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Transcript of variables constants (parameters)web.mit.edu/biophysics/sbio/PDFs/L17_notes.pdf · PDF...

  • 1

    ! P S

    #3 d

    ue to

    da y

    ! (h

    er e,

    in c

    la ss

    o r R

    m . 1

    3- 20

    42 b

    ef or

    e 3P

    M )

    2

    22 2

    2

    2 2

    xi D

    i a k

    ti xa

    D a

    ia k

    r ta

    i i

    i

    a a

    a a

    ∂∂ +

    − =

    ∂∂ ∂∂

    + −

    + =

    ∂∂

    γ

    γ

    R ev

    ie w

    Tu rin

    g- G

    ie re

    r- M

    ei nh

    ar dt

    m od

    el s

    Lo ca

    l e xc

    ita tio

    n, g

    lo ba

    l i nh

    ib iti

    on

    a: co

    nc en

    tra tio

    n ac

    tiv at

    or i:

    co nc

    en tra

    tio n

    in hi

    bi to

    r t:

    tim e

    x:

    po si

    tio n

    r a :

    ba sa

    l a ct

    iv at

    or s

    yn th

    es is

    ra te

    k a , k

    i: ra

    te c

    on st

    an t f

    or s

    yn th

    es is

    γ a ,γ

    i :

    de ca

    y ra

    te s

    D a,

    D i:

    di ffu

    si on

    c on

    st an

    ts

    va ria

    bl es

    co ns

    ta nt

    s (p

    ar am

    et er

    s)

    3

    22 2

    2

    2 2

    xi D

    i a k

    ti xa

    D a

    ia k

    r ta

    i i

    i

    a a

    a a

    ∂∂ +

    − =

    ∂∂ ∂∂

    + −

    + =

    ∂∂

    γ

    γ

    ( )

    2

    2 2

    2

    2 2

    1

    sI P

    I A

    Q τI

    sA A

    IA R

    τA

    ∂∂ +

    − =

    ∂∂ ∂∂

    + −

    + =

    ∂∂

    ch oo

    se di

    m en

    si on

    le ss

    va ria

    bl e

    no rm

    al iz

    e 4

    va ria

    bl es

    2 )1 (

    1 + =

    + =

    R I

    R A

    ho m

    og en

    eo us

    so lu

    tio n

    0 /

    / =

    ∂ ∂

    = ∂

    ∂ t

    s

    on ly

    o ne

    fi xe

    d po

    in t,

    si nc

    e bo

    th A

    a nd

    I >0

    4

    A I

    s s

    A I

    ho m

    og en

    eo us

    so lu

    tio n

    0 /

    / =

    ∂ ∂

    = ∂

    ∂ t

    s

  • 5

    st ab

    ili ty

    o f h

    om og

    en eo

    us s

    ol ut

    io n  

     

    − +

    + −

    +− =

     

     

    −− −

    Q Q

    R R R

    RR

    Q Q A

    IAR IAR

    )1 (2

    )1 (

    11

    2

    1 2

    2 2

    2

    tra ce

    < 0

    de t >

    0

    011 >

    < +−

    Q

    Q RR

    ) ,

    (' )

    , (

    ) ,

    (' )

    , (

    τ τ

    τ τ

    s I

    I s

    I s

    A A

    s A

    + =

    + =

    in ho

    m og

    en eo

    us so

    lu tio

    n:

    or in

    g en

    er al

    re al

    p ar

    t o f e

    ig en

    va lu

    es >

    0

    6

    A I

    s s

    A I

    in ho

    m og

    en eo

    us so

    lu tio

    n

    I’( s, τ)

    ) ,

    (' )

    , (

    ) ,

    (' )

    , (

    τ τ

    τ τ

    s I

    I s

    I s

    A A

    s A

    + =

    + =

    7

    2

    2

    2

    2

    2

    ' '

    ' )

    1( 2

    '

    ' '

    ) 1(

    ' 11

    '

    sI P

    Q I

    A R

    Q I

    sA I

    RR A

    RR A

    ∂∂ +

    − +

    = ∂∂

    ∂∂ +

    + −

    +− =

    ∂∂ ττ

    ) co

    s( )

    (ˆ )

    , ('

    ) co

    s( )

    (ˆ )

    , ('

    ll s I

    s I

    s A

    s A

    τ τ

    τ τ

    ==

    ) ,

    (' )

    , (

    ) ,

    (' )

    , (

    τ τ

    τ τ

    s I

    I s

    I s

    A A

    s A

    + =

    + =

    tri al

    s ol

    ut io

    n:

    8

    A I

    s s

    A I I’(

    s, τ)

    ) ,

    (' )

    , (

    ) ,

    (' )

    , (

    τ τ

    τ τ

    s I

    I s

    I s

    A A

    s A

    + =

    + =

    ) co

    s( )

    (ˆ )

    , ('

    ) co

    s( )

    (ˆ )

    , ('

    ll s I

    s I

    s A

    s A

    τ τ

    τ τ

    ==

  • 9

    I P

    Q A

    R Q

    dId

    I RR

    A RR

    dAd

    ˆ ˆ )

    1( 2

    ˆ

    ˆ )

    1( ˆ

    1 11

    ˆ

    2

    2 2

       

    + −

    + =

    + −

       

    − +−

    =

    l

    l

    ττ

    ) co

    s( )

    (ˆ )

    , ('

    ) co

    s( )

    (ˆ )

    , ('

    ll s I

    s I

    s A

    s A

    τ τ

    τ τ

    ==

    0 1

    11

    0 12

    1 11

    2 2

    2 2

    <    

    − +−

    − +

    > +

    +    

    +  

      −

    +− −

    l l

    l l RR

    P Q

    R Q R

    P Q

    RR st

    ab ili

    ty in

    ho m

    og en

    eo us

    so lu

    tio n

    11 +−

    > RR

    PQ

    10

    ho m

    og en

    eo us

    s ta

    bi lit

    y:

    st ab

    ili ty

    a ga

    in st

    s pa

    tia l d

    is tru

    ba nc

    e:

    11 +−

    > RR

    Q

    11 +−

    > RR

    PQ

    I

    s

    I I’(

    s, τ)

    if P

    < 1

    (D i<

    D a)

    , s ys

    te m

    s is

    a lw

    ay s

    st ab

    le , a

    ga in

    st a

    ny pe

    rtu rb

    at io

    n bo

    th s

    pa tia

    l a nd

    te m

    po ra

    l

    11

    I

    s

    I

    ho m

    og en

    eo us

    ly s

    ta bl

    e:

    I re

    la xe

    s ba

    ck to

    pr ev

    io us

    v al

    ue a

    fte r

    sm al

    l u ni

    fo rm

    d is

    tu rb

    an ce

    I

    s

    I

    st ab

    le a

    ga in

    st s

    pa tia

    l di

    st ur

    ba nc

    e:

    I’ re

    la xe

    s ba

    ck to

    af te

    r s m

    al l s

    pa tia

    l di

    st ur

    ba nc

    e

    I

    12

    To pi

    c I:

    Sy st

    em s

    C el

    l B io

    lo gy

    Sp at

    ia l o

    sc ill

    at io

    n in

    E . c

    ol i

    si m

    ila r t

    o ge

    ne tic

    o sc

    ill at

    or s,

    b ut

    n ow

    w e

    ca nn

    ot ig

    no re

    th e

    sp at

    ia l d

    im en

    si on

    s

    bi ol

    og ic

    al fu

    nc tio

    n:

    de te

    rm in

    e th

    e ce

    nt er

    o f t

    he c

    el l,

    to p

    re pa

    re fo

    r p ro

    pe r c

    el l d

    iv is

    io n

  • 13

    In tro

    du ci

    ng th

    e m

    ol ec

    ul es

    :

    - F ts

    Z fu

    nc tio

    n: A

    ss em

    bl y

    of a

    p ol

    ym er

    ic ri

    ng o

    f t he

    t ub

    ul in

    -li ke

    G TP

    as e

    Ft sZ

    (Z ri

    ng ).

    Th e

    Z- rin

    g is

    lo ca

    liz ed

    to th

    e ce

    nt er

    b y

    th e

    ac tio

    ns o

    f th

    e M

    in C

    , M in

    D , a

    nd M

    in E

    pr ot

    ei ns

    .

    - M in

    C in

    hi bi

    ts th

    e in

    iti at

    io n

    of th

    e Z

    rin g.

    M in

    C c

    ol oc

    al iz

    es w

    ith M

    in D

    . I

    n w

    ild -ty

    pe (W

    T) c

    el ls

    , M in

    C /D

    fo rm

    s a

    po la

    r p at

    te rn

    t ha

    t o sc

    ill at

    es b

    et w

    ee n

    th e

    po le

    s, k

    ee pi

    ng th

    e ce

    nt er

    f re

    e fo

    r i ni

    tia tio

    n of

    c el

    l d iv

    is io

    n.

    Th us

    , v irt

    ua lly

    a ll

    of M

    in C