3 B. Derivation of the evolution equations of the f$_{k}$ from the boundary integral formulation different orders. Only second order harmonics are present to the order of O ( ε ) in the equilibrium shape function of an almost spherical vesicle in the absence of flow. They are also the only harmonics that can interact with linear flow directly. For consistency considerations, pushing the expansion to next order implies that the fourth and ze- roth order harmonics cannot be neglected. They enter dynamics as a result of interactions between the vesicle shape and the Helfrich force and the flow; they are of or- der O ( ε $^{2}$) . The shape function does not contain spherical harmonics of odd orders owing to the centro-symmetry. The spherical harmonics of even orders higher than 4 have an amplitude O ( ε $^{3}$) and thus are neglected. As fol- lows from the inequality We shall adopt here a different spirit from that of pre- vious theories [8–11]. There the Lamb solution is used (a solution of the Stokes equations inside and outside the vesicle). We find it convenient to take the bound- ary integral formulation as a starting point. The present spirit is equivalent to using the Lamb solution, but it has the advantage that the boundary conditions at the mem- brane are already implemented in the boundary integral formulation. Indeed, the Stokes equations together with the boundary conditions on the membrane, and far away from it, can be converted into boundary integral formu- lation [19]. This leads to the following integral equation the average radius of the vesicle R$_{0}$ is not exactly 1. Therefore a negative correction must be added to it in order to satisfy (5). This correction is of order O ( ε $^{2}$) and leads to a non-zero coefficient for zero-order spher- ical harmonic in the expansion of f ( x ) , which will be denoted f$_{0}$ below. The expansion takes formally the form: Here v ( x ) is the actual velocity of point R ( x ) , u ( x ) is the velocity of point R ( x ) in the imposed flow, F ( x ) is the membrane rigidity force at point R ( x ) , N ( x ) is the outward pointing normal to the vesicle surface at point R ( x ) . The integration is taken over the surface of the vesicle and d $^{2}$R ( x $^{′}$) is the surface area element. The integration kernels have the following form: There are various definitions of spherical harmonics, here we use those that satisfy the following normalization conditions: Using the expression for the Helfrich force, we can find from (10) the velocity field on the surface of the vesicle. The Helfrich force is given in terms of the shape as However, all equations are written in universal form and are valid for any rescaling of spherical harmonics. The condition (5) together with the definition Δ = ε 2 provide two additional constraints on the coefficients of the expansion (7). We use the former to express f$_{0}$ through f$_{2}$$_{,l}$ where K is the Gaussian curvature and Δ$_{S}$ is the Laplace-Beltrami operator. It can easily be checked that expression (11) vanishes for a spherical shape, and thus is of order O ( ε ) . In order to balance (10) at order O (1) we need to assume the imposed flow to be of the same order as (11). This requirement can equivalently be ful- filled, following Refs. [8, 10], by the demand that κ scale as ε − $^{1}$. We then set κ = ε − $^{1}$¯ κ, with ¯ κ of order O (1) . and the latter to norm the coefficients f$_{2}$$_{,l}$ The precise technical details will be given elsewhere, while here we shall present only the spirit and some in- termediate steps. We expand v ( x ) , u ( x ) , and F ( x ) into vector spherical harmonics of x . For convenience we de- fine them as We retain one more order in these expressions compared to previous works for the sake of consistency. Note that the volume and the surface have been expanded to the order O ( ϵ $^{3}$) in order to obtain the equalities (8) and (9). 1 = 3 V 4 π = ∫ R ( x ) $_{3}$d$^{2}$x 4 π ≥ [∫ R ( x ) d $^{2}$x 4 π ] 3 = R 3 0 v$_{l}$ ( x )(1 + λ ) = = 2 u$_{l}$ ( x ) + 2 ∫ G$_{jl}$ ( R ( x $^{)}$− R ( x $^{′}$)) F$_{j}$ ( x $^{′}$) d $^{2}$R ( x $^{′}$)+ $^{+ (1}$− λ ) ∫ K$_{jlm}$ ( R ( x $^{)}$− R ( x $^{′}$)) v$_{j}$ ( x $^{′}$) N$_{m}$ ( x $^{′}$) d $^{2}$R ( x $^{′}$) . (10) f ( x ) = ε $^{2}$f$_{0}$ + εf$_{2}$ + ε $^{2}$f$_{4}$ + O ( ε $^{3}$) , (7) f$_{k}$ = k ∑ l = − k f$_{k,l}$ ( t ) Y$_{k,l}$ ( x ) . G$_{ij}$ ( R ) = 1 8 π ( δ$_{ij}$ R + R$_{i}$R$_{j}$ R 3 ) , K$_{ijk}$ ( R ) = 3 4 π R$_{i}$R$_{j}$R$_{k}$ R 5 . ∫ Y$_{k,l}$ ( x ) Y ∗ $_{k,l}$( x ) d $^{2}$x = 4 π ( k !) 2 (2 k + 1)( k + l )!( k − l )! F$_{i}$ $^{=}$− 2 ( κ [2 H ( H 2 − K ) + Δ$_{S}$ H $^{]}$− ZH ) N$_{i}$ + + ∂Z ( x ) ∂R$_{j}$ ( x ) ( δ$_{ij}$ − N$_{i}$N$_{j}$ ) , (11) ∫ f$_{0}$ ( x ) d $^{2}$x = − ∫ f$_{2}$ ( x ) $^{2}$d$^{2}$x − ε 3 ∫ f$_{2}$ ( x ) $^{3}$d$^{2}$x + O ( ε $^{2}$) (8) 2 ∫ f$_{2}$ ( x ) $^{2}$d$^{2}$x − 2 ε 3 ∫ f$_{2}$ ( x ) $^{3}$d$^{2}$x = 1 + O ( ε $^{2}$) . (9) Y i 1 $_{,k,l}$( x ) = e$_{imn}$x$_{m}$∂$_{n}$Y$_{k,l}$ ( x ) , Y i 2 $_{,k,l}$( x ) = (2 k + 1) x$_{i}$Y$_{k,l}$ ( x $^{)}$− ∂$_{i}$Y$_{k,l}$ ( x ) ,