lande’s model showed how biologicalmechanisms that bypass genetic constraints can divert thepath of evolution away from that under strict geneticinheritance.Empirical and theoretical evidence for such epigeneticinfluences grows:Without maternal effects the range of phenotypic evolution ispredicted using the principal components of G;

With additional biological inheritance systems predictions onthe response to selection based solely on G will necessarily beincomplete, ignoring, for example, the amplification(attenuation) from transient fluctuations that affect long-termadaptationWhat is unclear is how this attenuation alters the range ofphenotypes by bypassing gmax and whether the eccentricity ofthe phenotypic ellipses changes as the variation around themean evolutionary path is diverted by maternal inheritance.

Kirkpatrick and Lande’s model showed how biologicalmechanisms that bypass genetic constraints can divert thepath of evolution away from that under strict geneticinheritance.Empirical and theoretical evidence for such epigeneticinfluences grows:Without maternal effects the range of phenotypic evolution ispredicted using the principal components of G;With additional biological inheritance systems predictions onthe response to selection based solely on G will necessarily beincomplete, ignoring, for example, the amplification(attenuation) from transient fluctuations that affect long-termadaptation

What is unclear is how this attenuation alters the range ofphenotypes by bypassing gmax and whether the eccentricity ofthe phenotypic ellipses changes as the variation around themean evolutionary path is diverted by maternal inheritance.

Kirkpatrick and Lande’s model showed how biologicalmechanisms that bypass genetic constraints can divert thepath of evolution away from that under strict geneticinheritance.Empirical and theoretical evidence for such epigeneticinfluences grows:Without maternal effects the range of phenotypic evolution ispredicted using the principal components of G;With additional biological inheritance systems predictions onthe response to selection based solely on G will necessarily beincomplete, ignoring, for example, the amplification(attenuation) from transient fluctuations that affect long-termadaptationWhat is unclear is how this attenuation alters the range ofphenotypes by bypassing gmax and whether the eccentricity ofthe phenotypic ellipses changes as the variation around themean evolutionary path is diverted by maternal inheritance.

A Simulation Example to clarify What is Going On

4 2 00

Phenotype 1Ph

4 2 00

Phenotype 1

4 2 00

Phenotype 1

4 2 00

Phenotype 1

Figure: The path of evolution and the clouds of achievable phenotypeswith increasing strength of maternal effect -

M = sM0, s = 0.1, 0.4, 0.7, 1, M0 =

[−0.4 0.4−0.4 0.4

]. Simulated

phenotype resulting from disturbed selection around a nominal selectioncosπ t/4 acting on trait one only. Path of evolution given by nominal(Black line), variabillity of z(t), t = 1, . . . , 7 (Cyan), z(8) (Red)

In this example we can see that M influences the path ofphenotypic change in two distinct ways.

First, as M increases so the path unravels and explores agreater diversity of phenotype.Second, as M increases so the ellipses, which bound thephenotypic range in each generation, change shape andorientation.Using ideas from control theory, we show that theseinfluences of M are captured by two quantities – anassociated reachability matrix and its correspondingcontrollability gramian. The former controls the basic path;the latter controls the orientation and shape of the ellipses.As we will see, when selection is unconstrained we find thatM will often constrain phenotypic evolution. On the otherhand, when selection is constrained, we find that M isunconstraining, acting to open up the space of achievablephenotype

In this example we can see that M influences the path ofphenotypic change in two distinct ways.First, as M increases so the path unravels and explores agreater diversity of phenotype.

Second, as M increases so the ellipses, which bound thephenotypic range in each generation, change shape andorientation.Using ideas from control theory, we show that theseinfluences of M are captured by two quantities – anassociated reachability matrix and its correspondingcontrollability gramian. The former controls the basic path;the latter controls the orientation and shape of the ellipses.As we will see, when selection is unconstrained we find thatM will often constrain phenotypic evolution. On the otherhand, when selection is constrained, we find that M isunconstraining, acting to open up the space of achievablephenotype

In this example we can see that M influences the path ofphenotypic change in two distinct ways.First, as M increases so the path unravels and explores agreater diversity of phenotype.Second, as M increases so the ellipses, which bound thephenotypic range in each generation, change shape andorientation.

Using ideas from control theory, we show that theseinfluences of M are captured by two quantities – anassociated reachability matrix and its correspondingcontrollability gramian. The former controls the basic path;the latter controls the orientation and shape of the ellipses.As we will see, when selection is unconstrained we find thatM will often constrain phenotypic evolution. On the otherhand, when selection is constrained, we find that M isunconstraining, acting to open up the space of achievablephenotype

In this example we can see that M influences the path ofphenotypic change in two distinct ways.First, as M increases so the path unravels and explores agreater diversity of phenotype.Second, as M increases so the ellipses, which bound thephenotypic range in each generation, change shape andorientation.Using ideas from control theory, we show that theseinfluences of M are captured by two quantities – anassociated reachability matrix and its correspondingcontrollability gramian. The former controls the basic path;the latter controls the orientation and shape of the ellipses.

As we will see, when selection is unconstrained we find thatM will often constrain phenotypic evolution. On the otherhand, when selection is constrained, we find that M isunconstraining, acting to open up the space of achievablephenotype

In this example we can see that M influences the path ofphenotypic change in two distinct ways.First, as M increases so the path unravels and explores agreater diversity of phenotype.Second, as M increases so the ellipses, which bound thephenotypic range in each generation, change shape andorientation.Using ideas from control theory, we show that theseinfluences of M are captured by two quantities – anassociated reachability matrix and its correspondingcontrollability gramian. The former controls the basic path;the latter controls the orientation and shape of the ellipses.As we will see, when selection is unconstrained we find thatM will often constrain phenotypic evolution. On the otherhand, when selection is constrained, we find that M isunconstraining, acting to open up the space of achievablephenotype

From a control systems perspective the Lande–Kirkpatrick model isa second order, multivariable, discrete–time, linear system withmultivariable inputs β and state z. Let

B0 = −MP and B1 = (Caz + MP)

∆z(T ) = MT∆z(0) +RTUβ,T where Uβ,T =

β(T )...

RT :=[B1 (B0 + MB1) M(B0 + MB1) . . . MT−2(B0 + MB1) MT−1B0

z(T + 1) = z(0) + ∆z(1) + . . .+ ∆z(T ) .

Putting this together, we can characterise the evolution ofphenotype in compact form.

A Reachability matrix for achievable mean phenotype

The mean phenotype evolves from z(0), over T + 1 generations ofselection β(0), β(1), . . . , β(T ), to

z(T + 1) = z(0) +(M + . . .+ MT

)∆z(0) + ST Uβ,T . (2)

Here ST is the n(T + 1)× n reachability matrix

ST =[B1 |MB1 +B1 +B0 | . . . |MT−1B1 + . . .+B1 + MT−2B0+. . .+B0 |MT−1B0 + . . . B0

The equation (2) relates historical (i.e. trans-generational)selection pressures to current phenotype. With z(0) = z(1) = 0this simplifies to

z(T + 1) = ST Uβ,T A New MVB-Eqn for maternal effects

The principal singular vector smax of ST plays the role of gmax

The +ve/−ve Impacts of M on Clouds of EvolvingPhenotype

10 5 0 5 10

Phenotype 1

40 20 0 20 4040

Phenotype 1

5 0 56

Phenotype 1

10 0 1015

Phenotype 1

Figure: A comparison between ellipsoids of phenotype change overT = 2, 3, 4 and 5 generations of selection without (black) and with (red)maternal effects for four different M matrices. G = Diag[1 0.5];(

0.4 0.5−0.3 0.4

(0.4 0.50.3 0.4

(−0.4 0.3−0.5 −0.4

(−0.4 0.5

0.5 0.4

)Match up the negative maternal influences to relaxing constraintson change - the positive influences to the increasing constraints onchange

The impact of M on evolutionFor these four choices of M we see four distinct maternal effects:

Same path, less constraint For M1, the angle between thepaths of least genetic resistance without M (gmax) and withM (smax) is between −π/4 and π/4; the eccentricity of thecloud is reduced.

Same path, more constraint For M2, the angle betweengmax and smax is between −π/4 and π/4; the eccentricity ofthe cloud is increased by M2.

Altered path, less constraint For M3, the angle betweengmax and smax is between π/4 and 3π/4 and M3 rotates thepath of evolution; the eccentricity of the cloud is reduced byM3.

Altered path, more constraint For M4, the angle betweengmax and smax is between π/4 and 3π/4; the eccentricity ofthe cloud is increased by M4.

The impact of M on evolutionFor these four choices of M we see four distinct maternal effects:

Same path, less constraint For M1, the angle between thepaths of least genetic resistance without M (gmax) and withM (smax) is between −π/4 and π/4; the eccentricity of thecloud is reduced.

Same path, more constraint For M2, the angle betweengmax and smax is between −π/4 and π/4; the eccentricity ofthe cloud is increased by M2.

Altered path, less constraint For M3, the angle betweengmax and smax is between π/4 and 3π/4 and M3 rotates thepath of evolution; the eccentricity of the cloud is reduced byM3.

Altered path, more constraint For M4, the angle betweengmax and smax is between π/4 and 3π/4; the eccentricity ofthe cloud is increased by M4.

Can M constrain and/or alter the path of Evolution - YES

Figure: Left plots: Magnitude of the angle between gmax and smax. Thesolid black line is the contour where the angle is ±π/4. Right plots:Contours of constant eccentricity depicting whether M further constrainsor un-constrains phenotypic change. The solid black line is the contourcorresponding to eccentricity e = 0.9 of G. From top to bottom:

[0.4 m12

m21 0.4

[−0.4 m12

m21 0.4

[−0.4 m12

m21 −0.4

]with m12

and m21 in the range −0.5 to 0.5.

Calculating paths and clouds of trans-generationalevolutionary change

Phenotype 1

4 2 00

Phenotype 1

4 2 00

Phenotype 1

4 2 00

Phenotype 1Ph

4 2 00

Figure: M shapes both the path of evolution and the ellipses ofachievable phenotypes. Simulation of disturbed selection around a meanb(t) = cosπ t/4. Path of evolution given by mean (Black line) withclouds of uncertainty in phenotype. The path is computed by drawingfrom the normal distributions N(µt,Σt), µt = StUβ,t,Σt ∝ StSTt .

Tracking an optimum phenotype lying orthogonal to a pathof least genetic resistance

Phenotype 1

0 2 4 62

Phenotype 1

0 2 4 62

Phenotype 1

0 2 4 62

Phenotype 1

0 2 4 62

Figure: Steering phenotype in the direction of G matrix evolution withminimum effort and maternal effects. Notice the detour made by theevolutionary pathway away from the shortest, straight path between theinitial state (0, 0) and the new optimum at (5, 0), the width of thepathway and the change in shape and principal direction of the ellipses.

Lande & Kirkpatrick demonstrated the maternal effects model with

G = 0.25I,E = I . β(t) = u(t)b, with b = (1 0 0)T .

In the absence of maternal influences, the mean phenotype canmove only in the direction of b, i.e. phenotypes (z1, 0, 0).(a) In the case of the maternal matrix

0 0.5 0.250.5 0 00.25 0 0

we have

S3,b =

0.3945 0.9010 0.4368 −0.74921.0077 0.2650 0.5183 −1.29530.5039 0.1325 0.2591 −0.6476

.So S3,b has image spanned by (1, 0, 0)T and (0, 1, 0.5) So with

maternal influences we reach phenotypes (z1, 2z3, z3), withoutmaternal influences we only achieve (z1, 0, 0).Maternal inheritance strictly enhances phenotypic evolvability

(b) Suppose instead that we have a maternal effects matrix

0 0.25 00 0 00 0.5 0

0.2578 0.2578 0.2578 −0.00780.0312 0.0312 0.0312 00.0156 0.0156 0.0156 −0.0156

.In this case, the image of S3,b = {(z1, z2, 2z1 − 16z2), z1, z2 ∈ R}so that the space of mean phenotypes achievable with maternaleffects is not a superset of those achievable without.Not only do maternal effects increase phenotypic diversity, theycan profoundly change the course of evolution by enablingphenotypic evolution to bypass evolutionary constraint and followdirections that genetic processes cannot reach. For thisdemonstration example from Lande & Kirkpatrick it is clear thatthe impact of maternal influences goes much further than addingmomentum to evolutionary change.

Using control theory tools new to evolutionary biology we find:

When selection acts freely (in a small ball of radius r) M maynarrow the cloud of phenotype, further constraining evolution allonga line of least genetic resistance. Under this scenario, evolution withmaternal inheritance can deliver phenotypes that are an order ofmagnitude larger than without this additional path, but only inparticular directions.

When selection is constrained, e.g. acting on only one trait, thespace of achievable phenotypes may be enhanced by M. We seeevolutionary paths without maternal inheritance oscillates back andforth, whereas with maternal inheritance this tracking “spins out”.In this way, fitter phenotypes can be reached more quickly thanthrough genetic assimilation alone.

The space of achievable phenotypes with M may be transverse tothose achievable without M. Combining the genetic and epigeneticpathways therefore allows z2 to evolve in a direction that is notrepresented genetically. So M may actually alter the course of shortto medium term evolutionary change.

Post Doc Position at the University of Exeter – PenrynCampus, Cornwall, UK

Are Structured Life Histories Really Buffered AgainstEnvironmental Change?

with Dave Hodgson (Ecology), Stuart Townley (Maths),Miguel Franco (Plymouth)

Multi-disciplinary: joint between Mathematics, the Centre forEcology and Conservation, the Environment & SustainabilityInstitute in Exeter (Cornwall), and the University of Plymouth

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