Vesicle self-reproduction: the onset of the cell cycle

Saša Svetina

Ljubljana, Slovenia

KITPC, Beijing

May 10, 2012

Vesicle self-reproduction: the onset of the cell cycle

Saša Svetina

Ljubljana, Slovenia

KITPC, Beijing

May 10, 2012

Application of the shape equation in.the research on the origin of life

Some characteristics of vesicles that could be relevant for the life process

Vesicles:

• compartmentalize the space

• can grow by incorporation into the membrane of a new material and by the inflow of solution

• may exhibit the phenomenon of self-reproduction

• are, on the basis of the criterion for the self-reproduction, able to evolve

• have the capacity to increase their complexity

Many cellular processes that involve membrane transformations arose from processes that occur also at the level of vesicle.

During the evolution they were developed into deterministic machineries

A motto

(Svetina and Žekš, Anat. Rec. 2002)

An example is budding in vesicles and cells

Vesicles:

Cells:

An outline

Shapes of growing vesicles

Vesicle properties that are essential for the process of vesicle self-reproduction

The implications with regard to the cell cycle

Vesicles can grow and attain shapes at which they are apt to divide

Vesicles can be induced to grow by incorporating into their membranes new molecules and by transmembrane transport of the solution

Under some special circumstances such growth can lead to the formation of twin shapes, i.e. shapes composed of two spheres connected by a narrow neck

Experiments by Mojca Mally, Ljubljana

A vesicle growing at constant volume may exhibit a variety of budded shapes

spherical growth

sudden burst of buds

consecutive bud formation

invagination

evagination

(Peterlin et al., Phys Chem Lipids 2009)

There is a condition which determines whether a vesicle grows as a sphere or not

This condition can be derived by taking into consideration membrane bending energy

or ?

where C1 and C2 are principal curvatures, dA is the element of membrane area, kc membrane bending constant and C0 its spontaneous curvature,

and the transport of the material across the membrane

dACCCkW c

2

0212

1

Spontaneous curvature is the result of membrane asymmetry

W. Helfrich

Z. Naturforschung c 1973

2674 citations up to 27.4.2012

A membrane with spontaneous curvature C0

would tend to make a spherical vesicle with

the radius R0 = 2/C0

and thus attain zero bending energy (because

for the sphere C1 = C2= 1/R0)

dACCCkW c

2

0212

1

The non-spherical shapes can be theoretically predicted by the

minimization of the reduced bending energy (w =W/8πkc)

dacccw2

0214

1

with c1 = RsC1, c2 = RsC2, c0 = RsC0 and

Rs the radius of the sphere with the

membrane area A

Shapes are thus characterized by the

reduced spontaneous curvature c0 and the

reduced volume v

34 3 /R

volumevesicle

V

Vv

ss

The shape phase diagram of the spontaneous curvature model

Taken from Seifert et al., Phys. Rev. A 64 (1991)

c0 = RsC0

Vesicle bending energy in the vicinity of the sphere

Δwb (the reduced bending energy minus the reduced bending energy of the sphere) in dependence on v plotted for different values of c0 = C0Rs

The pressure due to the bending energy, Δpℓ, derived by Ou-Yang and Helfrich (1989) :

)6(2

03 ss

c RCR

kp

(Božič and Svetina, PRE 2009)

0)2(2 0023 sscss RCRCkRpR

The graphs show at which values of the pressure difference (Δp) and membrane

tension (σ) a vesicle is spherical

Ou-Yang and Helfrich (1989) also presented generalized Laplace equation:

p

Sphere is stable as long as

pp )6(

203 s

s

c RCR

kp

ck

A prototype model for vesicle growth

pALdt

AdVp

)(

It is assumed that membrane area (A) duplicates in time Td dT

t

02AtA

c0 = RsC0 is increasing in time because membrane area A is increasing in time and Rs = (A/4π)

Volume (V) changes are determined by the hydraulic permeability Lp

(Božič and Svetina, Eur Biophys J 2004)

Remember:

Δp is increasing while Δpℓ is decreasing in time:

Consequently, these two Δp-s eventually become equal.

pp

pd

s

LT

Rp

2

2ln )6(

203 s

s

c RCR

kp

Stability of the spherical shape of a growing vesicle

pALdt

AdVp

)(

The volume is changing according to the time dependence of the area which means that Δp depends on the flux

The relevant part of the shape phase diagram of the spontaneous curvature

model

Taken from Seifert et al., Phys. Rev. A 64 (1991)

c0 = RsC0

In the c0 – v shape diagram a vesicle has to transform

from v = 1, c0 = 2 into

v = 1/2 , c0 = 22

c0,cr

The trajectory from a sphere to the twin shape in the c0 – v shape phase diagram

Vesicle doubling cycle is divided into phases

Vesicle first grows as a sphere, and after it reaches the critical size (first arrow) its shape begins to change until it becomes a composion of two spheres connected by a narrow neck

Td membrane area doubling time Lp membrane hydraulic permeability kc membrane bending constant Co membrane spontaneous curvature time/Td

The criterion for vesicle self-reproduction

This criterion relates internal and external properties of the system and thus represents a condition for the selectivity.

pcpd CkLT 40

ℓp = 1.85pd

s

LT

Rp

2

2ln

)6(2

03 ss

c RCR

kp

ℓp > ℓp,min = 1.85

When ℓp > 1.85, the two spheres of the final shape have different radii. The average doubling time is larger than at ℓp,min = 1.85

pcpd CkLT 40

ℓp ℓp,min

Vesicle division needs not be symmetric

85.1CkLT 40cpd

Variability of vesicle doubling time at the asymmetrical division

Variable is the phase of spherical growth because smaller daughter vesicle needs more time to reach the critical size than larger daughter vesicle.

Rs =

√A/4π

ℓp =

ℓp,min ℓp

The addition of new components (e. g. a solute that can cross the membrane) increases the complexity of the system (Božič and Svetina, Eur Phys J 2007)

The concentration of solute (Φ) oscillates. During the first phase it decreases and during the second phase it increases. The opposite is valid for ΔP.

0V

Vv

0A

A

P

0

ℓp : reduced hydraulic permeabilityps : reduced solute permeabilityΦ0 : reduced outside solute concentration

The condition for vesicle self-reproduction in the case of added solute

The variability of the generation time is increased

The size of daughter vesicles after few generations attains a steady distribution with pronounced variability.

Basic facts about the cell cycle

The cell cycle is divided into phases. Its generation time is variable. The most variable is the G1 phase. The concentration of many cell cycle proteins is oscillating

Vesicle self-reproduction and the cell cycle have many common

featuresThe division of the cycle into phases

The start of the division phase by the commitment process

The variability of cycle generation times

The length of the growth phase is more variable

Both vesicle and cell constituents exhibit concentration oscillations(Svetina, chapter in Genesis 2012)

Most of the presented analysis was done in collaboration with Bojan Božič

Thank you for your attention!

Most of the presented analysis was done in collaboration with Bojan Božič