Two correlated sources emit a pair of sequences, each of which is observed by a different encoder. Each encoder produces a rate-limited description of the sequence it observes, and the two descriptions are presented to a guessing device that repeatedly produces sequence pairs until correct. The number of guesses until correct is random, and it is required that it have a moment (of some prespecified order) that tends to one as the length of the sequences tends to infinity. The description rate pairs that allow this are characterized in terms of the Rényi entropy and the Arimoto–Rényi conditional entropy of the joint law of the sources. This solves the guessing analog of the Slepian–Wolf distributed source-coding problem. The achievability is based on random binning, which is analyzed using a technique by Rosenthal. X is drawn from a finite set X equal to x?” until the guess is correct. The guessing order is determined by a guessing function G, which is a bijective function from G proceeds as follows: the first guess is the element X. Arıkan , then  follows ·a+b+c3p≤4p·ap+bpLemma 6.Let a, b, c, and d be nonnegative real numbers. Then, for all Proof. If 1], then  follows Lemma 7 .Let Proof. This is a special case of  p−1]≤E[X]·2p−E[Xp−1])≤2p−1E[X]We now consider two cases depending on which term on the RHS of  achieves implies  because p, then:E[Xp]≤2pE[X]p,so  holds alIn this section, we prove a nonasymptotic and an asymptotic converse result .Theorem 2.Let Proof. We view  as three because:E≤∑x,yPϕg(y′)ρ=∑x,yPρ]≤1+4ρ∑x,y])with:ψ=)ϕg(y′),β3=β31|U|ρ≤∑x,yP≤2−ρϵ,wh]ρ˜; and . In the −ρϵ.From , 82), ≤2−ρϵ.Frotisfying .We now consider  when ρ>1e did in . Insteade RHS of  by:93)ρ>1. Unlips as in –82),∑xx, y, f, and g is implicit in our notation.Bounding e RHS of , is moreosenthal  (Proof og(y˜)ρ−1=Ef,g∑x′≠2+γ3)ρ−1=∑x′≠x,y′2+γ3)ρ−1≤∑x′≠x,y′2+β3)ρ−1≤∑x′≠x,y′g[δ1ρ−1]=4ρ−1{+∑x]}with:γ1=γ1ϕf(x˜),δ2=δ2∑xWith the help of  and 120120, we nTo prove this, we consider four cases depending on which term on the RHS of  achievesum, then  holds bee LHS of  achievese LHS of  follows e LHS of  because e LHS of  achievese LHS of  follows β3]ρ,so  holds alHaving established , we now We now study the terms on the RHS of  separatee RHS of ,(127)∑xρ>1,∑x,yP·ρ−1ρ≤∑x,yP1ρx,y)]ρ˜; –90) and andρ>1,x,y)]ρ˜; –82). In. Inρ>1,x,y)]ρ˜; :(135)∑xFrom , 127), , 127), , 135), , 135), aFinally, , (99), . Then, –8) hold hold(RX,n, there exist encoders Using Theorem 3 with n tends to infinity, the RHS of (n tends to infinity, which implies that the rate pair Because e RHS of  tends to


We propose a new approach for the solution of initial value problems for integrable evolution equations in the periodic setting based on the unified transform. Using the nonlinear Schrödinger equation as a model example, we show that the solution of the initial value problem on the circle can be expressed in terms of the solution of a Riemann–Hilbert problem whose formulation involves quantities which are defined in terms of the initial data alone. Our approach provides an effective solution of the problem on the circle which is conceptually analogous to the solution of the problem on the line via the inverse scattering transform. In §In §(c)k-plane will be denoted by Dj, j = 1, …, 4, and Σ will denote the contour f on a contour from the left and right will be denoted by f+ and f−, respectively. We will use A will be denoted by [A]1 and [A]2, respectively.The four open quadrants of the complex 2.Before turning to the periodic problem, we recall some aspects of the analysis of the NLS equation on a finite interval presented in , which wμ is a 2 × 2-matrix-valued eigenfunction, the matrices Q and q of  is a smooth solution of  ∈  × , where 0 < L > ∞ and 0 < T < ∞ is some fixed final time. Following  × . The solution q can be obtained from m via the relationMoreover, m obeys the symmetriesP of poles is empty, then uniqueness follows by standard considerations because the jump matrix has unit determinant. The problem with a non-empty set P can be transformed into a problem for which P is empty. Indeed, if the set η and ξ can be regularized in the standard way; see, e.g. 1 = A[μ2]1/α has at least a simple zero at kj and it then follows from (m]2 has (at most) a simple pole at kj. This proves  as k → 0. The behaviour of m as k → ∞ is a consequence of   m and thmmetries  also impuence of . Finallyuence of  follows uence of . ▪4.m depends on the final time T via the function Γ = B/A. However, the solution q is independent of T. This suggests that it should be possible to eliminate the T dependence from the RH problem 3.3. In this section, we define a new T-independent RH problem—henceforth called the RH problem for m. The basic idea of this deformation is to replace Γ by a new T-independent function a(k) and b(k) alone, this will lead to our main result.The RH problem 3.3 for (a)q of the NLS equation on the interval  whose boundary values have decay as t → ∞. In this case, it can be shown that the functions A, B, T → ∞; we denote these limits by T-independent function x-periodic initial value problem.As motivation for the definition of the RH problem for relation  leads toWe first need to give a careful definition of  root in  is fully(b)k) is given byWe define the function The function fined in  is relat2:k) is the trace of the so-called monodromy matrix and features heavily in the classical approach to the x-periodic problem. In that framework, 1 and is well studied. In what follows, we collect some well-known facts about In order to make the definition  of Γ~ pree, e.g. , and defq0 ≡ 0, we have a = 1, thus Δ(k) = 2cos (kL), and hence 4 − Δ2 = 4sin 2(kL) has a double zero at each of the points nπ/L, 2 has a similar structure for large k for any initial datum q0 ∈ L2 in the following sense: For π/(4L) centred at nπ/L. Then, there is an integer N > 0 such that, counted with multiplicity, 4 − Δ2 has exactly two roots in each disk n| > N and exactly 4N + 2 roots in the disk {|k| < Nπ/L + π/(4L)}. Using this result and employing the lexicographic ordering of complex numbersn|. The symmetry λ = 1 and λ = −1 separately.In the case of vanishing initial datum λ = 1), the periodic spectrum nth spectral gap whenever it is non-empty. For λ = 1, we define nth interval n|.In the defocusing case (i.e. λ = −1), z1, z2) denote the open straight-line segment from z1 to z2, i.e.N > 0 be as in the Counting Lemma so that there are 4N + 2 roots counted with multiplicity in the disk {|k| < Nπ/L + π/(4L)}. An even number, say 2M, of these 4N + 2 roots have odd multiplicity; let kL) + o(1) as k → ±∞, an even number, say k| < Nπ/L + π/(4L)} ordered lexicographically:k = constant. We define n| > N:In the focusing case 2 in  is singl ∞ in R,  implies hat (see  for a mofined by  with the(c)g(k) bygj denotes the restriction of g to Dj for j = 1, …, 4. We introduce m is the solution of RH problem 3.3. The function g is defined in such a way that v except that Γ is replaced by v on λ = −1, then j = 1, …, 4. It is quite remarkable that all these jumps can be expressed completely in terms of a and b alone.Define the function pression  as v exc2 has roots of even multiplicity. Let Let k approaches where m, in general, has singularities at the poles of Γ, η and ξ, it turns out that Whereas the function We assume the following:—has no poles on the contour—In D1, has at most finitely many polesand these poles are all simple.—In D3, has at most finitely many polesand these poles are all simple.k0 is a zero of k tends to infinity in k0, we see that k0 to the same order as k near k0. It follows, in particular, that Regarding assumption 4.2, we note that there exist large families of initial conditions for which it can be shown explicitly that uence of ). Thus, Let ——m as k approaches The limits of —k → ∞, π/(4L) centred at nπ/L defined in  are nowhere zero.In order to show that finition  of the mThe following identities are valid:In particular, the functionis non-zero for allk, then Δ2 = Δ2 − 4 at k, which is a contradiction. Thus The identities follow by a direct computation using the definitions  and 4.44.4 of Γ~(d)q of the x-periodic NLS equation in terms of the solution of the RH problem 4.4. Since the formulation of this RH problem only involves quantities defined in terms of the initial datum, the theorem provides an effective solution of the IVP for the x-periodic NLS equation.The following theorem, which is the main result of the paper, provides an expression for the solution Suppose q is a smooth solution of (1.1) forwhich is x-periodic of period L > 0, i.e. q = q. Define a(k) and b(k) by (2.3) and letbe the function defined in terms of a and b by (4.3). Suppose assumption 4.2 holds.Then the RH problem 4.4 has a unique solutionfor each  ∈  ×  × [0, ∞). Choose T ∈  and define m and g defined using T as final time. We will show that x, t).Fix –(3.12) to show that μj as its starting point.Let us show that μj as follows:In light of the symmetries , it is einitions  and  as k → ∞ provided that we can show thatg, it is enough to prove for j = 1, 2 thatk → ∞ in the closed upper half-plane It only remains to show that symmetry  of g, itstimates  imply4.a = 1 + O(k−1) and k → ∞ in The estimates  also impombining  and (4.3g1. As k → ∞ in O(k−1) and g1. As for the non-zero off-diagonal elementB/A givesx ≤ L and T > t, this yieldsj = 1.Let us consider her with , this yishowing  for the relation  for Γ = ent (cf. )4−Δ2−4λ∈ZDn.By , 2.7), , g1. As ∈ZDn.By , 4.26)  g1. As k∈ZDn.By , we havetilizing , 4.28)  g1. As kproof of  for j = g2. We haveO(k−1) as x ≥ 0 and T > t, we conclude thatWe next consider nfinity,  givesa−over, by , a−λbΓ~¯ng as in , we deduproof of .g − I is exponentially small as k → ∞ along any ray Finally, by , 4.35),,4.35), aequation  then folequation . This co5.N is an integer and the constant q0 > 0 can be taken to be positive due to the phase invariance of N be an integer. Direct integration of the x-part of the Lax pair (q given by (a and b:r(k) denotes the square roota, b and Δ are entire functions of k even though r(k) has a branch cut. The periodic spectrum 2 = 4sin 2(Lr) and consists of the two simple zeros λ± defined byλ = 1, then all zeros are real; if λ = −1, then the zeros are real for |n| ≥ L q0/π and non-real for |n| < L q0/π. The function r so that r(k) = k + πN/L + O(k−1) as k → ∞. Then, using  as k approaches (a)λ = 1, thenIf 5.4) and .(b)λ = −1, thenIf 5.4) and .The limits of lows see :(a)If λ—k → ∞, —k → {0, − πN/L, λ±}, Find a 2 × 2-matrix valued function It is easy to verify that the jump matrices in RH problem 5.1 satisfy the following consistency conditions at the origin:(b)The RH problem 5.1 for λ−, λ+). Let us orient the contourλ = 1 and upward if λ = −1. It follows that k → ∞, and that f(k) ≡ f is defined byThe jump matrices ng (5.4)– show thak) by performing another transformation. Define δ(k) ≡ δ byδ satisfies the jump relation δ+δ− = f on  and lim k→∞δ(k) = δ∞, whereδ(k) = O((k − λ±)1/4) and δ(k)−1 = O((k − λ±)−1/4) as k → λ±. Consequently, k → ∞, (iii) k → λ±, and (iv) Q ∼ 1 as k → ∞. Since The jump matrix in  can be mq for all t. Indeed, . This givesλ = 1, then the substitutions s = −πN/L + σ and σ = q0sinθ giveλ = −1, then the substitutions s = −πN/L + iσ and σ = q0sinθ yieldδ∞ into  of  = q0 eπN/L) x of  correspoλ−, λ+). More generally, whenever the Riemann surface defined by We have shown that the single exponential solutions  can be colutions  is the g


Scientific Reports 10.1038/s41598-019-39832-9, published online 01 March 2019Correction to: θ = π/4’ should read ‘θ = π/2’.This Article contains an error in the legend of Figure 5, where ‘


Scientific Reports 10.1038/s41598-020-69659-8, published online 29 July 2020Correction to: This Article contains errors in Table 1.In the column, “Company”,“IRE EliT ”should read:“IRE EliT ”In the column, “Column matrix”,“Unspecified”should read:2”“TiO68Ga elution yield (%)”,In the column, “Initial “ > 65”should read:“ > 80”As a result, in the Introduction,–3% 68Ge breakthrough18. According to its brochure, metal content per elution is less than 1 ppm and ≤ 10 µg/GBq of 68Ga for Fe, Cu, Ni, Zn, Pb and Al. The Eckert & Ziegler GalliaPharm® and the IRE ELiT Galli Eo® generators are both GMP grade and have type II drug master files on file with the FDA.”“Another emerging generator is Galli Eo , however, the column contains an unspecified resin. The generator is eluted with 0.1 M HCl, and has more than 67% elution yield and less than 1 × 10should read:2. The generator is eluted with 0.1 M HCl, and has more than 67% elution yield and less than 1 × 10–3% 68Ge breakthrough18. According to its brochure, metal content per elution is less than 1 ppm and ≤ 10 µg/GBq of 68Ga for Fe, Cu, Ni, Zn, Pb and Al. The Eckert & Ziegler GalliaPharm® and the IRE ELiT Galli Eo® generators are both GMP grade and have type II drug master files on file with the FDA. Recently, the IRE ELiT generator has been authorized in several European countries  and its initial elution yield has been improved .”“Another emerging generator is Galli Eo  which is based on TiO


Scientific Reports 10.1038/s41598-019-55074-1, published online 11 December 2019Correction to: This Article contains a repeated typographical error where the unit “µm” is incorrectly given as “m” or “um” and “°” is incorrectly given as “deg”.


Scientific Reports 10.1038/s41598-019-42604-0, published online 17 April 2019.Correction to: The Article contains errors in Table 2, where the units used for components ‘EGF’ and ‘FGF’,“μg/m”should read:“ng/ml”Additionally, the units used for component ‘NRG-1′,“ng.ml”should read:“ng/ml”In the same table, the ‘Human colon epithelia’ entry for WCM,“—”should read:“0.5×”In addition, the ‘Human colon carcinoma’ entry for WCM,“0.5×”should read:“—”


Correction to: BMC Infect Dishttps://doi.org/10.1186/s12879-019-4668-xIn the Abstract:ng/L vs. 0.250 ng/L, p = 0.005)’ should be replaced with ‘testosterone ’‘testosterone ’ should be replaced with ‘testosterone ’‘testosterone :‘μg/mL’ should be replaced with ‘μg/dL’unit of DHEA-S in Table 4 (4th row):μg/mL’ should be replaced with ‘μg/dL’‘After publication of the original article , we were


Scientific Reports 10.1038/s41598-019-53693-2, published online 11 December 2019Correction to: This Article contains a typographical error in the Results and Discussion section under subheading ‘Experimental analysis of the optical properties’ where,“{L}_{w}  =  2.4 nm”should read:w = 2.4 nm”“L


Scientific Reports 10.1038/s41598-019-54688-9, published online 24 December 2019Correction to: This Article contains errors in Table 4.Under column ESR age (ka) in scenario 1,“1330±13”should read:“1330±134”Under column ESR age (ka) in scenario 2“128±95”should read:“1285±95”(c)” where,Finally, there is an error in the heading “% of total dose rate(c)”“% of total dose rateshould read:(b)”“% of total dose rate


Scientific Reports 10.1038/s41598-019-56215-2, published online 23 December 2019Correction to: This Article contains errors in the Methods sections under subheading ‘MRI protocol’.3, resolution = 125 × 125 × 372 µm3, TE/TR = 22/425 ms, gradient pulse duration/diffusion time (δ/Δ) = 4/12 ms, b-values = 2000 and 4000 s/mm2, 20 diffusion directions, number of averaged excitations = 2, and scan time = 7 hours and 40 minutes for each b-value.”“Images were acquired with a three-dimensional diffusion-weighted spin-echo sequence and the following parameters: field-of-view = 60 × 60 × 160 mmshould read:3 (adjusted to fit sample), resolution = 125 × 125 × 372 µm3, TE/TR = 22/425 ms, gradient pulse duration/diffusion time (δ/Δ) = 4/12 ms, b-values = 2000 and 4000 s/mm2, 20 diffusion directions, number of averaged excitations = 2, and scan time ≈ 7-10 hours for each b-value (depending on the field-of-view).”“Images were acquired with a three-dimensional diffusion-weighted spin-echo sequence and the following parameters: field-of-view ≈ 6 × 6 × 16 mm


Integral proteins can reach the plasma membrane via different routes. Here, Lucken-Ardjomande Häsler et al. identify three proteins that are associated with dynamic intracellular tubules, closely aligned with the ER, and involved in the transport of specific cargos whose export is particularly sensitive to ER stress. In addition to the classical pathway of secretion, some transmembrane proteins reach the plasma membrane through alternative routes. Several proteins transit through endosomes and are exported in a Rab8-, Rab10-, and/or Rab11-dependent manner. GRAFs are membrane-binding proteins associated with tubules and vesicles. We found extensive colocalization of GRAF1b/2 with Rab8a/b and partial with Rab10. We identified MICAL1 and WDR44 as direct GRAF-binding partners. MICAL1 links GRAF1b/2 to Rab8a/b and Rab10, and WDR44 binds Rab11. Endogenous WDR44 labels a subset of tubular endosomes, which are closely aligned with the ER via binding to VAPA/B. With its BAR domain, GRAF2 can tubulate membranes, and in its absence WDR44 tubules are not observed. We show that GRAF2 and WDR44 are essential for the export of neosynthesized E-cadherin, MMP14, and CFTR ΔF508, three proteins whose exocytosis is sensitive to ER stress. Overexpression of dominant negative mutants of GRAF1/2, WDR44, and MICAL1 also interferes with it, facilitating future studies of Rab8/10/11–dependent exocytic pathways of central importance in biology. In eukaryotic cells, the ER is the birthplace of the majority of membrane proteins, secreted proteins, and lipids. Despite the canonical belief that they follow the same route from the ER through the ER-Golgi intermediate compartment (ERGIC), Golgi, and TGN to reach the plasma membrane, differences between individual cargos exist. Lipids can be transferred between membranes at contact sites . Some prRabs are regulators of intracellular transport whose GTP-GDP cycle drives membrane trafficking processes forward. Rab8 controls the export of MMP14 , Rab11 mDrosophila plasmatocytes  can participate both in endocytic and exocytic routes. On the endocytic side, OPHN1 regulates clathrin- and Endophilin-dependent endocytosis in neuronal cells , while Gatocytes . Conversatocytes  and in catocytes ; GRAF1c atocytes ; GRAF1 aatocytes ; and GRAatocytes .Little is known about the molecular mechanisms governing GRAF1/2–mediated pathways as among the direct binding partners identified , which contains the PPKPP motif  was reported to inhibit Rab11-mediated recycling of the Transferrin receptor . In agreWe have shown that WDR44 and MICAL1 bind to the SH3 domain of GRAF1/2. But even though the membrane-binding regions of GRAF1/2 (GRAF1/2 BAR-PH) did not colocalize with endogenous MICAL1, endogenous WDR44 was found on the same structures . Live imMICAL1 G3W, WDR44 ΔC, GRAF2 BAR-PH, and GRAF1 BAR-PH are thus four dominant negative proteins affecting the normal functioning of different components of Rab8/10/11– and MICAL1/GRAF/WDR44–mediated trafficking.To gain more insight into the identity of MICAL1/GRAF/WDR44 tubules, we first examined colocalization of endogenous WDR44 with dextran as a marker of clathrin-independent endocytosis. There was none . EndogenIn transfected HeLa cells, WDR44 tubules and patches colocalized with VAPA and VAPB . WDR44 tBy similarly with WDR44, GRAF1b/2 tubules colocalized with VAPA and VAPB . ColocalTo identify the compartments with which WDR44 tubules communicate, we used BFA, a drug that leads to the tubulation of certain organelles and the mixing of groups of intracellular compartments . IncubatpF1KA0819, and overexpression of MICAL-L1 inhibited GRAF1b/2 tubulation , a G protein–coupled receptor that was reported to reach the plasma membrane in a Rab1-independent and Rab8-dependent manner -anchored protein GFP-GPI was not inhibited by overexpression of Rab8a T22N, Rab10 T23N, MICAL1 G3W, GRAF1/2 BAR-PH, or WDR44 ΔC  and with the fact that we did not see endogenous MICAL1 tubules under resting conditions. Alternatively, other proteins might substitute for MICAL1, although we have shown that none of the other MICAL family members interact with GRAF1b/2. Contrasting with our original expectation, but in agreement with others (2B-AR and of a GPI-anchored protein are unaffected by incubation of the cells with thapsigargin. As inhibition of ER to Golgi transport invariably leads to ER stress (2)–generating enzyme PIPKIγ, something that was reported to be abolished by ER stress  a route inhibited by ER stress followed by E-cadherin, MMP14, CFTR, and CFTR ΔF508 that may or may not use COPII-coated vesicles and at some stage depends on Rab8, Rab10, GRAF2, and WDR44; and (3) a stress-induced pathway followed by CFTR ΔF508 that bypasses the Golgi and is also dependent on Rab10.In this study, we report the identification of three novel proteins—GRAF2, WDR44, and MICAL1—participating in the export of a subset of neosynthesized proteins. E-cadherin, MMP14, CFTR, and CFTR ΔF508 were previously found to reach the plasma membrane in a Rab8- and/or Rab11-dependent manner . We now h others , we founR stress , we now R stress . NeverthIn recent years, proteins of the GRAF family have attracted attention because they are mutated in cancer  and haveOverexpression of Rab8 is known to induce a decrease in F-actin and focal adhesions in a RhoA-dependent manner . We now 2–enriched microdomains of the plasma membrane ; pCI GRAF2res ΔBAR-GFP was recombined from pDONR GRAF2res ΔBAR C-tag ; pCI GRAF1b-GFP, pCI GRAF1b-RFP, and pCI GRAF1b-myc were recombined from pDONR GRAF1b C-tag (made from a PCR of a human brain cDNA library using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCGAAGGAGATAGAACCATGGGGCTCCCAGCGCTCGAG-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCGAGGAACTCCACGTAATTCTCAGGG-3′); pGex 4T2 GRAF1 SH3 was recombined from pDONR GRAF1 SH3 N-tag ; pCI GRAF1 BAR-PH–GFP and pCI GRAF1 BAR-PH–RFP were recombined from pDONR GRAF1 BAR-PH C-tag ; pCI myc-WDR44, pCI GFP-WDR44, and pCI RFP-WDR44 were recombined from pDONR WDR44 N-tag (made from a PCR of IMAGE 4837674 using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCGCGTCGGAAAGCGACACC-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCCCTAAGATACATTTTTTCTTTTATTAACAAACACTTTG-3′); pCI WDR44-GFP and pCI WDR44-RFP were recombined from pDONR WDR44 C-tag (made from a PCR of IMAGE 4837674 using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCGAAGGAGATAGAACCATGGCGTCGGAAAGCGACACC-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCAGATACATTTTTTCTTTTATTAACAAACACTTTG-3′); pCI GFP-WDR44 ΔFFAT was recombined from pDONR WDR44 ΔFFAT N-tag ; pCI GFP-WDR44 ΔN1 was recombined from pDONR WDR44 ΔN1 N-tag ; pCI GFP-WDR44 ΔN2 and pGex 4T2 WDR44 ΔN2 were recombined from pDONR WDR44 ΔN2 N-tag ; pCI GFP-WDR44 ΔC, pCI myc-WDR44 ΔC, pCI RFP-WDR44 ΔC, and pGex 4T2 WDR44 ΔC were recombined from pDONR WDR44 ΔC N-tag ; pCI GFP-WDR44 ΔPro was recombined from pDONR WDR44 ΔPro N-tag (deletion of aa 210–257 of WDR44 by mutagenesis of pDONR WDR44 N-tag using 5′-GGCCACTTCTTCCACAGCGG-3′ and 5′-AGAAAAAGGAAAAGCGAATTGGAATTTG-3′); pCI RFP-WDR44res was recombined from pDONR WDR44res N-tag (mutagenesis of pDONR WDR44 N-tag using 5′-GTGGCTTTGTGGAACGAGGTAGACGGTCAGACAAAATTGATC-3′ and 5′-GATCAATTTTGTCTGACCGTCTACCTCGTTCCACAAAGCCAC-3′); pCI RFP-WDR44res ΔPro was recombined from pDONR WDR44res ΔPro N-tag (mutagenesis of pDONR WDR44 ΔPro N-tag using 5′-GTGGCTTTGTGGAACGAGGTAGACGGTCAGACAAAATTGATC-3′ and 5′-GATCAATTTTGTCTGACCGTCTACCTCGTTCCACAAAGCCAC-3′); pCI myc-MICAL1, pCI GFP-MICAL1, pCI RFP-MICAL1, and pGex 4T2 MICAL1 were recombined from pDONR MICAL1 N-tag (made from a PCR of IMAGE 5756097 using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCGCTTCACCTACCTCCACCAACC-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCCCTAGCCCTGGGCCCCTGTCC-3′); pCI MICAL1-GFP was recombined from pDONR MICAL1 C-tag (made from a PCR of IMAGE 5756097 using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCGAAGGAGATAGAACCATGGCTTCACCTACCTCCACCAACC-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCGCCCTGGGCCCCTGTCC-3′); pCI untagged MICAL1 (uMICAL1) was recombined from pDONR MICAL1 (made from a PCR of pDONR MICAL1 N-tag using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCGAAGGAGATAGAACCATGGCTTCACCTACCTCCACCAACC-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCCCTAGCCCTGGGCCCCTGTCC-3′); pGex 4T2 MICAL1 ΔN was recombined from pDONR MICAL1 ΔN N-tag ; pCI GFP-MICAL1 ΔMO was recombined from pDONR MICAL1 ΔMO N-tag ; pCI GFP-MICAL1 ΔMOΔCH and pGex 4T2 MICAL1 ΔMOΔCH were recombined from pDONR MICAL1 ΔMOΔCH N-tag ; pCI myc-MICAL1tail was recombined from pDONR MICAL1tail N-tag ; pCI myc-MICAL1Pro was recombined from pDONR MICAL1Pro N-tag ; pCI GFP-MICAL1 G3W, pCI RFP-MICAL1 G3W, and pCI myc-MICAL1 G3W were recombined from pDONR MICAL1 G3W N-tag (MICAL1 G91W G93W G96W by mutagenesis of pDONR MICAL1 N-tag using 5′-CAAGTGCCTGGTGGTGTGGGCTTGGCCTTGCTGGCTGCGGGTCGCTGTGG-3′ and 5′-CCACAGCGACCCGCAGCCAGCAAGGCCAAGCCCACACCACCAGGCACTTG-3′); pCI uMICAL1 G3W was recombined from pDONR MICAL1 G3W (mutagenesis of pDONR MICAL1 using 5′-CAAGTGCCTGGTGGTGTGGGCTTGGCCTTGCTGGCTGCGGGTCGCTGTGG-3′ and 5′-CCACAGCGACCCGCAGCCAGCAAGGCCAAGCCCACACCACCAGGCACTTG-3′); pCI myc-MICAL1 PPAPP, pCI GFP-MICAL1 PPAPP, and pCI RFP-MICAL1 PPAPP were recombined from pDONR MICAL1 PPAPP N-tag (MICAL1 K832A by mutagenesis of pDONR MICAL1 N-tag using 5′-GGAGCCTCCACCCGCGCCTCCCCGCAGC-3′ and 5′-GCTGCGGGGAGGCGCGGGTGGAGGCTCC-3′); pCI RFP-Rab8a, pCI GFP-Rab8a, and pCI myc-Rab8a were recombined from pDONR Rab8a N-tag (made from a PCR of pEGFP cRab8 using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCGCGAAGACCTACGATTACCTG-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCCCTACAGAAGAACACATCGGAAAAAGC-3′); pCI RFP-Rab8a T22N and pCI GFP-Rab8a T22N were recombined from pDONR Rab8a T22N N-tag (mutagenesis of pDONR Rab8a N-tag using 5′-GGGGGTGGGGAAGAACTGTGTCCTGTTCC-3′ and 5′-GGAACAGGACACAGTTCTTCCCCACCCCC-3′); pCI RFP-Rab8a Q67L and pCI GFP-Rab8a Q67L were recombined from pDONR Rab8a Q67L N-tag (made from a PCR of pEGFP cRab8 Q67L using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCGCGAAGACCTACGATTACCTG-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCCCTACAGAAGAACACATCGGAAAAAGC-3′); pCI RFP-Rab8b and pCI GFP-Rab8b were recombined from pDONR Rab8b N-tag (made from a PCR of IMAGE 4701429 using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCGCGAAGACGTACGATTATCTC-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCCCTAAAGTAGCGAGCAACGAAAG-3′); pCI RFP-Rab10 and pCI GFP-Rab10 were recombined from pDONR Rab10 N-tag (made from a PCR of a 293T cDNA library using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCGCGAAGAAGACGTACGACCTGC-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCCCTAGCAGCATTTGCTCTTCCAGCC-3′); pCI GFP-Rab10 T23N and pCI RFP-Rab10 T23N were recombined from pDONR Rab10 T23N N-tag (mutagenesis of pDONR Rab10 N-tag using 5′-GGAGTGGGGAAGAACTGCGTCCTTTTTC-3′ and 5′-GGAATCCCCGATCAGGAGCAGC-3′); pCI GFP-Rab10 Q68L was recombined from pDONR Rab10 Q68L N-tag (mutagenesis of pDONR Rab10 N-tag using 5′-GGGATACAGCAGGCCTGGAGCGATTTCAC-3′ and 5′-ATATCTGTAGCTTGATCTTCTTTCC-3′); pCI GFP-Rab11a and pCI RFP-Rab11a were recombined from pDONR Rab11a N-tag (PCR of IMAGE 3510339 using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCGGCACCCGCGACGACGAG-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCCCTAGATGTTCTGACAGCACTGCACC-3′); pCI GFP-Rab11a S25N, and pCI RFP-Rab11a S25N were recombined from pDONR Rab11a S25N N-tag (mutagenesis of pDONR Rab11a N-tag using 5′-GAGATTCTGGTGTTGGAAAGAATAATCTCCTGTCTCG-3′ and 5′-CGAGACAGGAGATTATTCTTTCCAACACCAGAATCTC-3′); pCI GFP-Rab11a Q70L was recombined from pDONR Rab11a Q70L N-tag (mutagenesis of pDONR Rab11a N-tag using 5′-GATATGGGACACAGCAGGGCTAGAGCGATATCGAGC-3′ and 5′-GCTCGATATCGCTCTAGCCCTGCTGTGTCCCATATC-3′); pCI GFP-Rab1 S25N was recombined from pDONR Rab1 S25N N-tag ] using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCTCCAGCATGAATCCCGAATATG-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCCCTAGCAGCAACCTCCACCTGAC-3′); pCI GFP-Rab1 and pCI RFP-Rab1 were recombined from pDONR Rab1 N-tag (mutagenesis of pDONR Rab1 S25N N-tag using 5′-GCGACTCTGGGGTTGGAAAGAGTTGCCTCCTTCTTAGG-3′ and 5′-CCTAAGAAGGAGGCAACTCTTTCCAACCCCAGAGTCGC-3′); pCI GFP-Rab1 Q70L was recombined from pDONR Rab1 Q70L N-tag (mutagenesis of pDONR Rab1 N-tag using 5′-CAAATATGGGACACAGCAGGCCTAGAAAGATTTCG-3′ and 5′-CGAAATCTTTCTAGGCCTGCTGTGTCCCATATTTG-3′); pCI myc-VAPA, pCI RFP-VAPA, and pCI GFP-VAPA were recombined from pDONR VAPA N-tag (made from a PCR of IMAGE 2822547 using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCGCGTCCGCCTCAGGGG-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCCCTACAAGATGAATTTCCCTAGAAAGAATCC-3′); pCI GFP-VAPA ΔCC was recombined from pDONR VAPA ΔCC N-tag (deletion of aa 151–225 of VAPA by mutagenesis of pDONR VAPA N-tag using 5′-CAGTGGAACAGCTTTGCTAGG-3′ and 5′-CTTCCTTCACTTCTTGTTGTAATTGC-3′); pCI GFP-VAPA ΔMSP was recombined from pDONR VAPA ΔMSP N-tag ; pCI GFP-VAPA ΔTM was recombined from pDONR VAPA ΔTM N-tag ; pCI GFP-VAPA DD was recombined from pDONR VAPA DD N-tag (VAPA K94D M96D by mutagenesis of pDONR VAPA N-tag using 5′-GAGTAAACACGACTTTGACGTACAGACAATTTTTGCTCCACCA-3′ and 5′-TGGTGGAGCAAAAATTGTCTGTACGTCAAAGTCGTGTTTACTC-3′); pCI GFP-VAPB and pCI myc-VAPB were recombined from pDONR VAPB N-tag (made from a PCR of IMAGE 3543354 using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCGCGAAGGTGGAGCAGG-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCCCTACAAGGCAATCTTCCCAATAATTAC-3′); pCI GFP-MICAL2 was recombined from pDONR MICAL2 N-tag (made from a PCR of IMAGE 5275364 using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCGGGGAAAACGAGGATGAGAAG-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCCCTAGCCAAGAAGTGGGTGTAGC-3′); pCI GFP-MICAL3 was recombined from pDONR MICAL3 N-tag (made from a PCR of IMAGE 100000333 using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCGAGGAGAGGAAGCATGAGACC-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCCCTAGGGCCGTGTGGCGGG-3′); pCI GFP-MICAL-L1 was recombined from pDONR MICAL-L1 N-tag (made from a PCR of IMAGE 100015878 using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCGCTGGGCCGCGGGG-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCCCTAGCTCTTGTCTCTGGGGGAC-3′); pCI GFP–MICAL-L2 was recombined from pDONR MICAL-L2 N-tag (made from a PCR of IMAGE 5521653 using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCGCGGCCATCAGGGCG-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCCCTACTGGGAGGGGCTGCTTTTGC-3′); pCI E-cadherin–GFP and pCI E-cadherin–RFP were recombined from pDONR E-cadherin C-tag (made from a PCR of a human kidney cDNA library using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCGAAGGAGATAGAACCATGGGCCCTTGGAGCCGC-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCGTCGTCCTCGCCGCCTCCG-3′); pCI Occludin-GFP was recombined from pDONR Occludin C-tag (made from a PCR of a human kidney cDNA library using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCGAAGGAGATAGAACCATGTCATCCAGGCCTCTTGAAAG-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCTGTTTTCTGTCTATCATAGTCTCCAACC-3′); pCI MMP14-GFP was recombined from pDONR MMP14 C-tag (made from a PCR of a human kidney cDNA library using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCGAAGGAGATAGAACCATGTCTCCCGCCCCAAG-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCGACCTTGTCCAGCAGGG-3′); HA–α2B-AR–GFP and HA–α2B-AR–RFP were recombined from pDONR HA–α2B-AR C-tag (made from a PCR on IMAGE 9020527 using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCGAAGGAGATAGAACCATGTACCCATACGATGTTCCAGATTACGCTGACCACCAGGACCCCTAC-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCCCAGGCCGTCTGGGTCCAC-3′); pDONR Sar1 N-tag was made from a PCR on IMAGE 712455 using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCTCTTTCATCTTTGAGTGGATCTAC-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCCCTAGTCAATATACTGGGAAAGCCAGC-3′); pCI myc-Sar1 H79G, pCI GFP-Sar1 H79G, and pCI RFP-Sar1 H79G were recombined from pDONR Sar1 H79G N-tag (mutagenesis of pDONR Sar1 N-tag using 5′-CTTTTGATCTTGGTGGGGGCGAGCAAGCACGTC-3′ and 5′-GACGTGCTTGCTCGCCCCCACCAAGATCAAAAG-3′); pCI Calnexin-GFP was recombined from pDONR Calnexin C-tag ; pCI GFP-STX16 was recombined from pDONR STX16 N-tag (made from a PCR of a 293T cDNA library using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCGCCACCAGGCGTTTAACCG-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCCCTATCGAGACTTCACGCCAACG-3′); pCI GFP-VAMP3 and pCI RFP-VAMP3 were recombined from pDONR VAMP3 N-tag (made from a PCR on IMAGE 3544610 using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCTCTACAGGTCCAACTGCTGC-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCCCTATGAAGAGACAACCCACACG-3′); pCI TGN46-RFP was recombined from pDONR TGN46 C-tag (made from a PCR on IMAGE 4272702 using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCGAAGGAGATAGAACCATGCGGTTCGTAGTTGCC-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCGGACTTCTGGTCCAAACGTTGGTA-3′); pCI GFP-EHD1 was recombined from pDONR EHD1 N-tag (made from a PCR on pEGFP C3 mEHD1 using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCTTCAGCTGGGTGAGCAAGG-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCCCTACTCGTGCCTCCGTTTGGAG-3′); and pCI GFP-EHD3 was recombined from pDONR EHD3 N-tag (made from a PCR on SKB-LNB hEHD3 using 5′-GGGGACAAGTTTGTACAAAAAAGCAGGCTTCTTCAGCTGGCTGGGTACGG-3′ and 5′-GGGGACCACTTTGTACAAGAAAGCTGGGTCCCTACTCGGCAACTTTCCTCTTGGACG-3′).In general, cDNAs were amplified from cDNA libraries, IMAGE clones, or plasmids and were integrated in the entry vector pDONR201 for cloning using the Gateway recombination system (Thermo Fisher Scientific). For expression in mammalian cells, the entry clones were recombined with modified pCI (Promega) vectors, engineered to be Gateway compatible and to express proteins with a GFP (EGFP protein), RFP (tagRFP protein), or myc tag in N-terminal or C-terminal position. The empty destination (DEST) vectors pCI N-GFP DEST and pCI N-RFP DEST were used for expression of soluble EGFP and tagRFP, respectively. For expression of GST-tagged proteins in bacteria, entry clones were recombined with a modified pGex 4T2 vector  engineered to be Gateway compatible. Mutagenesis was performed by PCR amplification of parent clones, digestion of the template with DpnI, ligation, and transformation. Apart from Rab8a and Rab1, which were canine, EHD1 which was from mouse, and Sar1 which was from rat, all the other constructs were human. pCI GRAF2-GFP and pCI GRAF2-RFP were recombined from pDONR GRAF2 C-tag  using the primers 5′-GTACAAGTCCGGACTCAGATCTCGAGCTCAGAGGTCGCCTCTGGAAAAGGCC-3′ and 5′-CCGGTGGATCCCGGGCCCGCGGTACCCTAAAGCCTTGTATCTTGCACCTCTTCTTCTGTCTCCTCTTT-3′ and inserted into pEGFP-C1 (Takara). The plasmids Str-KDEL_SBP-EGFP-Ecadherin, Str-KDEL_SBP-EGFP-GPI, and Str-Ii_VSVG-SBP-EGFP were gifts from Franck Perez . VSVG was excised with XmaI and EcoRI and replaced with human E-cadherin using primers 5′-CTTGCCACAACCCGGGAGGCGCGCCATGGGCCCTTGGAGCCGC-3′ and 5′-CTCGTCCATGGAATTCAGGTCGTCCTCGCCGCCTCC-3′ and the In-Fusion system (Takara Bio) to make Str-Ii_E-cadherin–SBP-EGFP. Str-Ii_VSVG-SBP–Fusion Red was made after excision of EGFP from Str-Ii_VSVG-SBP-EGFP using SbfI and XbaI and insertion of Fusion Red using primers 5′-TCAACGTGAACCACCTGCAGGTATGGTGAGCGAGCTGATTAAGG-3′ and 5′-TGATCAGTTATCTAGAGTTTCATTTACCTCCATCACCAGCGC-3′ and the In-Fusion system.pF1KA0819) was a gift from Anna Akhmanova ; HA-GLUT4-GFP was a gift from Sam Cushman ; pEGFP-wtCFTR, and pEGFP-ΔF508-CFTR were gifts from Ineke Braakman .GFP-GPI (GPI anchoring region of hCD58 fused to GFP) was a gift from Ben Nichols ; GFP-MICAL3FL . Corresponding oligonucleotides were annealed, phosphorylated, and ligated into pLKO.1-puro (Sigma), which had been predigested with BglII and HindIII. For each gene, at least three different shRNAs were cloned and tested. The best ones were used: shControl (5′-ccggCGTACGCGGAATACTTCGAttcaagagaTCGAAGTATTCCGCGTACGtttttg-3′ and 5′-aattcaaaaaCGTACGCGGAATACTTCGAtctcttgaaTCGAAGTATTCCGCGTACG-3′); shGRAF2a (5′-ccggGCACAGATCTGGAGGGAAAttcaagagaTTTCCCTCCAGATCTGTGCtttttg-3′ and 5′-aattcaaaaaGCACAGATCTGGAGGGAAAtctcttgaaTTTCCCTCCAGATCTGTGC-3′); shGRAF2b (5′-ccggCGTTGAAACACGAGGTATAttcaagagaTATACCTCGTGTTTCAACGtttttg-3′ and 5′-aattcaaaaaCGTTGAAACACGAGGTATAtctcttgaaTATACCTCGTGTTTCAACG-3′); shGRAF1 (5′-ccggAGGAAGTCCAAGAGAGAAAttcaagagaTTTCTCTCTTGGACTTCCttttttg-3′ and 5′-aattcaaaaaaGGAAGTCCAAGAGAGAAAtctcttgaaTTTCTCTCTTGGACTTCCT-3′); shWDR44a (5′-ccggGGAATGAAGTAGATGGTCAttcaagagaTGACCATCTACTTCATTCCtttttg-3′ and 5′-aattcaaaaaGGAATGAAGTAGATGGTCAtctcttgaaTGACCATCTACTTCATTCC-3′); shWDR44b (5′-ccggTGATGAAACCTGTGAGAAAttcaagagaTTTCTCACAGGTTTCATCAtttttg-3′ and 5′-aattcaaaaaTGATGAAACCTGTGAGAAATCTCTTGAATTTCTCACAGGTTTCATCA-3′); shMICAL1 (5′-ccggGCATTGATCTGGAGAACATttcaagagaATGTTCTCCAGATCAATGCtttttg-3′ and 5′-aattcaaaaaGCATTGATCTGGAGAACATtctcttgaaATGTTCTCCAGATCAATGC-3′); shRab8a1 (5′-ccggGAACAAGTGTGATGTGAATttcaagagaATTCACATCACACTTGTTCtttttg-3′ and 5′-aattcaaaaaGAACAAGTGTGATGTGAATtctcttgaaATTCACATCACACTTGTTC-3′) cotransfected with shRab8a2 (5′-ccggGAGAATTAAACTGCAGATAttcaagagaTATCTGCAGTTTAATTCTCtttttg-3′ and 5′-aattcaaaaaGAGAATTAAACTGCAGATAtctcttgaaTATCTGCAGTTTAATTCTC-3′); shRab10 (5′-ccggGAATAGACTTCAAGATCAAttcaagagaTTGATCTTGAAGTCTATTCtttttg-3′ and 5′-aattcaaaaaGAATAGACTTCAAGATCAAtctcttgaaTTGATCTTGAAGTCTATTC-3′); shRab11a (5′-ccggAGTTTAATCTGGAAAGCAAttcaagagaTTGCTTTCCAGATTAAACTtttttg-3′ and 5′-aattcaaaaaAGTTTAATCTGGAAAGCAAtctcttgaaTTGCTTTCCAGATTAAACT-3′); shVAPA (5′-ccggCCACAGACCTCAAATTCAAttcaagagaTTGAATTTGAGGTCTGTGGtttttg-3′ and 5′-aattcaaaaaCCACAGACCTCAAATTCAAtctcttgaaTTGAATTTGAGGTCTGTGG-3′); and shVAPB (5′-ccggCGGAAGACCTTATGGATTCttcaagagagaATCCATAAGGTCTTCCGtttttg-3′ and 5′-aattcaaaaaCGGAAGACCTTATGGATTCtctcttgaaGAATCCATAAGGTCTTCCG-3′).For lentivirus production, pMD2.G  and PSPAX2  were used. Both were gifts from Didier Trono .All constructs were verified by sequencing.Purified rabbit polyclonal α-GRAF1 and α-GRAF2 antibodies were described previously , as wellThe secondary reagents for Western blotting were goat α−mouse and α−rabbit IgG-HRP conjugates (Bio-Rad). HRP-Protein A (Thermo Fisher Scientific) was used for analysis of beads when antibodies used for the immunoprecipitation and for Western blotting were raised in the same species.The secondary antibodies used for immunofluorescence were Alexa Fluor 488, 514, 546, or 647 conjugates of goat α−mouse, α−rabbit, α−rat, and α−sheep IgG antibodies .The following reagents were used: Nocodazole, Cytochalasin D, BFA, D-biotin, methanol, and DMSO (Sigma); Alexa Fluor 546–Transferrin, Alexa Fluor 647–Transferrin, Alexa Fluor 546–dextran 10.000 MW fixable, and Alexa Fluor 546–phalloidin (Thermo Fisher Scientific); Puromycin and C4 ; VX-809 ; and thapsigargin (Santa Cruz Biotechnology).HeLa (ECACC 93021013), 293T , HT 1080 , U-87 MG (ECACC 89081402), COS-7 (ECACC 87021302), undifferentiated 3T3 L1 cells (ECACC 86052701), and BSC1 (ECACC 85011422) cells were grown in DMEM-GlutaMAX supplemented with 10% FBS. NIH 3T3 cells (ECACC 93061524) were grown in DMEM-GlutaMAX with 10% newborn calf serum. SH-SY5Y cells (ECACC 94030304) were grown in a 1:1 mixture of MEM and Ham’s F-12 supplemented with GlutaMAX, 1% nonessential amino acids, and 15% FBS. hTERT-RPE1 cells  were grown in a 1:1 mixture of DMEM and Ham’s F-12 with GlutaMAX, 0.25% sodium bicarbonate, and 10 µg/ml hygromycin. Unless otherwise stated, cell culture products were from Thermo Fisher Scientific. Cells were routinely checked for mycoplasma contamination. When export of MMP14 was tested, HT 1080 cells were grown on dishes coated with collagen from calf skin (Sigma). For biotinylation experiments using 293T cells, they were grown on dishes coated with Poly-L-Lysine (Sigma). HeLa, 293T, and HT 1080 cells were transfected with Genejuice (Merck Millipore) or polyethylenimine (PEI) Max Linear 40 kD (Generon). For PEI transfection of 35-mm dishes, in general, 1 µg of each DNA was diluted in 100 µl OptiMEM, then complexed with 3 µg PEI prediluted in 100 µl OptiMEM. The mix was vortexed, incubated at RT for 15–20 min, and added dropwise to cells. For colocalization purposes, cells were generally examined 16–24 h after transfection. For all transfections of CFTR ΔF508, C4 (5 µM) and VX-809 (5 µM) were added 12 h after transfection.res-RFP, pCI RFP-WDR44res, or pCI RFP–WDR44res ΔPro. Cells were split, transfected after 52–56 h, and collected 36 h later. For knockdown of protein expression in HT 1080 cells, cells were infected with lentivirus, selected 24 h later with puromycin , and split. Cells were transfected 72 h after the original infection with pCI MMP14-GFP, alone or together with pCI RFP-WDR44res or pCI RFP–WDR44res ΔPro, and analyzed 16 h later. To make lentivirus, 293T cells were grown on Poly-L-Lysine–coated plates and cotransfected with shRNA, pMD2.G, and PSPAX2. Medium was changed to DMEM supplemented with 4% FBS and 25 mM Hepes after 24 h, and lentivirus was harvested as tissue culture supernatant after a further 48–60 h.For knockdown of protein expression in HeLa cells, cells were transfected with an shRNA-encoding plasmid, selected 24 h later with puromycin , and then split. Experiments were performed after 88–96 h of knockdown. If shRNA cells were to be transfected, transfection was performed 72 h after the transfection of the shRNA. For knockdown of protein expression in 293T cells, cells were cotransfected with an shRNA-encoding plasmid and pCI N-RFP DEST, pCI GRAF2Cells grown on glass coverslips were fixed in 3.2% PFA diluted in culture medium . For immunostaining of cytoplasmic proteins, coverslips were then washed in PBS and blocked in 5% normal goat serum (NGS) and 0.1% saponin (Sigma) in PBS. All further incubations were done in 1% NGS and 0.1% saponin in PBS. Coverslips were mounted on slides in a buffered PVA glycerol mountant containing 2.5% DABCO (Sigma). For staining of F-actin, Alexa Fluor 546–labeled phalloidin was added to the primary antibody dilution . A similar protocol was followed for immunostaining of surface proteins under nonpermeabilizing conditions, but saponin was omitted from the buffers.2 95:5 and 100% humidity.For live imaging, cells were grown on glass-bottom culture dishes (MatTek). Immediately before imaging, medium was replaced with phenol red–free DMEM containing 5% FBS and placed in a temperature-controlled chamber on the microscope stage with air/COMost imaging data were acquired using a fully motorized inverted microscope  equipped with a CSU-X1 spinning disk confocal head  using a 60× oil immersion lens  under control of Volocity 6.0 (PerkinElmer). 14-bit digital images were acquired with a cooled EMCCD camera . Three 50-mW solid-state lasers  coupled to individual acoustic-optical tunable filters (AOTF) were used as a light source to excite fluorescent proteins and dyes, as appropriate. Imaging of RUSH proteins was performed with a Zeiss LSM 780 using standard photomultiplier tubes, a 63× oil immersion lens of NA 1.4, and ZEN acquisition software.ImageJ 1.48s  was used to process the images, adjusting only the brightness and contrast of the different channels. Due to the high dynamic range of the signals and the fact that we were particularly interested in the dim structures in the cell’s periphery, we adjusted the scaling of the images to bring this information into the dynamic range of a computer display. No filters were applied. We followed the principles of Color Universal Design. On the merged channel, images acquired in the red channel are colored in red; images acquired in the green channel are colored in cyan; and images acquired in the far red channel are colored in magenta. Representative images of live cells and fixed cells collected in a minimum of three independent experiments are shown. Unless otherwise mentioned, images are from a single focal plane. For confocal stacks, maximum intensity projections of images acquired at 0.3-µm intervals, in single channels and as a merged image, are shown.For quantification of the proportion of HeLa cells with RFP–Rab8a/10 tubules, cells were fixed 20 h after transfection. The number of cells with at least one tubule was counted and expressed as percentage of the total number of cells examined (minimum 25 cells per sample per experiment). For quantification of the total skeletal length of RFP–Rab8a/10 structures and for quantification of the total volume of GRAF1b/2–GFP structures per cell, cells were fixed 20 h after transfection. Confocal stacks were acquired. Each cell was identified as a region of interest. Volocity was used to identify fluorescent objects and measure their combined skeletal length or volume. Manders colocalization coefficients were quantified using Volocity, setting thresholds for each cell using a region of cytosol. For quantification of the proportion of HeLa cells with endogenous WDR44 tubules, cells were fixed 24 h (overexpression) or 96 h (shRNA) after transfection. The number of cells with at least one WDR44 tubule was counted and expressed as a percentage of the total number of cells examined (minimum 100 cells per sample per experiment). For quantification of the total WDR44 length per cell, confocal stacks were acquired. Each cell was identified as a region of interest. WDR44 objects with a nonspherical shape (shape factor <0.5) were identified using the Volocity software, and their skeletal length was measured. The total skeletal length per cell was quantified. For displaying fluorescence intensity profiles, fluorescence intensities were measured on a 1-px line using ImageJ software. Values were then normalized using the minimum and maximum values measured along that line. For detailed numbers of independent experiments, see the figure legends.For super-resolution imaging, cells were plated on high-precision coverslips . Coverslips were processed as for all other immunofluorescence experiments, but antibodies were used at a higher concentration: mouse α-myc (1:100), rabbit α-Calnexin (1:200), Alexa Fluor 488 conjugated α-mouse (1:100), and Alexa Fluor 514 α-rabbit (1:100). Coverslips were mounted in Prolong Diamond Antifade Mountant  curing for more than 48 h to achieve the best possible refractive index for STED microscopy.The microscope was a Leica TCS SP8 STED 3X equipped with a pulsed white light laser and HyD hybrid detectors enabling gating. Gating was adjusted to detect signal emitting between 1.7 and 6.8 ns after the excitation pulse. Excitation of Alexa Fluor 488 was performed at 488 nm and of Alexa Fluor 514 at 535 nm. Suitable clean-up filters were in place. The depletion laser was at 592 nm.The primary lens was a 100×/1.46 NA , chromatically corrected across the used spectrum. Sampling frequency was set to 20 nm, and the depletion laser was operated at settings that allowed a lateral resolution below 60 nm. As the photon budget was limited, line averaging (16×) and frame accumulation (8×) were used to achieve a suitable signal-to-noise ratio. ImageJ was used to process the images and to measure fluorescence intensity profiles across tubules.Escherichia coli cells (Bioline), and protein expression was induced with isopropyl-β-D-thiogalactoside . Cells were resuspended in bacterial lysis buffer  and lysed by freezing/thawing and sonication. Lysates were cleared by centrifugation  and incubated with glutathione sepharose beads for 45 min at 4°C under rotation. Beads were pelleted and washed three times in bacterial lysis buffer. Purified recombinant proteins were analyzed by electrophoresis on 4%–12% NuPAGE Novex Bis-Tris gels (Thermo Fisher Scientific) and Coomassie staining.pGex 4T2, pGex 4T2 GRAF1 SH3, pGex 4T2 GRAF2 SH3, pGex 4T2 WDR44 ΔC, pGex 4T2 WDR44 ΔN2, pGex 4T2 MICAL1, pGex 4T2 MICAL1 ΔN, and pGex 4T2 MICAL1 ΔMOΔCH were transformed in BL21(DE3) pLysS g), and protein concentration of the supernatants was determined with a Bradford assay (Bio-Rad). Equivalent protein amounts of GST, GST-GRAF1 SH3, or GST-GRAF2 SH3 bound to glutathione sepharose beads were incubated with a volume of lysate corresponding to ∼7 mg protein or with the same volume of cell lysis buffer for 30 min at 4°C under rotation. Beads were pelleted  and washed five times in cell lysis buffer without protease inhibitors. For identification of bound proteins by Western blot, beads were directly incubated with sample buffer, heated at 95°C, and loaded on gels. For identification of bound proteins by mass spectrometry, beads were incubated with Thrombin . Eluted proteins were collected in the supernatant after centrifugation , sample buffer was added, and proteins were separated using 4%–12% NuPAGE Novex Bis-Tris gels. Protein bands were revealed by Coomassie staining. Bands were excised, and proteins were identified by liquid chromatography-tandem mass spectrometry . Results were analyzed with Scaffold 3 (Proteome Software).For pull-downs of cell lysates, beads were washed three times in cell lysis buffer . In parallel, 15-cm dishes of HeLa cells or rat brains were lysed in the same buffer . Lysates were cleared by centrifugation  and collected in the supernatant after centrifugation. WDR44 and MICAL1 domains bound to glutathione sepharose beads were then incubated in the presence or absence of an equal amount of GRAF2 SH3 for 30 min at 4°C and washed three times in bacterial lysis buffer. Proteins bound to the beads were separated using 4%–12% NuPAGE Novex Bis-Tris gels and revealed by Coomassie staining.g), and protein concentration was determined using a Bradford assay. Rat brains were homogenized in the same buffer using a Dounce homogenizer, but lysates were cleared at 100,000 g, 30 min. When lysates were to be directly analyzed, sample buffer was added to equal protein amounts (75 µg protein per sample); in general, for immunoprecipitation of endogenous proteins, 150 µg protein was incubated with purified antibodies or serum in a total volume of 100 µl and incubated 3 h at 4°C under rotation. 12.5 µl of a 50% suspension of Protein G Sepharose 4 Fast Flow beads  was added for another 30 min before beads were spun down , washed three times in lysis buffer, and finally resuspended in sample buffer. Samples were heated 3 min at 95°C, loaded on 4%–12% NuPAGE Novex Bis-Tris gels, and analyzed by Western blot. Proteins were transferred to polyvinylidene fluoride membranes; powder milk was used as blocking agent. Secondary antibodies were detected using Amersham ECL detection reagent , and results were developed on x-ray film. The same protocol was followed for the coimmunoprecipitation of endogenous proteins from SH-SY5Y cells but using a volume of lysate corresponding to 100 cm2 of a confluent layer of cells for each immunoprecipitation.Untreated cells or shRNA-expressing cells (96 h after transfection) were scraped in cold PBS and pelleted. They were resuspended in lysis buffer  and incubated 30 min at 4°C. Lysates were cleared . The pellets were resuspended in the same volume of lysis buffer and cleared at low speed , and a volume equivalent to 125 cm2 of a confluent layer of cells was used for each immunoprecipitation.For coimmunoprecipitation of endogenous proteins from HeLa cells, cells were resuspended in Triton X-100–containing lysis buffer  and incubated 30 min at 4°C. Lysates were centrifuged . 10 µl of each supernatant was collected, and 5 µl sample buffer was added (lysates). The rest of the supernatants were incubated with 12.5 µl of a 50% suspension of Protein G Sepharose 4 Fast Flow beads for 30 min at 4°C under rotation. Beads were then spun down  and washed three times with 500 µl lysis buffer. Beads were resuspended in 10 µl sample buffer. Beads and lysates were heated 3 min at 95°C, loaded on 4%–12% NuPAGE Novex Bis-Tris gels, and analyzed in parallel by Western blot. Membranes from immunoprecipitations done with rabbit α-GFP antibody were first probed with mouse α-myc antibody. Corresponding beads were then probed with mouse α-GFP antibody while lysates were probed with rabbit α-GFP antibody. Membranes from immunoprecipitations done with mouse α-myc antibody were first probed with rabbit α-GFP antibody. Corresponding beads were then probed with rabbit α-myc antibody while lysates were probed with mouse α-myc antibody. For coimmunoprecipitation of uGRAF2 with rabbit α-GFP, uGRAF2 was detected in the beads with rabbit α-GRAF2 and HRP–Protein A. In general, the membranes corresponding to the beads were chosen to show the level of expression of the baits, but when the background was too strong, the lysates were shown. For immunoprecipitations using α-myc–coated beads, a similar procedure was followed, but 100-mm dishes were used. Cells were resuspended in 200 µl lysis buffer, 15 µl of each lysate was loaded per well, and 5 µl of α-myc–coated magnetic beads  were used per sample. In all cases, results displayed are representative of a minimum of three independent experiments.On the first day, 293T cells were plated in six-well dishes and a few hours later were transfected with the plasmid coding for the prey. The next day, cells were transfected with plasmids encoding the baits. 20–24 h later, the medium was removed, and cells were detached by pipetting in 1 ml cold PBS. Cells were spun down, and pellets were resuspended in 100 µl lysis buffer  and incubated on ice for 30 min. Lysates were cleared  and resuspended in 1% NGS in PBS containing DyLight 650 α-HA antibody (5 µg/ml) or unconjugated α-HA (1 µg/ml) and incubated for 90 min at RT. Cells were spun down, washed three times in 1% NGS in PBS, and resuspended in Alexa Fluor 647–conjugated α-mouse antibody in 1% NGS in PBS. Cells were incubated for another hour, washed three times, fixed once more in 3.2% PFA, and resuspended in PBS. Cells were analyzed by FACS using an LSRFortessa analyzer (BD Biosciences), and results were analyzed with FlowLogic 7.2.1 (Inivai Technologies Pty. Ltd.). After comparison with undifferentiated cells, differentiated cells were selected based on their light-scattering properties. GFP or GFP/RFP–positive cells were selected. Background was subtracted from the GFP and Alexa Fluor 647 fluorescence intensities. The ratio of the geometric mean of the Alexa Fluor 647 fluorescence intensity to the geometric mean of the GFP fluorescence intensity was calculated and expressed as a ratio to the value obtained for the corresponding insulin-treated control. This was used as a measurement of HA-GLUT4-GFP export efficiency. For all cotransfections of nonfluorescent proteins, coexpression of myc-tagged proteins or of MICAL1 was verified in parallel following a similar protocol but using buffers containing 0.1% saponin and α-myc (2 µg/ml) or α-MICAL1 (1.9 µg/ml). 50–1,000 differentiated cells were analyzed in each independent experiment.To induce differentiation, 3T3 L1 cells were grown to confluence in DMEM-GlutaMAX with 10% newborn calf serum. They were then fed with DMEM-GlutaMAX with 10% FBS, 0.5 mM 3-isobutyl-1-methylxanthine (IBMX), 0.25 µM dexamethasone, 1 µg/ml bovine insulin , and 5 µM troglitazone . 2–3 d later, medium was replaced with the above without IBMX for another 2–3 d. Cells were then maintained in DMEM-GlutaMAX with 10% FBS for 2–3 d. HA-GLUT4-GFP was used to examine export of GLUT4 . For trag) and incubated in blocking solution (5% NGS in PBS) for 30 min at RT. All further incubations and washes were done in PBS with 1% NGS. Cells were pelleted and incubated with unconjugated α-Integrin-β1 antibody (5 µg/ml) for 90 min and washed three times. They were then incubated with Alexa Fluor 546–conjugated α-mouse antibody (2 µg/ml) for 1 h, washed three times, and fixed once more in 3.2% PFA before being resuspended in PBS. Cells were analyzed by FACS using an LSRFortessa analyzer, and results were analyzed with FlowLogic 7.2.1. Within each experiment, the geometric means of the Alexa Fluor 647 and the Alexa Fluor 546 intensities were normalized by the values obtained for the shControl uptake sample. For each shRNA, the ratio of the normalized intensity of internalized Integrin-β1 (Alexa Fluor 647) to the one of surface Integrin-β1 (Alexa Fluor 546) was calculated as a measurement of the internalized Integrin-β1 ratio. Any defect in degradation of the internalized Integrin-β1 antibody or in the recycling of the receptor should lead to a higher internalized Integrin-β1 ratio. Within each experiment, between 1,000 and 30,000 cells were analyzed in each sample.shRNA-expressing HeLa cells (96 h after transfection) were incubated for 1 h with Alexa Fluor 647–conjugated α-Integrin-β1 antibody (1 µg/ml) at 37°C. Cells were then washed in PBS, acid stripped  for 45 s , and eit2B-AR–GFP, HA–α2B-AR–RFP, GFP-GPI, SBP-GFP–E-cadherin, SBP-GFP-GPI, E-cadherin–SBP-GFP, and VSVG-SBP-GFP in HeLa cells, of MMP14-GFP in HT 1080 cells, and of GFP–Extope CFTR and GFP–Extope CFTR ΔF508 in 293T cells. HeLa and HT 1080 cells were grown in 35-mm dishes. Unless otherwise specified in the figure legend, they were delicately scraped in 1 ml culture medium 16 h after transfection. 293T cells were grown in 12-well dishes and collected 36 h after transfection by pipetting. In all cases, cells were fixed in PFA . Following fixation, all further incubations and centrifugations were done at RT. Cells were spun down , resuspended in blocking solution (5% NGS in PBS), and incubated 30 min. Cells were then pelleted and resuspended in 50 µl of buffer (1% NGS in PBS) containing the appropriate primary antibody  and incubated for 90 min under gentle rocking. Cells were washed three times in 250 µl wash buffer (0.1% BSA in PBS). When unlabeled primary antibodies were used, cells were then incubated in a dilution of Alexa Fluor 647–labeled secondary antibody (2 µg/ml) in buffer (1% NGS in PBS) and incubated for 1 h under gentle rocking; cells were then washed three times in 250 µl wash buffer. After the final washes, cells were fixed once more in 3.2% PFA in PBS (20 min), washed, and resuspended in PBS. Cells were analyzed by FACS using an LSRFortessa analyser, and results were analyzed with FlowLogic 7.2.1. For single transfections, GFP-positive or RFP-positive cells were selected; for cotransfections, GFP- and RFP-positive cells were selected. Background was subtracted from the GFP, RFP, and Alexa Fluor 647 fluorescence intensities. The ratio of the geometric mean of the Alexa Fluor 647 fluorescence intensity to the geometric mean of the GFP or RFP (depending on the tag of the export candidate) fluorescence intensity was calculated and expressed as a ratio to the value obtained for the control. This was used as a measurement of protein export efficiency. Within each experiment, between 1,000 and 20,000 cells were analyzed in each sample.This assay was used to quantify the export of E-cadherin–GFP, E-cadherin–RFP, HA–αg), and 5 µl of each supernatant was removed (lysate). 10 µl of a 50% suspension of Neutravidin agarose beads (Thermo Fisher Scientific) was added to each sample, and they were then incubated 30 min at 4°C under rotation. Beads were pelleted , washed three times in lysis buffer, and resuspended in sample buffer. Lysates and beads were heated 3 min at 95°C and analyzed by Western blot. For quantification of protein export efficiency, band intensities in the beads and in the lysates were measured in triplicate using ImageJ software; background was removed. Within each experiment, the intensity of the band in the beads was normalized by the corresponding value in the lysates and expressed as a ratio to the chosen reference.Cells were delicately washed three times in cold PBS. Fresh EZ-Link Sulfo-NHS-LC-Biotin  was added to each well, and cells were gently rocked for 30 min at 4°C. Reaction was stopped by adding 1 ml of 100 mM glycine + 0.1% BSA in PBS. Cells were scraped, pelleted, washed delicately in Tris buffer , and lysed  for 30 min on ice. Lysates were cleared  is detailed in the figure legends. In all cases, significant adjusted P values are represented as *, P < 0.05; **, P < 0.01; ***, P < 0.001; and ****, P < 0.0001.Statistical data analysis was performed using Prism version 8.0.0 for Mac, GraphPad Software. For all quantifications provided, the means and SEM are shown. Unless otherwise stated, statistical data analysis was performed with one-way ANOVA of unmatched data. Gaussian distribution was assumed. Homoscedasticity was tested using a Brown-Forsythe test. The number of independent experiments (n = 24 for GFP-transfected cells. In n = 24 for shControl-transfected cells. In t tests between shGRAF2a cells cotransfected with RFP and RFP-GRAF2res and between shWDR44a cells cotransfected with RFP and RFP-WDR44res or RFP–WDR44res ΔPro were used. In t test between untreated and BFA-treated GFP-transfected cells and Dunnett’s multiple comparisons tests between BFA-treated transfected cells using BFA-treated GFP-transfected cells as control were used.In


The gender of rats used for the experiment was wrongly mentioned as “female.” This should be corrected as “male.”μg/ml” should be corrected to “20.93 ± 6.98 μg/ml.”In Figure 4 legend, the value of serum uric acid in normal rats “21.93 ± 6.98 In the article titled “Excretory Function of Intestinal Tract Enhanced in Kidney Impaired Rats Caused by Adenine” , there wThe authors would like to apologize for any inconvenience caused.


A theory of cultural structures predicts the objects observed by anthropologists. We here define those which use kinship relationships to define systems. A finite structure we call a partially defined quasigroup  on a dictionary  allows prediction of certain anthropological descriptions, using homomorphisms of pdqs onto finite groups. A viable history (defined using pdqs) states how an individual in a population following such history may perform culturally allowed associations, which allows a viable history to continue to survive. The vector states on sets of viable histories identify demographic observables on descent sequences. Paths of vector states on sets of viable histories may determine which histories can exist empirically. The structures described here and their consequences imply much of what may be predicted about empirical cultures. Anthropologists very often draw illustrations of structures using methods discussed here, but based on intuition, thus have little notion of what their commonly used diagrams might predict. While ,2 defineOur notion of studying viable minimal structures—which are the smallest minimal cultural structures that can “reproduce” the ascribed social relations in one generation—follows from . Our hiss = 4. The minimal structure is not intended to illustrate the actual empirical relations of each empirical generation of individuals, instead it shows “how the rule operates”—it describes the minimal representation of the kinship and marriage rules  and let * be a partially defined binary operation on D, such that when x, y ∈ D:(1) If there exists an a ∈ D such that a*x and a*y are defined and a*x = b and a*y = b then x = y, we call such object  a partially defined quasigroup, or pdq.(2) If  is a pdq and * is fully defined on D, then  is a (complete) quasigroup.(3) The pair L =  is a natural language with dictionary D whenever  is a pdq.(4) If L =  is a natural language, a kinship terminology is a quasigroup subset k ⊆ L.Definition 2.Let X, Y and Z be non-empty finite sets and let ,  and  be quasigroups with binary relations *, ° and ▪ respectively. Then:(1) A function f: X → Y is a homomorphism if f for all b, c ∈ X, f(b * c) = f(b) ° f(c).(2) If f: X → Z and g: Y → Z are homomorphisms then f and g are isotopic.k of a quasigroup (a kinship terminology k) onto itself forms a group (the symmetric group on k). If ,  and  are complete pdqs then (isotopic) homomorphisms classify kinship terminologies by the form of the pdq onto which they are mapped. For example, the isotopic terminologies classified as Dravidian /2δy = 2 − 2αχn where y = v(t). When sα ≠ sχ and at least one of αs, χs is > 3, αχn > 2, then exp[r(t)] is concave down; so, r(t) is also concave down. ☐To show Proof 3.H. Since sn increases as s increases, then Lemma 1 part 1 shows that the largest value of r(t) will be set by that two-history subset α, χ of H having the largest difference between their structural numbers, hence the largest αχn in Equation (3). Combining the sn of the largest s with any other combination of sn values will result in a smaller αχn hence smaller er(t), from Equation (3). ☐There is a finite number of two-history pairs in Observation 1.H acting on a finite non-empty descent sequence G. Let α, β, χ ∈ tH ⊆ H act on generation Gt ∈ G. See Assume a finite non-empty set of viable histories np = 2, so includes the modal demography of each history in H; that is, the modal demography for each of α, β and χ each appear on the line np = 2, since spsn = 2. If for any α, αv(t) = 1, then n(t) = αn, p(t) = αp, so ret)( = ½n(t)p(t) = ½αpαn = ½2 = 1 so r(t) = 0; this occurs if all histories in tH have the same structural number . When that does not occur we have a set of two or more histories in tH each with 0 < αv(t) < 1 and thus r(t) > 0; see also Lemma 1. When H is full, assume α, β and χ have distinct structural numbers, at least two of α, β, χ have structural numbers >3, and α is has the lowest structural number of those three histories. Then the modal demography of α, β, χ have  ≠  ≠ , and the computation of r(t) appears in the values in the triangle area of tH is not full then events in the triangle area might not occur. Even if paths allow α with β, α with χ and β with χ (thus the boundaries of the triangle area), values of r(t) within the triangle only occur if all three histories are allowed by H, which may be prohibited by not-full H. We call such area an un-accessed region. Thus, we study change using both full and non-full sets of histories. The curved bottom-line in Definition 6.α, pα) be the modal demography for history α ∈ H. Let G be a finite non-empty descent sequence using H, and let Gt ∈ G be the generation at time t. Let Ht be a face of H specified at t. Let St := { sα|α ∈ Hi} be the set of structural numbers St ⊆ S of histories Ht available at t. Let At := {|sα ∈ Si, α ∈ Hi,  = } be the set of modal demographies At of the histories in Ht; and let v(t) be the vector state of Gt. List the histories in H in a defined order from α to χ. Then for all α ∈ Ht and all  ∈ At, let:Let H be a finite non-empty set of histories and let  be a row vector;(n2. t) :=  be a column vector;|p3. for all α, χt|pt) as a square matrix then for all α, χ∈ H, arranging the sum of the inner product  at t;∈ H, H(t): = [n4. T as V(t) := [vαχ(t)] is a probability picture  of the vector state of a descent sequence at t;the square matrix we get by arranging the products of v(t)v(t)5. αχ(t + ε) − vαχ(t)] := [Δαχ(t)]. (Notice that −1 ≤ Δαχ(t) ≤ 1)for ε ≥ 0 let V(Δ(t)) := V(t + ε) − V(t) =  at all entriesLet H be a finite non-empty set of full viable histories. Let G be a non-trivial descent sequence using H, let H. Proof of Lemma 2.iΣjvi(t)jv(t)ipjn. From the definition of modal demography, ip = 2/in and jp = 2/jn. The values on the diagonal of ½H(t) are for each history α, ½αpαn = 1. We thus examine the off-diagonal products αpχn and χpαn. Then ipjn = 2in/jn and jpin = 2jn/in. Thus, 2ni/jn = 2jn/in occurs only if in = jn, in which case ipjn = jpin = 2. This occurs only if all histories i and j have the same structural number or both have s = 2 or 3, and thus ½H(t) = [1] in all entries. Otherwise stated, in this case the value from Equation (3) is αχn = 2. ☐Assume the premises. Equations (4)–(7) simply rearrange terms in ½ΣH we can compute a proposed population growth rate r(t).αn = χn occurs if structural numbers αs, χs are <4 or whenever αs = χs; so r(t) = 0. Otherwise, then Lemma 2 implies Lemma 1, which says that ret) > 0. This occurs since αpχn does not equal χpαn; thus from Lemma 1 and  where r11 = 1 + r3, r22 = 1 − r3, r21 = r1 + ir2, r12 = r1 − ir2. That isLet r:z0, z1, z2, z3, be complex numbers such that R = z01 + z1Σ1 + z2Σ2 + z3Σ3 where:Following  1, Σ2, and Σ3 are the standard Pauli spin matrices, where for R then z0 = 1, z1 = r1, z2 = r2 and z3 = r3. Note that −1 ≤ r1, r2, r3 ≤ 1. From Definition 8.t ∈ H, let Gt ∈ G be a generation of G using a face Ht ∈ H at t, and let v(t) be the vector state of GtLet H be a finite non-empty set of viable histories, let G be a non-trivial viable descent sequence using histories H. 2tH = {α, χ} t⊆ H be a two-history subset of tH. Let R(t) = ½[ijr(t)] be a projection, let r1(t), r2(t), and r3(t) be real numbers such that r1(t)2+r2(t)2 + r3(t)2 = 1, such that 0 ≤ r1(t) < 1, 0 ≤ r2(t) < 1, and such that αv(t) = ½r1(t) = ½(1 + r3(t)). Then R(t) is the status of tG.Let unit circle C is meant a set of points  in the plane R2 which satisfy the equation x2 + y2 = 1.A Theorem 1.t, and let v(t) be the vector state of Ht. Let 2Ht = {α, χ} ⊆ Ht be a non-empty subset of Ht. Let t2 > t1 > t0 define a path of vα(t) from t = t0 to t = t2 such that vα(t0) = 1 changes monotonically to vα(t1) = 0 and then monotonically back to vα(t2) = 1. That is, let r3 move from r3(t0) = 1 to r3(t1) = − 1 and then back to r3(t2) = 1. Let O(t) = (n(t), p(t), r(t)). Then:Assume the premises of Definition 8. Let H be a finite non-empty set of viable histories having structural numbers s < 152 (see [(1) trR(t) = 1;(2) χ(t) = ½(1 − r3(t));v(3) 2Ht is given by the main diagonal of R(t);the vector state v(t) of (4) 3 = 0r(t) is a maximum when r. Theorem 2.1(t) = 0. Then: (i) R(t) has ΣΣijrij(t) = 1; and (ii) the sum Σ∫r(t)dv(t) = 0 when summed over all paths  of for pairs 2HtLet r.Theorem 3.2(t) = 0. Then Σ∫r(t)dv(t) = 0 when summed over all paths  of all pairs 2HtLet r.Proof of Theorem 1.2tH = {α, χ} ⊆ tH is the vector state v(t) = (αv(t), χv(t)) where χv(t) = 1 − αv(t). R(t) is a status and since in a status αv(t) = ½r11 = ½(1 + r3), and since αv(t) + χv(t) = 1 in a two-history state, then χv(t) = ½r22 = ½(1 − r3). In addition, also then αv(t) + χv(t) = ½(1 + r3) + ½(1 − r3) = 1 = trR(t), which establishes Theorems 1, 2, and 3. Establishing 4: We find r(t) is a maximum when r3 = 0, given Theorem 1(1) and 1(3), and Lemma 1(2), so when r3 = 0 then v(t) = . Assume the premises of Theorem 1. In a two history system, r1(t) = 0 soiΣjrij(t) = ½2 = 1 which establishes 1(1). Let tH = {α, χ}. At time t, tH picks a set of modal demographies At = {, } and v(t) acts as a linear operator on At; so we getLet O(t) = (n(t), p(t), e(t)) are the predicted results at t; when αs ≠ χs then  ≠ . R(t) is an idempotent Hermitian matrix per Definition 7, and under the premises has r1 = 0. Then r22 + r32 = 1. We have a two history system with vector state v(t) = (αv(t), χv(t)) where χv(t) = 1 − αv(t), and where αv(t) = ½r11 = ½(1 + r3(t)). We let t2 > t1 > t0 define a path from t = t0 to t = t2 such that αv(t0) = 1 changes monotonically to αv(t1) = 0 and then monotonically again to αv(t2) = 1, which occurs as r3(t) moves monotonically from r3(t) = 1 to r3(t) = −1 and then back to r3(t) = 1. At each t, given r3(t), we compute r2(t)2 = 1 − r3(t)2. Then r2(t)2 + r3(t)2 = 1 traces a unit circle C. Theorems 2 and 3 then follow from Green’s theorem. □From Lemma 2, Observation 2.t ∈ G with the sub-populations Gt using the set of histories using face Ht ∈ H, and let v(t) be the vector state of Gt. Assume history α ∈ Ht, α ∈ Ht+1, and vα(t) = vα(t + 1) = 1. Then from t to t + 1, v(t) forms a loop. That is, the minimal descent sequence of any viable pure system α also forms a loop, indicated also since the minimal structure of α is a group. So any pure system is a loop. Diagrams like Let G be a population, G.sn, sp) pairs by the structural number of the identified history; or systems undergoing change in their culture. In that case the  pairs are changing and we can predict that rate of change yielding both the n(t) and p(t) for the given t, and the value of the adiabatic growth rate r(t) at t. An example of this prediction of rate of change in western Europe for about 1000 years from about AD 1000 to 1950 is given in [Following the examples of ,2 we hern(t), p(t) and r(t) of the society per generation. The methods of [Thus, study of the homotopy groups resulting from Definitions 5, 7 and 8 may thus tell us a lot about the possible paths of the empirical demography of cultures. Definitions 7 and 8 assume no physical model, but we can use their math to study the changes in vector states on histories on the predicted thods of ,29 and mthods of  in sociathods of , which ithods of , are quithods of , Postulathods of ,31 are aIn this paper, in , and in Our study also helps identify what cannot be predicted by this method. For example, sociologists and anthropologists use relationship studies to describe how individuals are “related”; the minimal structure defined here based on assignments made based on the “principles” used to arrange or avoid marriages, given the natural language and the history; they do not define which specific individuals are in fact assigned to each relation. In contrast, in genetic inbreeding experiments Sewall Wright  at diagrThe ability to derive population measures from the language-based statement of rules is something new to science, and should be explored. Many other things also affect population change, and are not explored here.


N = p2q where p and q are balanced large primes. Suppose e, ϕ(N)) = 1 where ϕ(N) = p(p − 1)(q − 1) and d < Nδ be its multiplicative inverse. From ed − kϕ(N) = 1, by utilizing the extended strategy of Jochemsz and May, our attack works when the primes share a known amount of Least Significant Bits(LSBs). This is achievable since we obtain the small roots of our specially constructed integer polynomial which leads to the factorization of N. More specifically we show that N can be factored when the bound N = p2q.This paper presents a cryptanalytic approach on the variants of the RSA which utilizes the modulus  Next, let e be a random integer such that gcd) = 1 where ϕ(N) = (p − 1)(q − 1) is the Euler totient function. Let d be its multiplicative inverse of e such that ed ≡ 1 mod ϕ(N). Let  be publicised for encryption purpose while p, q, ϕ(N), d are kept private. For decryption process, private parameter d is needed. The mathematical difficulty of the RSA cryptosystem relies on the hardness of solving the integer factorization problem on N = pq, solving the key equation ed − kϕ(N) = 1 and solving the RSA diophantine key equation that is, C ≡ Me mod N. Up until today, the RSA cryptosystem has remained secure.Secure communication up till the 70’s was executed through symmetrical ways. In other word, both of the encryption and decryption processes used the same key. Later in 1978, the first assymetric cryptosystem went public and solved the problematic issue of distributing keys. This cryptosystem used different keys to encrypt and decrypt the data. It is known as the RSA cryptosystem d could bN = prq for r ≥ 2 is utilized. This type of modulus provides advantage for both key generation and the decryption algorithms provided the Chinese Remainder Theorem is utilized  fo foω,n∈Z+Theorem 1 [Letbe the lattice generated by a set of basisand has the dimension ω. The reduced basisproduced by the LLL algorithm satisfiesfor all 1 ≤ i ≤ ω.heorem 1  LetLbeLLL algorithm has been extensively applied in order to find reduced basis vectors in a lattice. For instance, [LLL algorithm to find a reduced basis of the lattice generated by the modular polynomial, [Since its invention, nstance,  introducynomial,  managed ynomial,  describeTheorem 2 [Letbe a polynomial with at ω monomials. Suppose thatwhereandThenholds over integers.heorem 2  Leth. These results will be used to approximate the bound for one of the variables in our polynomial.The following results by  show an Lemma 1Let N = p2q with q < p < 2q. ThenLemma 2Let N = p2q with q < p < 2q. ThenN = p2q consists of two primes that share a known amount of their LSBs.The following lemma is reformulated from result . It consLemma 3Let N = p2q be the modulus and suppose that p − q = 2bu for a known value of b. Let p = 2bp1 + u0and q = 2bq1 + u0where u0is a solution to p3 ≡ N (mod 2b). Ifp2 + pq − p = 2b3s + s0 − v whereProof. Suppose that p − q = 2bu. Then q = p − 2bu andp3 ≡ N (mod 2b). Let u0 be a solution of the modular form p3 ≡ N (mod 2b). Then, p ≡ u0 (mod 2b) is a solution which implies that p = 2bp1 + u0 where p1 is a positive integer. Now we have,q1 = p1 − u. Using N = p2q, we getp3−2bup2.Hence, fu0, 2b3) = 1, we multiply (From  we deducN = p2q which works when there has a known amount of LSBs shared between the primes p and q.This section presents the attack on modulus Theorem 3Let N = p2q be modulus such that p − q = 2bu where 2b ≈ Nα. Let e be a public exponents satisfying e ≈ Nγand ed − kϕ(N) = 1. Suppose that d < Nδ. Then N can be factored in time polynomial ifProof. Suppose we have public exponent e and key equationp − q = 2bu. Then, from Lemma 3, p2 + pq − p can be rewritten in the form p2 + pq − p = 2b3s + s0 − v where u0 is a solution of the modular equation p3 ≡ N (mod 2b). Thus, substitutes  is a root of f and can be solved by using Coppersmith’s technique [e1, e2) = Nγ.max(d) < X1 = Nδ.max(p − q = 2bu with 2b ≈ Nα and p2 + pq − p = 2b3s + s0 − v with 2b3s − v < X3 = N2/3.stitutes  from Lempq−p))=1.Suppose s−v)−1=0.We transechnique  due to iS and M be defined as:fm−1 satisfiesM areW satisfiesa4 is coprime with R. We want to work with a polynomial that has constant term 1, thus we define g and h as:g and h with dimensionm = 3 and t = 1.The bounds of the variables are fixed as follows:1x2i2x3i3The monog and h and their combinations share the root  modulo R. A new basis with short vectors is produced after applying the LLL algorithm to the lattice i = 1, 2, let fi be two short vectors of the reduced basis. Each fi shares the roots . Then by Theorem 3, we havefi for i = 1, 2 to fulfill the condition ofNext, defineposed by , we forcω = |M| and |M| − |M∖S| = |S|, we gett = τm, then,m3, the inequality . By Assumption 1 in Section 2, the solution of the roots can be extracted using resultant technique. By using the third root 2b3s − v, we compute p2 + pq − p = 2b3s + s0 − v. This value is then used to find ϕ(N) and since ϕ(N) = p(p − 1)(q − 1) we can factor out p by taking the gcd). By knowing the value of p, we can factor the modulus N.Using 0)Using , we getjsj<W|S|.Using  = 1 where ϕ(N) = pr−1(pr − 1)(q − 1). Note that in these former attacks, their primes do not share any amount of LSBs. Their bounds depend only on the value of r. We compare the results with various values of γ = logN(e). Our corollary is as follow.We compare our bounds with these three former attacks,  and 19 tCorollary 1Let N = p2q be the modulus where q < p < 2q. Let e be a public exponent satisfying ed − kϕ(N) = 1 for ϕ(N) = p(p − 1)(q − 1). Suppose that d < Nδ. Then N can be factored in time polynomial ifδ of [r = 2. We describe their bound for d as in the Note that the bounds for δ of  and 19 rδ increases inversely proportional to the value of γ.Remark 1 In Corollary 1, it states that N can be factored ife = Nγ. We haveγ. A direct calculation shows that γ, this translates into d ≈ N.N = p2q where the primes share an amount of LSB(s). Based on the result of Nitaj et al. [p2 + pq − p which is the value of N − ϕ(N). We use the result from our lemma in our theorem which then yields a set of integer polynomials. By applying the extended strategy of Jochemsz and May, one is able to determine the small roots of our integer polynomial and thus factoring the modulus N. We show that N can be factored when d < Nδ for N = p2q.We describe an attack related to partial key exposure. Our attack works upon the modulus j et al. , we refoS1 Appendix(PDF)Click here for additional data file.


Heaney LG, Busby J, Hanratty CE, et al. Composite type-2 biomarker strategy versus a symptom–risk-based algorithm to adjust corticosteroid dose in patients with severe asthma: a multicentre, single-blind, parallel group, randomised controlled trial. Lancet Respir Med 2020; 9: 57–68—In this Article, in table 1, column “1”, the threshold ranges quoted should read “≥15–29”, “≥150–299”, and “≥45–54”, and in column “2”, the ranges should read “≥30”, “≥300”, and “≥55”. These corrections have been made to the online version as of Jan 4, 2021 and will be made to the printed version.