Supplementary material

Electrostatic switching of nuclear basket conformations provides a potential mechanism for nuclear

mechanotransduction

Shaobao Liu, Haiqian Yang, Tian Jian Lu, Guy M. Genin, Feng Xu

Methods: electrostatic interaction energy

The scaling for the electrostatic interaction energy W ( x ) between charged basket filaments separated by a

distance x followed the Ray-Manning solution (2000) for attraction and repulsion between lines of point

charges. The potential had three regimes that were determined by the Debye screening parameter, κ (Fig.

1): a long-range repulsion up to an energy barrier ER for κx>1; an intermediate range of attraction due to

condensation of Debye clouds, down to an energy well of depth EA for 1/e≤ κ x ≤1; and a short-range

repulsion for κx ≤ 1/e. We linearized the interaction energy in the intermediate region, so that:

W ( x )≈ { EA ( c−K 0 (κx )c−K0 (1/e ) ), κx ≤ 1/e

ER−( ER−EA ) 1−κx1−1/e

, 1/e<κx ≤1

ER

K 0 (κx )K0 (1 )

, κx>1

(S1)

where K 0(x ) is zeroth order modified Bessel function of the second kind, and the constant c and

interaction energies EA and ER can be related to the parameters of the Ray-Manning solution as follows.

The Ray-Manning parameters are c1=1/2Z2 ξ, c2=ln κb, and c3=4 Zξ−1, where ξ= λB /b is the

dimensionless charge density, λB is the Bjerrum length, b is the line charge spacing, and Z is the

counterion valence. The peak attraction energy EA and the peak repulsion energy ER can be written

by noting that W (1/e )=EA and W (1 )=ER:

ER=2 c1 ( c3−1 ) K 0 (1 ) (S2)

EA=−2 c1 (1+c2 )+c1 c3 ( c2+K 0 (1/e ) ) (S3)

By rewriting the first term in the form of the Ray-Manning solution:

W ( x )=−c1 (2 (1+c2 )−c3 (c2+K0 (κx )) ) , κx ≤ 1/e (S4)

we can equate the expressions:

−c1(2 (1+c2 )−c3 (c2+K0 (κx ) ) )=E A( c−K0 (κx )c−K 0 (1/e ) ) (S5)

Solving for c yields:

c=2 (1+c2 )

c3−c2 (S6)

Methods: calculation of the size of the largest rigid sphere that could pass through a transverse pore.

The sizes of the transverse pores formed by the nuclear basket filaments and segments of the distal and

nucleoplasmic rings were estimated by approximating the largest sphere that could pass through the pore.

The vertices of three-dimensional (3D) closed curve were denoted M 1 , M 2 ,M 3 ,∧M 4, each a point in a

3D space.

We defined the line bisecting the angle made by edges M⃑1 M 3 and M⃑1 M 2 as α⃑ 1 and that bisecting the

angle made by the second edge M⃑2 M 1 and the subsequent edge M⃑2 M 4 asα⃑ 2. We defined the normal

vector of the plane containing α⃑ 2 and perpendicular to the plane formed by M⃑2 M 1 and M⃑2 M 4 as n⃑p 2.

The intersection of the bisecting line and the bisecting plane could then be written as:

X⃑1=n⃑ p 2 ·⃑ M 1 M 2

n⃑p 2 · α⃑1α⃑1+M 1 (S7)

The distances d i between X⃑1 and each of the four edges were calculated, and the maximum size of the

rigid sphere that could pass through that transverse pore was estimated as:

r1=min {d i }i=1,2,3,4 (S10)

There are eight intersections of bisecting lines and bisecting plane, which yields eight possible rigid

spheres. The transverse pore radius was taken as the maximum of these eight possible radii:

r=max {r i } i=1,2 , …8 (S11)

Accuracy of this efficient, approximate algorithm was checked carefully, and was evident in Video S2.

Figure S1. Geometrical shape and sizes of nuclear basket. Red sphere: largest molecule that can pass

through the transverse pore.

Figure S2. Principal stresses of the nuclear envelope with biaxial membrane stretch of 15%, normalized with respect to the average stress in the membrane σ ∞.

Figure S3. NPCs provide little resistance to the expansion of the nucleoplasmic ring except at very high levels of strain. Red: normalized radius of an empty circular hole in an elastic membrane. Blue normalized radius of a NPC. At low strain, the weak flexural resistance of the NPC causes it to behave like a rigid, hinged rod.

Figure S4. NPC opening with different sizes of nuclear basket filaments. (a) κd=0.4 ,1.2 ,2.0 , κb=0.5 , κl=6. (b) scaled-up κd ,κb ,κl. With a smaller Debye screening length or a

larger nuclear basket (b), a second bifurcated region occurs in the blue plot.

Figure S5. Normalied radius of tarnsverse pore. κl=3 , κb=0.25. Boundaries: β 1:

κ (a+b )sin π8=1 /e; β 2: κ (a−b ) sin π

8=1/e; β 3: κ (a+b )sin π

8=1; β 4: κ (a−b ) sin π

8=1; β 5:

a+b=l. In the case of bi-stable states, the background shading patches represented the radius of the smaller of the two possible transverse pore sizes.

Video S1. FEM Simulation of nuclear pore opening.

Video S2. Stretch-dependent phase changes in the configurations of the nuclear basket. b=5 nm, l=60 nm, 1/κ=10 nm, d=4 nm.