Markov and semi-Markov processes describe the dynamics of biological ion channels

Professor Alan G HawkesSwansea University

Professor David Colquhoun:

Professor of Pharmacology, University College London

Assad JalaliAnton Merlushkin

Alan Hawkes

Swansea

ion-channel team

Sir Andrew Huxley died on May 30th 2012, aged 94

Basic resultsBursting behaviourTime Interval Omission (TIO)Joint distributions – maximum likelihood estimationMultiple levelsBursting behaviour with TIO

Channel is modelled as a finite-state Markov processwith transition rate matrix

é ùê ú= ê úê úë û

Q QQ

Q QAA AF

FA FF

Transition probability matrix

[ ]

1

( ) ( ( ) | (0) )

( ) , 0

( ) e i

ij

t

kt

ii

t P X t j X i

t e t

t l-

=

= = =

= ³

= å

Q

T

T

T A

il are eigenvalues of - Q

1nT

+%

nJ% 1nJ +%

kJ 1kJ +

1kT

+

In this example four actual open intervals make one

{( , )}r rJ T

kernel density being a matrix whose ijth element is

[ ]0

( ) lim (( ) ( ) | (0) )/nij ht P t h T t X t j X i h

®= - < < Ç = =G

is a semi-Markov process with

( )( )

( )

tt

t

é ùê ú=ê úê úë û

0 GG

G 0

AF

FA

* 1

0

( ) ( ) ( )te t dtqq q¥

- -= = -òG G I Q QAF AF AA AF

AA FF

AF AF FA FA( ) ; ( )t tt e t e= =Q QG Q G Q

*

0

( ) (0)t dt¥

= =òG G G

{ }nJ Is a Markov chain with transition matrix

Taking alternate events, the open ones, we have a Markov chain with equilibrium probability vector satisfying

A A AF FA A A, 1= =G G uff f

The equilibrium distribution of observed open times is

A AF F( )tG uf

Bursting behaviour

Q-matrix for model CH82

19000 400

0 3000 0

500 0 0

0 15000

15

0 0

50 2065 20

0 0

000

0 10

3050 50

0.667 500.667

10

-

-

-

é ùê úê úê úê ú= ê úê úê úêê û

-

úë

-

ú

Q

Burst lengthGap between burstsGaps within burstsTotal open time per burstTotal shut time per burstNumber of openings per burstLength of the kth opening in a burst with r openings

1( ) exp( )( ) / ( )

k r k

b bf t t e P r

G G Q Q G GAB BA AA AA AB BA

1nT

+%

nJ% 1nJ +%

kJ 1kJ +

1kT

+

In this example four actual open intervals make one

Time interval omission (TIO)

{( , )}nnJ T% %

is a semi-Markov process with kernel density

( )( )

( )

tt

t

é ùê ú=ê úê úë û

0 GG

G 0

AF

FA

%%

%

*

*

*0

( )( ) ( )

( )te t dtq

qq

q

¥-

é ùê ú= = ê úê úë û

ò0 G

G GG 0

AF

FA

%% %

%

Modified kernelsInstead of embedding a semi-Markov process at time after an observed interval begins, it is more natural to do so at the start of each such interval. The trouble is that, at such moments, we do not know that the first interval is going to last for at least .The probability that it does last that long, conditional on the starting state is given by vectors

and e et tQ Qu uAA FFA F

for open and closed intervals, respectively. Then the new semi-Markov kernels are given by

( )1 ( ) diag( )diag e e t et t tt- -Q Q Qu R Q uAA AA FFAA AF F

for open intervals and a similar expression for closed intervals.

*

0

( ) (0)t dt¥

= =òG G G% % %

{ }nJ% Is a Markov chain with transition matrix

Taking alternate events, the open ones, we have a Markov chain with equilibrium probability vector satisfying

, 1= =G G uff fA A AF FA A A% %

The equilibrium distribution of observed open times is

( )tG uf A AF F%

define

( ) ( ( ) (0, ) | (0) )ij

t P X t j D t X ié ù = = Ç =ë ûRAF

For ,i j Î A

where (0, )D tFIs the event that no shut period is detected over (0, t)

( ) ( )t t e tt= - QG R Q FFAAF AF

%

Let

0

( ) ( 1) ( )mm

m

t t mt¥

=

= - -åR MA

Theorem. If - Q has eigenvalues 1 20, kl l l= K

1

( ) ( ) , 0

0, 0

i

kt

m imi

t t e t

t

l-

=

= >

= <

åM B

Where ( )im tB

1

( )m

rim imr

r

t t=

= åB C

Is a polynomial of degree m in t with matrix-valued coefficients

So, in the interval ( 1)m t mt t< < + The exponentials are

multiplied by polynomials of degree m.

Asymptotic resultsWe can use the algebra of partitioned matrices to get an alternative Laplace transform expression, which can be also be obtained by the followingmore appealing direct argument

( ) 1*( ) ( )q q-

=R WA

( ) ( )q q q= -W I H

( ) 1( ) ( )( )e q tq q- - -= - -I QH Q I I Q QFFAF FF FA

Theorem: When Q is reversible, det W() = 0 has exactly kA real roots

1 1 0.A Ak kq q q-£ £ £ <L

If Q is irreducible and the roots are distinct, then, as t ® ¥

1

( ) ~ /i

kt

i i i ii

t eq=

¢åR cr rW cA

A

where ,i ic r are the right (column) and left (row) eigenvectors of ( )iqH corresponding to eigenvalue iq

Det W() for CH82 model: = 0.2

Observed open time density for CH82 model: = 0.2

Observed shut time density for CH82 model: = 0.2

ApplicationsJoint distributions: it is interesting to study the joint behaviour of neighbouring open/shut pairs of intervals, looking at conditional distributions, means etc. This can be done from the product

( ) ( )o ct tG GAF FA% %

Likelihood: The likelihood for a whole sequence can found, and maximised to provide parameter estimates from the product

( ) ( )oi ciit tÕ G G uf AF FA

% %

Jumps and pulses: The techniques discussed can be used to study the first few events following a jump or a pulse change in agonist concentrations or voltage level, which modify the Q-matrix in known ways.

Results of simulation with critical resolution 25 µs

We have found that a limitation of ML analyses based on records at a single agonist concentration is the statistical correlation between the estimates of the channel opening rate, b, and the shutting rate, a . The correlation coefficient between these estimates is often greater than 0.9, found from the off-diagonals of the Hessian matrix of the likelihood evaluated at its maximum. There is a corresponding diagonal ridge in the likelihood surface.

This corresponds to the difficulty in distinguishing between long openings with few interruptions (small , a b ) and many shorter openings separated by very short shuttings (large a , b ) which combine to form a large apparent opening..

However, good estimates can be made if data from recordings at more than one concentration are combined to form an overall likelihood.

Multiple Levels

Some channels exhibit more than one conductance level when open. and this raises some complication.

The main kernel densities can be found in a manner similar to the two-level case.

( )rs tG%

The transition rate matrix can then be partitioned in the form

11 1

1

l

l ll

é ùê úê úê ú= ê úê úê úê úë û

Q Q

Q

Q Q

L L

M O M

M O M

L L

We look at an embedded semi-Markov process for which we note the duration of periods of time spent at each level and the “gateway state”, the state in which an occupancy begins. This has a density kernel.

12 1

21 2

1 2

( ) ( )

( ) 0 ( )( ) .

( ) ( )

l

l

l l

t t

t tt

t t

é ùê úê úê ú= ê úê úê úê úë û

0 G G

G GG

G G 0

L

L

M M O M

L

( ) exp( ) , rs rr rst t s r= ¹G Q Qwhere

The difficulty arises because, while we may be sure that the channel has left a particular level for a period in excess of ,We may not be sure where it has gone to: it may hop around rapidly between two or more levels before settling on one of them.The trick is to introduce some ‘indeterminacy intervals’ and augment the state space of the semi-Markov process to include states of the form (r, i), which indicates that the channel is in state i at the start of an indeterminacy that follows an observed sojourn at level r.

Burst behaviour and TIO