Makhlouf Hadji

Ming Chau

On Unicyclic Graphs Spectra: New Results

Keywords: Unicyclic graph, Laplacian eigenvalues, Algebraic connectivity, Circulant matrices

Let G = (V, E) be a unicyclic simple undirected graph. In this paper, we investigate the spectra of a particular class of unicyclic graphs G(q, n1) where q is the size of the unique cycle. Each vertex of the unique cycle is attached to n1 vertices. We provide the "exact values" of the extremal eigenvalues of the adjacency matrix A and the Laplacian matrix L of G, in contrast to lower and upper bounds reported in the literature.

I. INTRODUCTION

Given an undirected graph G = (V, E), in which V represents the set of vertices and E the set of edges, if G is connected with the property |V | = |E|, then G is said to be a unicyclic graph. In other words, if G is connected and has the same number of vertices and edges then G is unicyclic (i.e. contains one and unique cycle).

Unicyclic graphs have applications in different research areas and domains. For example, unicyclic graphs are often used in telecommunications. They allow end-users connected in the same unicyclic component or graph to communicate using the two directions of the cycle. The cycle ensures a certain level of survivability to link failure that occurs on the edges of this cycle (commonly known as a "ring" in telecommunications). The traffic demands between nodes on the same cycle are then fully protected against failures while the other demands can be disrupted.

This paper focuses on a particular class of unicyclic graphs G(q, n 1 ) where q is the size of the unique cycle and n 1 represents the number of nodes connected to each vertex on the cycle. The degree of each vertex of the cycle is n 1 + 2 (i.e., each vertex of the cycle has n 1 neighbors not on the cycle). We can easily deduce that the total number of vertices of this class of unicyclic graph is given by q + (q × n 1 ) (see Figure 1 for more details through a simple example).

Given a graph G with n vertices and m edges, we denote by A the adjacency matrix of G and by L the Laplacian matrix of G. The matrix L is the result of the difference between the degree matrix D (D is a diagonal matrix, and each value represents the degree of a vertex in G) and the adjacency matrix A, inducing L = D -A.

Let d 1 , d 2 , . . . , d n be the degrees of the vertices 1, 2, . . . , n respectively in graph G. Without loss of generality, we Let λ 1 , λ 2 , . . . , λ n be the eigenvalues of the adjacency matrix A of G (we suppose that λ 1 ≥ λ 2 ≥ . . . ≥ λ n ) and µ 1 , µ 2 , . . . , µ n-1 , µ n = 0 are the eigenvalues of the Laplacian matrix (we suppose also that µ 1 ≥ µ 2 ≥ . . . µ n-1 ≥ µ n = 0). Note that we are interested only by connected graphs, and according to [START_REF] Cvetkovic | Spectra of graphs, theory and application[END_REF], the smallest eigenvalue of the Laplacian matrix of a connected graph, is equal to zero. Graphs with different connected components are not in the scope of this paper.

Our key contributions consist of proposing exact values of the extremal eigenvalues of the Laplacian and the adjacency matrices of a particular class of unicyclic graphs G(q, n 1 ), in contrast to lower and upper bounds provided by the current literature. Our results improve the quality of the spectral lower and upper bounds for any graphs.

Section II of this paper is dedicated to related work on lower and upper bounds of the eigenvalues of the adjacency and Laplacian matrices of a given graph. Section III provides the extremal and exact eigenvalues of A and L of unicyclic graphs G = (q, n 1 ). Scalability analysis and performance evaluations of our results are found in Section IV. Further work is presented in Section V.

II. RELATED WORK

A. Bounds on λ 1 and λ n Computation of extremal eigenvalues of graph spectra is a classical problem, investigated using different numerical methods (see references [START_REF] Lehoucq | An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices[END_REF] and [START_REF] Lehoucq | Arpack users guide: Solution of large scale eigenvalue problems by implicitly restarted arnoldi methods[END_REF], for example) to characterize the exact values for small graphs. Since characterizing the eigenvalues becomes harder for large graphs, prior art resorted to lower and upper bounds to estimate the spectra of these graphs. We review some of these upper and lower bounds results from the literature.

If A is a symmetric matrix with values in R and x is a vector with norm 1, then a function f (x) = x t Ax reaches its maximum (resp. minimum) value at the eigenvector associated to the largest (respectively. smallest) eigenvalue.

Proposition II.1. [START_REF] Cvetkovic | Spectra of graphs, theory and application[END_REF] If G is a graph with n vertices and m edges and we note d = n i=1 di n , then we can write:

∆ ≥ λ 1 ≥ d n i=1 λ i = 0 n i=1 λ 2 i = 2m λ 1 ≥ 2cos π n + 1
In order to propose tighter lower and upper bounds, previous work proposed different bounds improving the distance to the exact value. For instance, reference [START_REF] Stevanovic | Bounding the largest eigenvalue of trees in terms of the largest vertex degree[END_REF], proposed a new tight bound on λ 1 as established in the following proposition, for a given graph: Proposition II.2. Let λ 1 be the largest eigenvalue (index) of the adjacency matrix of a graph G. Then

λ 1 ≥ √ ∆
In case of unicyclic graph, reference [START_REF] Hu | The largest eigenvalue of unicyclic graphs[END_REF] proposed an upper bound on λ 1 , given as follows:

Proposition II.3. If G is a unicyclic graph then:

λ 1 ≤ 2 √ ∆ -1
References [START_REF] Cvetkovic | Spectra of graphs, theory and application[END_REF] and [START_REF] Das | Sharp upper bounds on the spectral radius of the laplacian matrix of graphs[END_REF] proposed also upper bounds on λ 1 using different parameters of graph G. Their results are given in Propositions II.4 and II.5 respectively.

Proposition II.4. Let G be a connected graph with n vertices and m edges, this new bound is valid :

λ 1 ≤ 2m -(n -1)δ + (δ -1)m max
with m max = max i=1,n m i and m i = 1 di j∼i d j Proposition II.5. Let G be a graph with n vertices and m edges. Then we have :

• 1 ≤ λ 1 ≤ n -1 • -λ 1 ≤ λ n ≤ -1
In the same manner, authors of [START_REF]Some new bounds on the spectral radius of graphs[END_REF] proposed new bounds on λ n to reduce the separation of the obtained bounds and the exact value. This result is indicated in Proposition II.6.

Proposition II.6. Let G be a connected graph with n vertices and m edges. Then

λ n ≥ -2m -(n -1)δ + (δ -1)∆.
Another reference dealing with spectral radius of some unicyclic graphs is proposed in [START_REF] Rojo | New upper bounds on the spectral radius of unicyclic graphs[END_REF]. Authors of this reference investigate new upper bounds on µ 1 and λ 1 as proposed in the following proposition:

Proposition II.7. Let G be a simple undirected unicyclic graph, then:

µ 1 < ∆ + 2 √ ∆ -1 cos π 2k(G) + 1 λ 1 < 2 √ ∆ -1 cos π 2k(G) + 1
where k(G) is the maximum distance between all couples of vertices in the forest obtained by G \ E(C), where C is the unique cycle of G.

B. Bounds on µ 1 and µ n-1

In this section, we briefly review some interesting results on tight lower and upper bounds of the algebraic connectivity of any graph G, and we specify some particular results when G is unicyclic.

Authors of reference [START_REF] Fiedler | Algebraic connectivity of graphs[END_REF] proposed a lower bound on the algebraic connectivity (i.e. the smallest eigenvalue of the matrix L). This bound depends on the edge connectivity which is the minimum number of edges whose deletion from a graph G disconnects G.

Proposition II.8. Let G be a connected graph with n vertices and m edges. We denote by e(G) the edge connectivity. Then :

µ n-1 ≥ 2e(G) 1 -cos π n
A well-known result on the exact value of the algebraic connectivity is given in Proposition II.9. A detailed proof can be found in [START_REF] Fiedler | Algebraic connectivity of graphs[END_REF].

Proposition II.9. If G is a graph with n vertices, then :

µ n-1 = 2n min x =0 uv∈E a uv (x u -x v ) 2 u∈V v∈V (x u -x v ) 2
The two following propositions that can be found respectively in [START_REF] Mohar | Some applications of laplace eigenvalues of graphs[END_REF] and [START_REF] Chung | Spectral graph theory[END_REF], provide lower and upper bounds on µ 1 and µ n-1 , and give relations between the two eigenvalues. More details and information on proofs of the previous propositions, can be found in [START_REF] Mohar | Some applications of laplace eigenvalues of graphs[END_REF], [START_REF] Cvetkovic | A new spectral method for determining the number of spanning trees[END_REF], [START_REF] Cvetkovic | On graphs with second largest eigenvalue does not exceed (sqrt(5)-1)/2[END_REF], [START_REF] Henrot | Minimization problem for eignevalues of the laplacian[END_REF] and [START_REF] Hong | A sharp upper bound of the spectral radius of graphs[END_REF].

Proposition II.10.

µ n-1 ≤ n n -1 δ ≤ n n -1 ∆ ≤ µ 1 ≤ 2∆.
If G is a unicyclic graph, then

∆ + 1 ≤ µ 1 ≤ ∆ + 2 √ ∆ -1
Proposition II.11. Let G be a graph with a diameter D ≥ 4. Then

µ n-1 ≤ 1 - 2 √ ∆ -1 ∆ 1 - 2 D + 2 D III. EXTREMAL EIGENVALUES OF G(q, n 1 )
In this Section, we consider unicyclic graphs G(q, n 1 ) as already shown in Figure 1, and propose extremal exact eigenvalues of A and L of G.

Lemma III.1. Let G be a unicyclic graph of the class G(q, n 1 = ∆ -2), then

λ 1 = 1 + √ ∆ -1 (1)
Proof. Let x be an eigenvector associated to λ 1 . Each component of x is associated to each vertex of the graph G.

The vertices of the unique cycle are indexed by c 1 , ..., c q . For each vertex c i on the cycle (i = 1, . . . , q), there are ∆ -2 leaves connected to c i . Each leaf j connected to c i is denoted by

f j i where 1 ≤ j ≤ ∆ -2.
Assuming that x is an eigenvector, we obtain the two following equations (∀j = 1, . . . , (∆ -2) and i = 1, . . . , q.):

λ 1 x f j i = x ci (2)
λ 1 x ci = x ci+1 + x ci-1 + ∆-2 j=1 x f j i (3)
By combining these two equations, we obtain :

(λ 1 - ∆ -2 λ 1 )x ci = x ci+1 + x ci-1 (4)
By summing the two sides of (4) on each vertex on the cycle, we obtain:

λ 1 - ∆ -2 λ 1 -2 × i x ci = 0 (5)
Following the Perron-Frobenius theorem (Theorem III.2 given bellow), we know that i x ci = 0.

Theorem III.2. [START_REF] Godsil | Algebraic graph theory[END_REF] Let A be a n × n real and non-negative matrix of a connected graph G. Then the index of G is simple (its multiplicity is equal to one). Each component of the eigenvector x associated to this eigenvalue is different from zero and has the same sign.

From the equality (5) we have:

λ 1 - ∆ -2 λ 1 = 2
We multiply the two sides of (5) by λ 1 , and we obtain :

λ 2 1 -(∆ -2) = 2λ 1 .
The discriminant of the equation III is then given by Disc = 1 + ∆ -2 = ∆ -1 > 0 (∆ ≥ 3), and we deduce the two following solutions :

λ (1) 1 = 1 - √ ∆ -1 λ (2) 1 = 1 + √ ∆ -1.
It is well known that the index of the adjacency matrix A verifies λ 1 ≥ 1, and that ∆ ≥ 3 (see Figure 1), then λ

(1) 1 is not a feasible solution. We can conclude that:

λ 1 = 1 + √ ∆ -1
In the same manner, we propose the exact and smallest eigenvalue of the adjacency matrix of the graph G(q, n 1 ).

Lemma III.3. Let G be a unicyclic graph of the class G(q, n 1 ). Then

• For even values of q:

λ n = -1 - √ ∆ -1 (6)
• For odd values of q:

λ n = -cos π q -cos 2 π q + ∆ -2 (7)
Proof. To prove this result, we can use the same approach as in the proof of Lemma 1. We find (∀j = 1, . . . , (∆ -2) and i = 1, . . . , q):

λ n x f j i = x ci (8)
λ n x ci = x ci+1 + x ci-1 + ∆-2 j=1 x f j i (9)
We obtain:

λ n - ∆ -2 λ n x ci = x ci+1 + x ci-1
We recall that λ n ≤ -1, then λ n = 0 . Then, we obtain a system of q equations associated to a circulant matrix [START_REF] Gray | Toeplitz and circulant matrices: A review[END_REF], which is given by:

C =         γ -1 -1 0 ... 0 -1 γ -1 0 ... 0 0 -1 γ -1 ... 0 ... 0 -1 γ -1 0 0 ... 0 -1 γ -1 -1 0 ... 0 -1 γ         with γ = λ n -∆-2
λn , and this leads to an eigenvalue γ -2 cos 2kπ q ∀k = 1, . . . , q.

It is easy to see that 0 should be an eigenvalue of C, which is equivalent to justify the existence of k, such that γ -2 cos 2kπ q = 0 (see [START_REF] Cvetkovic | Spectra of graphs, theory and application[END_REF], [START_REF] Cvetkovic | A new spectral method for determining the number of spanning trees[END_REF], [START_REF] Gray | Toeplitz and circulant matrices: A review[END_REF]).

Finally, we solve the following equation:

λ n - ∆ -2 λ n = 2 cos 2kπ q
which is a second degree equation equivalent to :

λ 2 n -2 cos 2kπ q λ n -(∆ -2) = 0
The discriminant of equation (III) is given by:

Disc = cos 2 ( 2kπ q ) + ∆ -2 > 0, ∀k = 0, . . . , q
the solution of this equation is given by the two following roots, ∀k = 0, . . . , q -1

λ (1) n = cos 2kπ q -cos 2 ( 2kπ q ) + ∆ -2 λ (2) n = cos 2kπ q + cos 2 ( 2kπ q ) + ∆ -2
As ∆ ≥ 3 then cos 2 ( 2kπ q ) + ∆ -2 ≥ 1 and we can deduce that

cos 2 ( 2kπ q ) + ∆ -2 + cos 2kπ q ≥ 0
This is in contradiction with λ n < 0 (see [START_REF] Cvetkovic | Spectra of graphs, theory and application[END_REF]). Thus, the solution is given by:

λ (1) n = cos 2kπ q -cos 2 ( 2kπ q ) + ∆ -2 Let f (k) = cos 2kπ q -cos 2 ( 2kπ q ) + ∆ -2.
The derivative form of f is given by :

f (k) = sin( 2kπ q )   cos 2kπ q cos 2 ( 2kπ q ) + ∆ -2 -1  
the expression between the square brackets is negative, because

cos 2kπ q < cos 2 ( 2kπ q ) + ∆ -2
this derivative is equal to zero for the values of k = 0, and q 2 . According to [START_REF] Cvetkovic | Spectra of graphs, theory and application[END_REF], one can see that λ n is bounded. The function f (k) is then :

• decreasing : for k ∈ [0, q 2 ]. • increasing : for k ∈ [ q 2 , q]
and reaches its minimum value for k = q 2 . This gives us : • q even : λ

(1) n = -1 - √ ∆ -1 • q odd : λ (1) n = -cos π q -cos 2 ( π q ) + ∆ -2
With these values of λ n , we can verify that 0 is an eigenvalue of C. Then, we have an eigenvector x C of C.

It is not difficult to reconstruct the rest of the vector using the relations ( 8) and ( 9).

In the following, and according to the unicyclic graph G(q, n 1 ), we investigate exact and extremal values of the eigenvalues of the Laplacian matrix L.

Lemma III.4. Let G be a unicyclic graph of the class G(q, n 1 ). The exact value of the algebraic connectivity is given by

µ n-1 = 1 2 × ∆ -1 + 4 sin 2 ( π q ) -[∆ -1 + 4 sin 2 ( π q )] 2 -16 sin 2 ( π q ) (10)
Proof. Starting from the definition of an eigenvalue of a Laplacian matrix, we have ( ∀j = 1, . . . , ∆ -2 and ∀i = 1, . . . , q):

1) i / ∈ C q : (1 -µ n-1 )x f j i = x ci 2) i ∈ C q : (∆ -µ n-1 )x ci = x ci+1 + x ci-1 + ∆-2 j=1 x f j i by substituting x f j i
in the second case, we obtain:

∆ -µ n-1 - ∆ -2 1 -µ n-1 x ci = x ci+1 + x ci-1
which is a system of q equations, and q variables. To solve it, we set γ = ∆ -µ n-1 -∆-2 1-µn-1 , and we can remark that the associated matrix to this system is also circulant :

C =         γ -1 -1 0 ... 0 -1 γ -1 0 ... 0 0 -1 γ -1 ... 0 ... 0 -1 γ -1 0 0 ... 0 -1 γ -1 -1 0 ... 0 -1 γ        
and this leads to an eigenvalue γ -2 cos 2kπ q ∀k = 1, . . . , q such that ∃k, γ -2 cos 2kπ q = 0. From the proposition II.11, we see that µ n-1 = 1.

Thus, we solve the following:

∆ -µ n-1 - ∆ -2 1 -µ n-1 = 2 cos 2kπ q
It is a second degree equation which can be reformulated as follows:

∆ -( µ n-1 -µ 2 n-1 + ∆ -2 1 -µ n-1 ) = 2 cos 2kπ q ⇒ ∆ -∆µ n-1 + µ 2 n-1 -µ n-1 -∆ + 2 = = 2 cos 2kπ q -2 cos 2kπ q µ n-1
with µ n-1 = 1 and then

µ 2 n-1 -(∆+1-2 cos 2kπ q )µ n-1 +2(1-cos 2kπ q ) = 0
The discriminant (Disc) of this last equation is given by

Disc = (∆ + 1 -2 cos 2kπ q ) 2 -8 + 8 cos 2kπ q Disc = [∆ + 1 -2 cos 2kπ q ] 2 -16 sin 2 ( kπ q )
Thus, we obtain two solutions ( 11) and ( 12):

µ (1) n-1 = 1 2 ∆ + 1 -2(1 -2 sin 2 ( kπ q )) -[∆ + 1 -2(1 -2 sin 2 ( kπ q ))] 2 -16 sin 2 ( kπ q ) (11) µ (2) n-1 = 1 2 ∆ + 1 -2(1 -2 sin 2 ( kπ q )) + [∆ + 1 -2(1 -2 sin 2 ( kπ q ))] 2 -16 sin 2 ( kπ q ) ( 12
)
after some simplification, we can deduce:

µ (1) n-1 = 1 2 ∆ -1 + 4 sin 2 ( kπ q ) -[∆ -1 + 4 sin 2 ( kπ q )] 2 -16 sin 2 ( kπ q ) (13) µ (2)
n-1 = 1 2 ∆ -1 + 4 sin 2 ( kπ q ) + [∆ -1 + 4 sin 2 ( kπ q )] 2 -16 sin 2 ( kπ q ) (14)
As ∆ ≥ 3, then ∆ -1 + 4 sin 2 ( kπ q ) ≥ 2, then we can write µ

n-1 ≥ 1 + [∆ -1 + 4 sin 2 ( kπ q )] 2 -16 sin 2 ( kπ q ) 2 > 1 knowing that µ n-1 < 1, we can reject µ (2)
n-1 which is greater than 1, and we keep only the solution given by ( 15)

µ n-1 = µ (1) n-1 = 1 2   ∆ -1 + 4 sin 2 ( kπ q ) - ∆ -1 + 4 sin 2 ( kπ q ) 2 -16 sin 2 ( kπ q )   (15)
We investigate the value k (k ranging from k = 0, . . . , q) that minimizes the function µ n-1 (k). We remark that if k = 0, or k = q, we find µ n-1 = 0, which is absurd. Thus, we are interested in integer values of k ∈ [1, q -1].

The derivative function of µ n-1 (k) is given by:

µ n-1 = 8π q sin( kπ q ) cos kπ q ×   1 - 4 sin 2 ( kπ q ) + ∆ -3 (∆ -1) 2 + 16 sin 4 ( kπ q ) + 8(∆ -3) sin 2 ( kπ q )   (16)
(∆ -1) 2 + 16 sin 4 ( kπ q ) + 8(∆ -3) sin 2 ( kπ q ) > 4 sin 2 ( kπ q ) + ∆ -3 (∆ ≥ 3) (17)
thus if we write (with ∆ ≥ 3):

(∆ -1) 2 > (∆ -3) 2 ⇒ (∆ -1) 2 + 16 sin 4 ( kπ q ) > (∆ -3) 2 + 16 sin 4 ( kπ q ) ⇒ (∆ -1) 2 + 16 sin 4 ( kπ q ) + 8(∆ -3) sin 2 ( kπ q ) > (∆ -3) 2 + 16 sin 4 ( kπ q ) + 8(∆ -3) sin 2 ( kπ q )
we observe that the square of the two sides of the inequality is given by

(∆ -1) 2 + 16 sin 4 ( kπ q ) + 8(∆ -3) sin 2 ( kπ q ) > 4 sin 2 ( kπ q ) + ∆ -3
The derivative function is equal to zero only if sin( kπ q ) = 0 or cos kπ q = 0, so it is equal to 0 for the following values of k: k = 0, q, q 2 , 3q 2 and as k ∈ [1, q -1], so it is equal to 0 only for k = q 2 , with µ n-1

(k = q 2 ) = ∆+3- √ ∆ 2 +6∆-7 2 < 1.
The function µ is bounded and has the following behaviour:

• increasing : pour k ∈ [1, q 2 ]. • decreasing : pour k ∈ [ q 2 , q -1]
We conclude that µ n-1 (k = q -1) = µ n-1 (k = 1) and k = 1 (or k = q -1), and obtain:

µ n-1 = 1 2 × ∆ -1 + 4 sin 2 ( π q ) -[∆ -1 + 4 sin 2 ( π q )] 2 -16 sin 2 ( π q )
In the following, we provide the exact and largest eigenvalue of L of the unicyclic graph G(q, n 1 ).

Lemma III.5. Let G be a unicyclic graph of the class G(q, n 1 ). Then 1) If q is even:

µ 1 = ∆ + 3 + (∆ + 3) 2 -16 2 (18)
The expression between the square brackets is non-negative as we can see it in [START_REF] Gray | Toeplitz and circulant matrices: A review[END_REF] 2) If q is odd:

µ 1 = (∆ + 1) + 2 cos π q 2 + [(∆ + 1) + 2 cos π q ] 2 -8(1 + cos( π q )) 2 (19)
Proof. According to the proof of Lemma [START_REF] Mohar | Some applications of laplace eigenvalues of graphs[END_REF], and by using the associated circulant matrix of the current case, we obtain the following equation :

µ 2 1 -∆ + 1 -2 cos 2kπ q µ 1 +2 1 -cos 2kπ q = 0
and we recall that for a unicyclic graph, we have:

∆ + 1 ≤ µ 1 ≤ ∆ + 2 √ ∆ -1 (20)
The discriminant of this equation is non-negative (see proof of the Lemma [START_REF] Mohar | Some applications of laplace eigenvalues of graphs[END_REF]) and this leads to two solutions :

1) µ (1) 1 = ∆+1-2 cos( 2kπ q )-[∆+1-2 cos( 2kπ q )] 2 -8[1-cos( 2kπ q )] 2 2) µ (2) 1 = ∆+1-2 cos( 2kπ q )+ [∆+1-2 cos( 2kπ q )] 2 -8[1-cos( 2kπ q )] 2
It is easy to verify that µ violates the requirement (20), and this is due to formulas (21):

µ (1) 1 = ∆ + 1 2 -( 2 cos 2kπ q + [∆ + 1 -2 cos 2kπ q ] 2 -8[1 -cos( 2kπ q )] 2 ) < ∆ + 1 (21)
thus, we reject it and consider the solution given by:

µ 1 = ∆ + 1 -2 cos 2kπ q 2 + [∆ + 1 -2 cos 2kπ q ] 2 -8[1 -cos( 2kπ q )] 2 (22)
µ 1 is a function of k, which is symmetric around q 2 . In other words, we find the same value of µ 1 for instances of k = 0 and k = q, then for k = 1 and k = q -1,etc. According to this remark, we illustrate the behaviour of µ 1 for some values of k = 0, . . . , q (see Figure [START_REF] Lehoucq | An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices[END_REF]). We note that: cos 2(q-1)π 2q = cos(π -π q ) = -cos( π q ). The derivative formula of µ 1 is given by ( 23) and simplified in (24):

µ 1 = 1 2   4π q sin( 2kπ q ) + 2(∆ + 1 -2 cos( 2kπ q ))( 4π q sin( 2kπ q )) -16π q sin( 2kπ q ) 2 [∆ + 1 -2 cos( 2kπ q )] 2 -8[1 -cos( 2kπ q )]   (23) µ 1 = 2π q sin( 2kπ q )   1 + (∆ + 1 -2 cos( 2kπ q )) -2 [∆ + 1 -2 cos( 2kπ q )] 2 -8[1 -cos( 2kπ q )]   (24)
as ∆ ≥ 3, the expression between the square brackets is non-negative. The derivative function is equal to zero for the values of k = 0 and q 2 , and as we are interested by integer values of k ∈ [1, q -1], then we keep only k = q 2 . The function reaches its maximum for k = q 2 , and we find:

1) q even : k = q 2 , and

µ 1 = ∆ + 3 + (∆ + 3) 2 - 2
2) q odd : k = q-1 2 , and

µ 1 = (∆ + 1) + 2 cos π q + [(∆ + 1) + 2 cos π q ] 2 -8(1 + cos( π q )) 2

IV. NUMERICAL RESULTS AND VALIDATION

In this Section, we verify and validate numerically the provided results on different unicyclic graph instances of type G(q, n 1 ), compared to a well-known numerical method that can be found in [START_REF] Anderson | Lapack: A portable linear algebra library for high-performance computers[END_REF] for example.

We use an instance of unicyclic graphs of type G(q, n 1 ) as shown in Table I and Table II. In this instance, we consider odd cycle sizes, when varying n 1 values from 1 to 10.

We used the LAPACK method [START_REF] Anderson | Lapack: A portable linear algebra library for high-performance computers[END_REF] as a numerical analysis benchmark to get exact eigenvalues of the considered graphs. Thus, one can observe in Tables I and II that our proposed results are exactly the same as LAPACK. This reinforces and validates our provided proofs in the previous sections, and leads to use our results for large unicyclic graph instances, when the numerical methods suffer from scaling-up and to provide exact eigenvalues solutions.

V. CONCLUSION

In this paper, we proposed exact values of the extremal eigenvalues of the adjacency matrix and the Laplacian matrix of a particular class of unicyclic graphs. The proposed results are often given by solving linear equation systems involving circulant matrices. The provided results are new and reinforce the lower and upper bounds of the spectra of any graphs with a topology close to the unicyclic graphs considered in our work.

In future work, we will investigate new results on the spectra of the adjacency and Laplacian matrices for larger classes of unicyclic graphs noted by G(q, n 1 , n 2 ). Figure 3 illustrates an example of such a class of unicyclic graphs.

Fig. 1 .

 1 Fig. 1. Example of a unicyclic graph G(7, 3)

Fig. 2 .

 2 Fig. 2. µ 1 's behavior with k evolution

Fig. 3 .

 3 Fig. 3. Example of a unicyclic graph G(7, 2, 3)

TABLE I COMPARISON

 I OF EXTREMAL EIGENVALUES OF A: UNCYCLIC GRAPHS G(q, n 1 ) WITH q = 3

	n 1	(1)	num λ 1	(III.3)	num λn
	1	2.414	2.414	-1.618	-1.618
	2	2.732	2.732	-2	-2
	3	3	3	-2.302	-2.302
	4	3.236	3.236	-2.561	-2.561
	5	3.449	3.449	-2.791	-2.791
	6	3.645	3.645	-3	-3
	7	3.828	3.828	-3.192	-3.192
	8	4	3.999	-3.372	-3.372
	9	4.162	4.162	-3.541	-3.541
	10	4.316	4.316	-3.701	-3.701
			TABLE II
	COMPARISON OF EXTREMAL EIGENVALUES OF L: UNCYCLIC GRAPHS
			G(q, n 1 ) WITH q = 3
	n 1	(19)	num µ 1	(10)	num
					µ n-1
	1	4.302	4.302	0.697	0.697
	2	5.449	5.449	0.550	0.550
	3	6.541	6.541	0.458	0.458
	4	7.605	7.605	0.394	0.394
	5	8.653	8.653	0.346	0.346
	6	9.690	9.690	0.309	0.309
	7	10.720	10.720	0.279	0.279
	8	11.744	11.744	0.255	0.255
	9	12.764	12.764	0.235	0.235
	10	13.782	13.782	0.217	0.217

ACKNOWLEDGMENT

This research work has been carried out in the framework of the Technological Research Institute SystemX, and therefore granted with public funds within the scope of the French Program "Investissements d'Avenir".


Bertram Daum

Tessa E F Quax

Martin Sachse

Deryck J Mills

Julia Reimann

Özkan Yildiz

Sabine Häder

Cosmin Saveanu

Patrick Forterre

Sonja-Verena Albers

Werner Kühlbrandt
email: werner.kuehlbrandt@biophys.mpg.de

David Prangishvili
email: david.prangishvili@pasteur.fr

Self-assembly of the general membrane-remodeling protein PVAP into sevenfold virus-associated pyramids

Keywords: archaea, virus, viral egress

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Introduction

Release of virus particles from infected cells is the last essential step of the viral replication cycle. In the course of this process, virions face the challenging task of crossing the cell envelope.

Viruses have developed an arsenal of diverse strategies to overcome this problem. Most bacterial viruses are lytic and induce lysis of the infected cell with help of the holin-endolysin system (1), while others disrupt the host cell envelope via inhibition of the murein biosynthesis pathway (2). The morphological and genomic properties of archaeal viruses (3) suggested that their egress from host cells may have unusual traits that are different from those of bacterial viruses. Indeed, while most archaeal viruses exit cells without lysis, some, in particular the rodshaped or icosahedral viruses that respectively infect Sulfolobus islandicus (SIRV2) and Sulfolobus turreted icosahedral virus 1 (STIV1), are lytic and exploit a special mechanism of virion egress (4)(5)(6)(7)(8). During the infection cycle of these viruses, pyramidal protrusions with sevenfold rotational symmetry form in the host cell membrane. As the final step of the infection cycle the VAPs open outwards along the seams of their seven facets, creating ~100 nm apertures through which the newly formed virions escape from the host cell (4,7). VAPs consist of multiple copies of a ~10 kDa virus-encoded protein, which we term PVAP (SIRV2_P98) (7)(8)(9). Surprisingly, PVAP assembles into membrane pyramids even when expressed heterologously in archaeal and bacterial expression systems, demonstrating that no other viral proteins are required for VAP formation (7). The mechanism by which VAPs self-assembles in the membrane remains unknown.

In the present study we used electron cryo-tomography to investigate morphological features of SIRV2 replication and the formation of VAPs at different time points after infection. By subtomogram averaging, we determined a first 3D map of the VAP. This map, in combination with secondary structure predictions of PVAP and expression of wild type PVAP or a variety of truncation mutants in archaeal, bacterial and eukaryotic cells allows us to propose a model of how PVAP arranges to form the sevenfold pyramids. These insights are fundamental for understanding how this mechanism can be exploited as a universal tool to engineer the formation and controlled opening of large pores in biological or artificial lipid bilayers.

Results

SIRV2 induces morphological changes of the host cell

We analyzed morphological changes in S. islandicus during SIRV2 infection and the time points of VAP formation and opening by whole cell electron cryo-tomography at 0.5, 3, 6 and 12 hours post infection (h.p.i.). This allowed us to monitor morphological changes at high resolution and to compare these with previous results obtained by thin-sectioning of chemically fixed cells during the final stages of SIRV2 infection (4). Up to 3 h.p.i. infected cells were indistinguishable from uninfected control cells and no virions were visible in the cytoplasm (Figs. 1a). This suggests that SIRV2 does not enter the cell as an intact virus particle, in accordance with a previous study, in which we have directly observed infection of S. islandicus by SIRV2 in the electron microscope (10). Electron-dense ~100 nm globules were the only conspicuous features in the cytoplasm of controls and infected cells (Fig. 1a), similar to those reported for S. solfataricus cells (11). However, each cell, infected or not, contained one or a few of these globules, which are therefore unrelated to virus replication. Their size and density above the cytoplasmic background indicates that they may be storage granules (12). At 3-6 h.p.i., about half way through the infection cycle, newly assembled virions became visible in the cytoplasm as described earlier (4). They were organized in up to three bundles per cell, each consisting of roughly 50 rod-shaped particles (Fig. 1b). Starting from 3 h.p.i., VAPs of various sizes formed in the plasma membrane of the host cells, most of which had penetrated the S-layer (Fig. 1b, 2 a-d). Each cell contained on average about 10 VAPs. The height of the VAPs (measured from the membrane to the tip of the pyramid) ranged from ~20nm to 150nm. At an early stage of formation, the VAPs in the plasma membrane had not yet punctured the S-layer of the host cell (Supplementary Fig. 1), but they already had the distinct features of hollow heptagonal pyramids, corroborating the earlier assumption that VAPs grow from the base by gradual expansion of their triangular facets (7). Very occasionally, VAPs contained a spherical storage granule (Fig. 2c). These granules likely correspond to the previously described intrapyramidal bodies (IPB) in STIV-induced VAPs (11). Analysis of cells at distinct time points post infection showed that at 6 h.p.i. a number of VAPs had opened. The fraction of open VAPs increased, until at 12 h.p.i. all VAPs had unfolded, like the petals of a flower (Fig. 1c, 2e-h). The

VAPs appeared to open as the pyramidal structures broke along the seams of the triangular pyramid faces (Fig. 2f). The VAP facets curved outwards with counter-clockwise handedness when viewed from the cell exterior (Fig. 2f), as suggested by electron micrographs of isolated VAPs (7). As the VAPs opened, the virion bundles disintegrated and individual virions diffused into the medium through the open VAPs (Fig. 3).

VAP structure

The shape of isolated VAPs has previously been investigated by negative staining and electron microscopy (7). To gain insight in the structure of VAPs in situ, closed or open VAPs were imaged by whole-cell electron cryo-tomography (Fig. 4a-c). The triangular faces of all VAPs appeared to consist of two distinct layers, irrespective of their conformation or stage of assembly. The ~4.5±1 nm outer layer was continuous with, and indistinguishable from, the cell membrane. The inner layer had a thickness of ~4.0±1 nm (Fig. 4). The centre-to-centre distance between the two layers was 10.0±1 nm, leaving a 5.8±1 nm gap of lower density. At the base of each VAP, the inner layer extended up to 15 nm beyond the outer layer into the cytoplasm. This suggests that the inner layer consists of a protein sheet that is attached and runs parallel to the cytoplasmic membrane surface (Fig. 4).

In order to verify that both layers of the VAP consist only of PVAP protomers, we analysed archaeal (S. acidocaldarius) or bacterial (E. coli) PVAP expression mutants constructed in a previous study (7). Whole-cell electron cryo-tomography of transformed cells revealed the distinctive two layers in all VAPs (Fig. 4c,S3), indicating that both consist solely of PVAP (Supplementary Fig. 3).

Subtomogram averaging of VAPs

A 3D map of the VAP was obtained by averaging 57 tomographic subvolumes of closed VAPs in the membrane of PVAP-expressing E. coli cells. Sections through the averaged volume parallel to the membrane showed clear sevenfold symmetry (Supplementary Fig. 4), which was applied to the final average to further improve the signal-to-noise ratio (Fig. 4d-k, Supplementary movie 1).

The local resolution estimated by the program ResMap (13) was between 36 and 58 Å, with a mean resolution of ~43Å (Supplementary Fig. 5). Since VAPs come in different sizes, only the upper parts of the pyramids, to a height of ~62 nm and an outer diameter of ~96 nm, were used for averaging. At the tip, the inner opening angle of the pyramid was ~80° (Fig. 4d). As observed in the raw tomograms, the structure consisted of two layers (Fig. 4d,e,). The outer layer formed a continuous envelope, consisting of seven triangular facets (Fig. 4f-k). As seen from the outside, the facets had a perceptible counter-clockwise handedness (Fig. 4f) and each facet was slightly convex towards the inside (Fig. 4f,g,j,k). The angle at the tip of the triangular facets was 35°, in accordance with previous measurements on isolated VAP fragments (7). Overall, the entire structure had the appearance of a tent or teepee (Fig. 4 g). The inner, cytosolic layer of the pyramid consisted of seven triangular sheets parallel to the membrane on the outside (Fig. 4i-k).

Cross-sections through the sevenfold averaged volume revealed narrow connecting densities between the two layers of the pyramid, suggesting that they are physically linked (Fig. 4k).

Membrane remodelling by PVAP

In silico secondary structure predictions (14) and hydrophobicity analysis (15) of PVAP suggest that the protein consists of an N-terminal transmembrane helix (residues 5-34), followed by three hydrophilic α-helices of two to three turns each, separated by short linker regions (Supplementary Fig. 2). The PSORT-server ( 16) indicated that PVAP does not contain a presequence. This is consistent with a previous study of the N-terminal amino acid sequence of purified PVAP of SIRV2 with Edman degradation, which did not indicate a cleavable signal sequence (9). Thus, we assumed that the hydrophobicity of its predicted N-terminal transmembrane segment drives the spontaneous insertion of PVAP into the lipid bilayer. To test this hypothesis, PVAP was expressed in the Saccharomyces cerevisiae, an eukaryotic organism.

Cells were harvested 16 hours after induction of PVAP expression, high-pressure frozen and freeze-substituted. In addition, sections prepared by the Tokuyasu method (17) were immunolabeled with antibodies raised against PVAP. All samples were analysed by transmission electron microscopy. Surprisingly, VAPs were found in most, if not all, cellular membranes. In terms of size and appearance, the pyramids were indistinguishable from those that assembled in E.coli after PVAP expression or in S. islandicus after SIRV2 infection. PVAP-specific antibodies labeled VAPs in the nuclear envelope, the endoplasmic reticulum, Golgi apparatus, intracellular vesicles and mitochondria (Fig. 5).

Role of PVAP domains in VAP assembly

To identify which parts of the PVAP are required for VAP assembly, truncated mutants lacking the last 10, 20, 30, 40 or 70 C-terminal residues (ΔC10, ΔC20, ΔC30, ΔC40 or ΔC70) were constructed (Fig. 6). EM analysis of E.coli cells transfected with these constructs revealed VAPs only in case of the ΔC10 mutant (Fig. 6).

By contrast, VAPs did not form after truncation of 20 to 70 C-terminal residues (PVAPΔC20, ΔC30, ΔC40 or ΔC70, Fig. 6), corresponding to one to three C-terminal α-helical segments.

Instead, expression of these constructs resulted mostly in protein aggregates. In addition, constructs lacking 20-40 C-terminal residues caused the inner membrane of E.coli to form large invaginations, suggesting that these variants still interact with the membrane (Fig 6b). The effect was most pronounced for PVAPΔC20. In contrast, PVAPΔC70 did not produce any membrane invaginations (Fig 6b).

Expression of a PVAP construct lacking the predicted N-terminal transmembrane helix (PVAPΔN30) likewise did not result in VAP formation. There was no sign of any interaction with the membrane (Fig. 6b), indicating that the N-terminal transmembrane domain is indeed required for membrane insertion of PVAP protomers.

We asked if the PVAP transmembrane domain is essential for VAP formation or could be replaced by any other transmembrane domain. To characterize the role of the PVAP transmembrane domain in VAP formation we constructed a chimera by fusing the E. coli flagellar regulator Flk, a gene encoding a single trans-membrane helix inner membrane protein [START_REF] Borgese | Remote origins of tail-anchored proteins[END_REF] to PVAPΔN30, replacing the N-terminal transmembrane helix (residues 1-30) of PVAP [START_REF] Borgese | Remote origins of tail-anchored proteins[END_REF]. After expression, this fusion construct (PVAPtmFlk) was indeed inserted into the membrane, as judged by Western blot analysis of cell fractions with SIRV2-PVAP antibody (Supplementary Fig. 6). However, there was no evidence of VAPs in these cells (Fig 6b).

Taken together these findings indicate that the N-terminal domain is essential for membrane insertion of PVAP and for the interaction between PVAP protomers, which results in the assembly of a protein sheet on the inner membrane surface. The C-terminal domain of PVAP (except the last 10 residues, which are predicted to be disordered) is required for VAP formation. Without this domain, the protein aggregates instead of forming VAPs.

PVAP oligomers

To characterize the oligomerization of PVAP in vitro, we fused a His-tag to the C-terminus and expressed the protein heterologously in E. coli. Isolated membranes were solubilized with the detergent N-laurylsarcosine. PVAP was purified by nickel affinity chromatography and size exclusion chromatography. The single peak in the gel filtration profile corresponds to a molecular mass of ~70 kDa (Supplementary Fig. 7). SDS-PAGE analysis of peak fractions show discrete PVAP bands at ~10, 20, 30 and 70 kDa (Supplementary Fig. 7), indicating that in detergent solution PVAP forms different oligomers, the largest of which is most likely a heptamer. A PVAP heptamer is also suggested by gel filtration chromatography.

Discussion

The VAP, an archeoviral egress structure that takes the shape of a large sevenfold pyramid in the host membrane, is without parallel in biology. It consists of multiple copies of PVAP, a 10 kDa membrane protein, which forms VAPs in the membrane, evidently without the need for any other cellular component.

How does the 10 kDa PVAP assemble to form sevenfold pyramids in the membrane? To address this question, we investigated the VAP structure itself by electron cryo-tomography, and have studied the membrane insertion and biochemical properties of PVAP. We have demonstrated that PVAP forms VAPs in archaeal, bacterial and eukaryotic membranes, into which it inserts indiscriminately, and that, with the exception of the last 10 C-terminal residues, the entire length of the protein is required for VAP assembly. Finally, we have shown that PVAP forms oligomers, most likely heptamers, in detergent solution.

PVAP is a universal membrane remodeling system

Sequence analysis of PVAP suggested that the protein does not contain a signal sequence and thus most likely integrates spontaneously into the archaeal membrane. A similar mechanism of membrane insertion has been found for tail-anchored (TA) proteins [START_REF] Borgese | Targeting pathways of C-tail-anchored proteins[END_REF] and for bacterial poreforming toxins (bPTFs) [START_REF] Bischofberger | Pathogenic pore-forming proteins: function and host response[END_REF]. TA proteins are indigenous proteins, which contain a single Cterminal transmembrane segment. They are inserted into their target membrane in a Secindependent, but organelle-specific manner, occasionally aided by cytoplasmic chaperones [START_REF] Borgese | Targeting pathways of C-tail-anchored proteins[END_REF].

Similar to PVAP, bPTFs are expressed as monomers and insert into the target membrane, where they assemble into pore-forming oligomers, either to kill other bacteria or, in case of pathogens, to lyse the host membrane and thus to aid bacterial proliferation [START_REF] Bischofberger | Pathogenic pore-forming proteins: function and host response[END_REF].

Overexpression of PVAP in the archaeon S. acidocaldarius, the bacterium E. coli and the eukaryote S. cerevisiae resulted in the formation of VAPs in the plasma membranes of all hosts.

Even more remarkably, VAPs were observed in virtually all cellular membranes of the eukaryote S. cerevisiae, including the nuclear envelope, the ER, the mitochondrial outer membrane and the plasma membrane. This demonstrates that, in contrast to other known type of protein spontaneously integrating into membranes, PVAP is able to insert into practically any biological lipid bilayer, solely by virtue of its N-terminal transmembrane segment. Once inserted into the bilayer, it forms sevenfold pyramids, irrespective of fundamental differences in lipid or protein composition of the target membrane. These characteristics render PVAP a unique, universal membrane remodeling tool.

Supramolecular organisation of VAPs

Whole-cell electron cryo-tomography and subtomogram averaging revealed that the VAPs consist of the same two layers of roughly equal thickness in all endogenous and heterologous expression hosts. The outer layer was continuous with the plasma membrane, whereas the inner layer formed a discontinuous sheet at the cytoplasmic membrane surface. As PVAP must insert with its N-terminal TM segment into the plasma membrane, the outer layer most likely consists of multiple copies of this part of PVAP plus interspersed membrane lipid. In turn, the cytoplasmic protein sheet must consist of the tightly associated C-terminal domains of the protein. The low-density region between the two layers would then account for the linker region between the cytoplasmic domains and the N-terminal trans-membrane segments (Fig. 7a).

Recently it was suggested that the opening of STIV-induced VAPs depends on polymerization of ESCRT-III homologs, resulting in the 'stripping' of VAPs from their cytoplasmic membrane [START_REF] Snyder | Functional interplay between a virus and the ESCRT machinery in archaea[END_REF].

Our observation that the outer layer of the double-layered VAP structure is continuous with the plasma membrane (Fig 4 ) does not support this model. Moreover, global analysis of host gene expression during the SIRV2 infection cycle documented that ESCRT-III-like proteins were down-regulated in infected cells [START_REF] Quax | Massive Activation of Archaeal Defense Genes during Viral Infection[END_REF]. Thus, it is possible, although not very likely, that mechanisms of VAP opening differ in SIRV2-infected and STIV-infected cells. This suggests that in the closed state, the VAP structure is under mechanical tension. This tension is likely to provide the energy required for VAP opening. As the tip of the VAP displays the point of strongest membrane curvature in the whole assembly, it most likely serves as predetermined breaking point, at which the pyramid would begin to unfold along its seams.

Our tomograms of closed and open

VAP assembly and opening

We propose a model for VAP assembly (Fig. 7). Upon synthesis in the cytosol, PVAP integrates spontaneously into the cell membrane, depending solely on the hydrophobicity of its N-terminal transmembrane helix segment. In the membrane, tight interactions between PVAP protomers result in the formation of protein sheets, which consist of the observed two layers. We propose that this interaction involves both the N-terminal transmembrane helices and the C-terminal hydrophilic PVAP domains, as indicated by the expression of truncation constructs.

In order to assemble into a pyramid instead of a flat sheet, at least two different kinds of interactions between PVAPs are necessary, one in-plane interaction within the triangular facets, and one out-of-plane interaction, at the edges of the pyramid. The interaction at the edges is most likely weaker than the in-plane interaction, so that the pyramids open along these lines. Since all VAPs observed in S. islandicus or in heterologous systems grow to roughly the same size, the expansion of VAP must be constrained in some way.

At present it is unknown if the pyramids are built in a one-by-one self-assembly process from individual PVAP protomers or if, upon membrane insertion, the protomers assemble into heptamers, which then combine into pyramids in a second stage of assembly. The prior formation of heptamers in the membrane is suggested by the gel filtration experiments, which show one homogenous peak of ~70 kDa in detergent solution. Given that detergent mimics the hydrophobic membrane environment, it is not unlikely that the same interactions that give rise to the heptamer in a detergent micelle would also promote the formation of heptamers in the membrane, which may thus be the building blocks of the pyramids.

VAP opening presumably involves a host or virus-specific factor, since the pyramids only open in virus-infected Sulfolobus cells but remain closed in PVAP-expressing bacteria and yeast. Once the mechanism that triggers VAP opening is elucidated, this system could be utilised to introduce ~100 nm apertures in any lipid bilayer. VAPs might then be used for targeted drug delivery, releasing compounds from liposomes upon a specific signal. In addition, the PVAP transmembrane domain has the ability to insert into all types of biological membranes and may therefore be fused to proteins that otherwise cannot be reconstituted into lipid bilayers. This system thus has interesting potential applications in basic research, biotechnology and therapy.

Materials and Methods

Virus and host strains S. islandicus LAL 14/1 cells were grown, synchronized and infected with SIRV2 as described previously (7) and in the Supplementary Information (SI).

Plasmid constructs and transformation of S. acidocaldarius SIRV2_ORF98 (NCBI RefSeq ID: NP_666583) was amplified from SIRV2 genomic DNA and cloned into the pSVA1450 plasmid behind an araS promoter, which yielded pTQ26. pTQ26 was transformed to S. acidocaldarius M31 as described in the SI.

Plasmid constructs and transformation of E.coli SIRV2_ORF98 was amplified from SIRV2

genomic DNA with different primers resulting in 3' truncated PCR products of 267, 237, 207, 177 and 87 bp. A 5' truncation of ORF98 was created by amplification of a 216 bp product starting at position 81. The same sequence was fused with the 75 bp transmembrane segment of the E.coli Flk gene as described in the SI. All PVAP gene mutants were cloned into the T7 promoter-driven expression vector pSA4. Expression was induced with isopropyl β-d-1-thiogalactopyranoside (IPTG). Analysis of PVAP expressing cultures by high pressure freezing, freeze substitution and western blotting was performed as described in ref ( 9) and (7).

Plasmid constructs and transformation of S. cerevisiae SIRV2_ORF98 was amplified from SIRV2 genomic DNA and cloned in the expression vector pCM190. S. cerivisiae was transformed with this plasmid. A pre-culture was grown in selective medium as described in the SI. After one day, cells were diluted 1/1000 in medium without doxycyclin.

Immuno-electron microscopy Yeast cells were fixed, washed, pelleted in gelatin and the gelatine pellet was solidifed on ice and cut into small blocks as described in the SI. These were infiltrated 2.3M sucrose, mounted on aluminium pins and frozen in liquid nitrogen. Thin sections were cut and picked up in a 1:1 mixture of 2.3M sucrose and 2% methylcellulose. Labelling for PVAP was done as described previously (7).

High-pressure freezing and freeze-substitution E.coli cells were taken up in cellulose capillary tubes and S. cerevisiae cultures were concentrated by filtration. Samples were highpressure frozen and freeze-substitution was performed in anhydrous acetone containing 2% osmium tetroxide. Afterwards the samples were washed with dry acetone and embedded stepwise in EPON. After heat polymerization thin sections were cut, collected on 200 mesh Formvar-coated copper grids and post-stained as described in the SI. Images were recorded with a JEOL 1010 electron microscope equipped with an Olympus Keen View camera.

Whole cell cryo-tomography S. islandicus cells were harvested, concentrated by low-speed centrifugation and plunge-frozen. E. coli cells overexpressing PVAP were harvested at the same conditions, washed once in 50mM Tris, 300mM NaCl, pH7 and plunge-frozen in the same buffer.

Before freezing, suspensions were mixed with an equal volume of 10 nm colloidal protein-A gold suspension. Tomograms were recorded with a Polara G2 Tecnai field emission transmission electron microscope equipped with a Gatan Tridiem energy filter and 2x2 k CCD camera.

Tomographic tilt series of zero-loss filtered images were recorded and tomograms reconstructed as described in the SI.

Subtomogram averaging For subtomogram averaging of VAPs, 57 pyramid volumes were cut out from a single tomogram of a PVAP overexpressing E. coli cell, aligned and averaged applying 7-fold rotational symmetry using the PEET software as described in the SI. The resolution of the map was estimated using the ResMap software (13).

PVAP purification A codon-optimized SIRV_ORF98 gene was synthetized and inserted in the plasmid pET26b. E. coli BL21DE3/Rosetta/pLysS cells were transformed with this plasmid.

Protein expression was induced with 1mM IPTG. After 2 hours, cells were pelleted, resuspended in lysis buffer and disrupted with a Microfluidizer. The membrane fraction was pelleted by centrifugation, and diluted in 50 mM Tris pH 7.0, 300 mM NaCl to a protein concentration of 5 mg/ml. 1.5% N-laurylsarcosine was added. After high-speed centrifugation the supernatant was loaded onto a Ni-NTA column followed by several washing steps. The protein was eluted in buffer containing 500 mM imidazole and concentrated using Amicon spin columns with a 30 kDa cutoff prior to loading onto a gel filtration column, as described in more detail in the SI. Addition of PVAP units at the base pushes the pyramid outwards.

  VAPs throw some light on the mechanism by which the pyramids open to facilitate viral egress. The seven edges of the closed pyramid are slightly curved in a right-handed fashion (Fig. 4 f). In the open state, the edges of the seven individual facets are curved in the same way (Fig. 4 f). In the open VAP, each facet curls outwards (Fig. 4 f).

Fig. 1 .

 1 Fig. 1. Morphological changes of S. islandicus during infection with SIRV2. Tomographic slices of typical archaeal cells at 0.5 (a), 3-6 (b) and 9 (c) hours post infection with SIRV2. Black arrowheads, closed VAPs; white arrowheads, open VAPs. Scale bars, 500 nm.

Fig. 2 .Fig. 3 .

 23 Fig. 2. VAPs in closed and open conformation. (a, c, e, g) Tomographic slice and segmented, surface-rendered volumes (b, d, f, h) of VAPs in the membrane of SIRV2-infected S. islandicus cells. VAPs are either closed (a-d) or open (e-h). The S-layer is purple, the cell membrane blue and the VAP is yellow. Scale bars, 200nm

Fig. 4 .

 4 Fig. 4. VAP structure. Tomographic slices through closed (a) and open (b) SIRV2-induced VAPs of S.islandicus and VAPs formed after PVAP expression in E.coli (c), indicating two layers, one of which is continuous with the cell membrane (black arrowheads), while the other (white arrowheads) forms a sheet at the cytoplasmic surface of the membrane. VAPs in S. islandicus protrude trough the S-layer (white arrows). (d-k) 3D map of VAP obtained by subtomogram averaging, with sevenfold symmetry applied. Tomographic slice perpendicular to the pyramid base (d) and successive tomographic slices parallel to the base (e) show the two layers in the walls of the pyramid. (f) Top-view of the 3D map in solid representation shows that the edges of the of the seven pyramidal facets are slightly curved counter-clockwise (dotted lines). (g-k) Different orientations of the 3D map in surface representation. Transparent mesh and golden surface show different threshold levels. Black and white arrowheads indicate outer and inner layer, respectively. Open arrowheads indicate connections between inner and outer layers of the VAP. Scale bars, 200 nm (a-c), 50 nm (d-k).

Fig. 5 .

 5 Fig. 5. VAP formation in S. cerevisiae. PVAP expression in S. cerevisiae causes VAP formation in various cellular membranes. (a,b) Immuno-labelling of unfrozen cryo-sections with anti-PVAP antibodies. (a) VAP in the endoplasmic reticulum. (b) VAPs in mitochondrial membranes (c)

Fig. 6 .

 6 Fig. 6. Expression of PVAP variants in E.coli. Several PVAP truncation mutants were constructed. (a) Schematic representation of PVAP constructs expressed in E.coli. (PVAP) wild type PVAP. (PVAPΔC10) PVAP lacking 10, (PVAPΔC20) 20, (PVAPΔC30) 30, (PVAPΔC40) 40, or (PVAPΔC70) 70 C-terminal residues. (PVAPtmFlk) PVAP construct in which the transmembrane segment is replaced by that of the E.coli membrane protein Flk. (PVAPΔN30) PVAP lacking the Nterminal transmembrane segment. (b) Electron micrographs of thin sections trough E.coli cells expressing PVAP constructs as in A. VAPs (open arrows) and invagination of the membrane (black arrow) are shown; Bars, 200 nm; bar in inset, 100 nm.

Fig. 7 .

 7 Fig. 7. Model of VAP assembly. (a) Predicted secondary structure of PVAP, with N-terminal trans-membrane helix and three short C-terminal α-helices. (b) Schematic top view of the VAP, with its seven facets in different colours. Strong interactions between individual PVAP protomers stabilise the integrity of each facet, whereas weak interactions at the seams (white dashed lines) between two neighbouring facets form predetermined break points. The black dashed indicates the cross section through the VAP in in c. (c) PVAP protomers (green) insert spontaneously into the plasma membrane (blue), with the short C-terminal helices exposed to the cytoplasm. Close contacts between PVAP molecules in each facet (aqua and orange outline) exclude S-layer proteins (grey), and combine into a protein sheet below the plasma membrane.

Acknowledgements

We thank John van der Oost and Alain Jacquier for helpful discussions and Alp Kuckelbir for help with ResMap. DP and TQ acknowledge financial support of L' Agence nationale de la recherche.

WK and BD acknowledge financial support from the Max Planck Society.

Data deposition: The map and fitted model reported in this paper have been deposited in the Electron Microscopy Data Bank (accession no. 5844).

Supplementary

Supplementary Methods

Virus and host strains The SIRV2 virus stock was prepared and the S. islandicus LAL 14/1 strain grown as described previously (1). For analysis of morphological changes upon viral infection, cells were synchronized by dilution of precultures in fresh medium. The cultures were grown for ~12 hours until an OD of 0.1-0.2 was reached and SIRV2 was added directly to the cultures as described (1).

Plasmid constructs and transformation of S. acidocaldarius For the expression of PVAP in S.

acidocaldarius M31, SIRV2_ORF98 (NCBI RefSeq ID: NP_666583) was amplified from SIRV2 genomic DNA and cloned into the lacS gene locus in the pMZ1 plasmid (2) using the NcoI and BamHI sites. SIRV2_ORF98 (PVAP) and the araS promoter were transferred from pMZ1 to the pSVA1450 plasmid (3) using the NcoI and EagI sites, which yielded pTQ26. pTQ26 was transformed to S. acidocaldarius M31. The preparation of competent cells, methylation of the plasmid, and electroporation were carried out as described (4) using a Bio-Rad Gene Pulser Xcell electroplater with 1-mm cuvettes at 1,500 V, 600 Ω, and 25 μF. Selection of PVAP expressing colonies and induction of expression were performed as before (5).

Plasmid constructs and transformation of E.coli For the overexpression of SIRV2_ORF98

(PVAP) C-terminal truncation mutants in E. coli Rosetta(DE3)pLys (Novagen Merck), SIRV2_ORF98 was amplified from SIRV2 genomic DNA with different reverse primers resulting in PCR products of 297 (full length), 267 (-10 AA), 237 (-20 AA), 207 (-30 AA), 177 (-40 AA), 87 (-70 AA) bp. PCR amplification of a 216 bp product starting 81 bp downstream of the ATG resulted in a PVAP mutant lacking the N-terminal transmembrane domain. The same sequence was used for a fusion with the 75 bp transmembrane segment of the E.coli Flk gene, preceded by the 27 N-terminal bp of PVAP. This oligonucleotide sequence was synthesized by GeneArt® (Invitrogen). All PVAP gene mutants were cloned into the T7 promoter-driven expression vector pSA4 using the NcoI and BamHI sites (6). The pSA4 vector contains an isopropyl β-d-1thiogalactopyranoside-inducible promoter that was used for the expression of a C-terminally His-tagged protein. Analysis of PVAP expression cultures by high pressure freezing and freeze substitution was performed as described (5). The location of the Flk-PVAP fusion was analyzed by isolation of membranes as described in ref (11), followed by SDS PAGE, western blotting and immunolabelling with antibodies raised against PVAP as described (5).

Plasmid constructs and transformation of S. cerevisiae

For the overexpression of SIRV2_PVAP (PVAP) in Saccharomyces cerivisiae, SIRV2_PVAP was amplified from SIRV2 genomic DNA and cloned in the expression vector pCM190 (7) using PstI and XbaI sites. S. cerivisiae was transformed with the plasmid according to Gari et al. (7). After selection on plates without uracil a single colony was picked and grown at 30 °C overnight in a pre-culture in selective uracil-free medium, and with 10 microgram/ml doxycyclin. After one day, cells were diluted 1/1000 in medium without doxycyclin.

Immuno-electron microscopy Yeast cells were fixed with 4% paraformaldehyde in 0.1M Hepes, pH 5.4 for 2 h at RT. The cells were then washed with 50 mM NH4Cl in PBS to quench free aldehyde groups and pelleted in 12% gelatin in PBS. The gelatine pellet was solidifed on ice and cut into small blocks, which were infiltrated over night at 4°C with 2.3M sucrose for cryoprotection, mounted on aluminium pins and frozen in liquid nitrogen. Thin sections were cut with a UC6/FC6 (Leica microsystems, Vienna, Austria) and picked up in a 1:1 mixture of 2.3M sucrose and 2% methylcellulose (8). Labelling for PVAP was done as described previously (5).

High-pressure freezing and freeze-substitution E.coli cells were taken up in cellulose capillary tubes (Leica Microsystems GmbH, Vienna, Austria) as described in (9). S. cerevisiae cultures were concentrated by filtration. Tubes or cell concentrates were placed into brass planchettes filled with 1-hexadecen (Agar Scientific, Stansted, United Kingdom). Samples were frozen in a HPM 010 high pressure freezer (Baltec, now Abra Fluid AG, Widnau, Switzerland).

Freeze-substitution was performed in anhydrous acetone containing 2% osmium tetroxide (Merck, Germany). Small cracks were introduced under liquid nitrogen in solid 1-hexadecen by pre-cooled fine point forceps (No 5, Dumont, Switzerland) to allow perfusion of the substitution mix. Freeze-substitution was carried out at -90°C for 24h, and at -60°C and -30°C for 8h each in a freeze substitution device (Leica Microsystems GmbH, Vienna, Austria). Afterwards the temperature was raised to 0°C and the samples were washed with dry acetone and embedded stepwise in EPON. After heat polymerization thin sections were cut with an Ultracut UCT microtome (Leica Microsystems GmbH, Vienna, Austria). Sections were collected on 200 mesh Formvar-coated copper grids and post-stained with 4% uranylacetate and Reynold's lead citrate.

Images were recorded with a JEOL 1010 electron microscope at 80 kV equipped with an Olympus Keen View camera (Olympus Soft imaging systems, Münster, Germany).

Whole cell cryo-tomography S. islandicus cells were harvested shortly after infection, concentrated by low-speed centrifugation (3000 rpm for 10 min) and plunge-frozen directly in the growth medium. For this, cell pellets were resuspended in an equal volume of fresh medium.

E. coli cells overexpressing PVAP were harvested at the same conditions, washed once in 50mM

Tris, 300mM NaCl, pH7 and plunge-frozen in the same buffer. Before freezing, suspensions were mixed with an equal volume of 10 nm colloidal protein-A gold suspension (Aurion, Wageningen, The Netherlands). 3 μl of this mixture were added to a 300 mesh R2/2 glow-discharged Quantifoil grid, blotted and rapidly injected into liquid ethane using a homemade plunge freezer.

Tomograms were recorded with a Polara G2 Tecnai field emission transmission electron microscope (FEI, Hillsboro, USA) operated at 300 kV, equipped with a Gatan Tridiem energy filter and 2x2 k CCD camera (Gatan Inc., Pleasanton, USA). Zero-loss filtered images were collected using the Digital Micrograph software (Gatan Inc., Pleasanton, USA). Tomographic tilt series were recorded with the FEI tomography software (FEI Company, Hillsboro, USA). Tilt series were generally collected in a range of -60° to +60° in steps of 1.5° or 2°, at 6-9 μm defocus and magnifications of 41.000x, 34.000x or 27.500x, corresponding to a pixel size of 0.5766nm, 0.709nm or 1.073nm, respectively. Tomograms were reconstructed using the IMOD software package (10) and de-noised by non-linear anisotropic diffusion (NAD) (11).

Subtomogram averaging For subtomogram averaging of VAPs, 57 pyramid volumes were cut out from a single tomogram of a PVAP overexpressing E. coli cell, aligned and averaged using the PEET software (12,13) as described before (14). The average volumes were averaged further by applying 7-fold rotational symmetry. 3D maps obtained by subtomogram averaging were displayed and analyzed in 3Dmod (IMOD,(34)) or UCSF chimera (15). The resolution of the map was estimated using the ResMap software (16).

PVAP purification

The codon-optimized gene coding for PVAP was synthetized (Genscript) and inserted in to the expression plasmid pET26b (Novagene) using the restriction sites NdeI and XhoI. E. coli BL21DE3/Rosetta/pLysS cells were transformed with the resulting plasmid and transformants were selected using kanamycin (Kan, 50µg/ml) and chloramphenicol (Cam, 34µg/ml). An overnight pre-culture of a single colony was transferred to 12L "terrific broth" (TB) medium containing Kan and Cam and incubated in the shaker at 160 rpm until the optical density at 600nm (OD600) reached 0.8 -1.0. Protein expression was induced with 1mM IPTG (isopropyl--D-1-thio-galacatopyranoside) at 37°C. After 2 hours, cells were pelleted, resuspended in lysis buffer (50 mM Tris pH 7.0, 300 mM NaCl, 0.5 mM PMSF) and disrupted with a Microfluidizer (M-110L, Microfluidics Corp., Newton, MA). Unbroken cells were removed by 30 min. centrifugation at 14,000g. The membrane fraction was pelleted by centrifugation of the supernatant at 100.000g for 90 min. at 4°C ,resuspended in 50 mM Tris pH 7.0, 300 mM NaCl and diluted to a protein concentration of 5 mg/ml. Membranes were solubilized by adding of Nlaurylsarcosine to a final concentration of 1.5%. Non-solubilized protein was removed by centrifugation for 60 minutes at 100.000g. The supernatant containing the His-tagged protein was loaded onto a Ni-NTA column and unspecifically bound proteins were removed in several washing steps. The protein was eluted in buffer containing 500 mM imidazole and concentrated using Amicon spin columns (Millipore) with a 30 kDa cutoff prior to loading onto a gel filtration column (Superdex75). Purified protein was eluted using 50 mM Tris pH 7.0, 300 mM NaCl and 0.05 % N-laurylsarcosine as running buffer.


Doped tin dioxide aerogels as alternative catalyst supports for Proton Exchange Membrane Fuel Cells

). This value is even higher than that of the chosen carbon based electrocatalyst reference, TEC10E40E, measured in the same conditions (23.4 mA. mg Pt

 -1).

The work presented here is funded by the European Union's Seventh Framework Program for the Fuel Cells and Hydrogen Joint Technology Initiative under grant agreement n325239 (FCH-JU project Nano-CAT) and the French National Research Agency PROGELEC programme, (ANR-12-PRGE-007 project SURICAT). It was supported by Capenergies and Tenerrdis.

Figure 1. TEM image of a Pt/Sb-doped SnO 2 aerogel obtained from method A


Many work situations require the use of knee protectors, which are tested according to European Standard EN 14404. The objective of this study was to analyse the relevance of certain parameters for testing bearing force distribution, in particular the three force measurement points, the artificial test knee used, and the limit value of 30 N.

The three sensor positions defined in the standard do not guarantee that the maximum pressure will be measured. The grid of pressure sensors implemented in this study maps the bearing force distribution fully and immediately.

Three artificial knees of different sizes and a simplified knee were defined and compared: depending on the type of knee protector, the force distribution can vary. But, the pressures obtained with the simplified knee are within the range of those recorded with the other knees.

Use of the simplified knee can therefore be considered in place of the currently used test knee, which is difficult to fabricate and is not representative of morphological diversity.

Finally, based on the sample group of protectors evaluated during subjective tests, we can conclude that the 30 N limit specified by the standard is consistent with the degree of comfort felt by the test subjects. iv

Reasons and objectives

Prepatellar bursitis is a periarticular disorder that produces an inflammation of the subcutaneous tissue of the bearing zones of the knee where weight is put during kneeling. It mainly affects people who often work on their knees (floor tilers, floor covering fitters, electricians, roofers, plumbers, etc.). In 2012, about 400 occupational disease cases were declared in France (Table N 57D).

To limit these risks, workers are equipped with knee protectors that are tested according to European Standard EN 14404 [1]. That standard defines the specifications and testing methods for knee protectors. In particular, it gives a limit value for the bearing force at the knee and a test method for measuring it. The test consists in applying a force of 750 N to an artificial test knee equipped with the knee protector to be tested. Three force sensors are disposed on the knee and they make it possible to measure the force distribution, which should not exceed 30 N per sensor (cf. Figure 1).

For the tests, the standard defines a reference knee, whose dimensions were determined by moulding on a human subject. We do not know the criteria that led to that subject being chosen, and, to our knowledge, no validation of that knee was performed to verify that it is representative in terms of force distribution and of bearing contact area. We might question whether a human knee exists that is representative of an average population, and whether it is possible to design a knee with geometrical shapes that are simple and readily reproducible, and with which results can be obtained that are fully relevant.

The standardised method of measuring the force distribution specifies only 3 force sensors located at the most critical places, but that method does not make it possible to obtain sufficient mapping of the force distribution, and in particular to determine the maximum value of the force, in the zone in which the bursitis is likely to develop. For example, if the surface of the protector is designed to reduce the pressure at the 3 sensors, a protector made of a material of poor quality would comply with the standard, despite not procuring satisfactory comfort for users. Given the variability of knees and of the places that are critical for prepatellar bursitis, this study takes into account the maximum force, regardless of the place at which it is applied.

The limit value of 30 N was defined from a consensus between the various participants in the standardisation group on the basis of knee protectors that were deemed by the group to be comfortable, without a study being conducted on subjects to endorse that value.

Metal surface

Artificial test knee

Velocity of descent 50 ± 5 mm/min Knee pads

Three force sensors

The objective of this study is to analyse the relevance of the following test parameters:

the 3 positions of the points at which the bearing force is measured and the measurement means used; -the artificial test knee defined in the standard; and the limit value of 30 N. The relevance of the other parameters of that test is not called into question.

Equipment and method

Prior to the study, and in collaboration with INRS's physiology experts, a bibliographic search was conducted to identify studies on prepatellar bursitis and the means for preventing it [2,3]. All that was found were means for treating the disease. We did not identify any study showing a relationship of cause and effect between the kneeling position without protectors and the appearance of prepatellar bursitis. It appears relevant to consider that the lower and the wider-spread the pressure on the knee, the more limited the risk of prepatellar bursitis will be. But in the absence of scientific data on the topic, validation of the specification of the standard was based on the degree of comfort felt by the test subjects. In the remainder of this document, all of the values given by the force sensors are converted into pressure so that it is possible to compare the results of the various measuring means. Thus, the limit value of 30 N defined by Standard EN14404 and associated with a force sensor whose bearing contact area is constituted by a disk that is 1 cm in diameter corresponds to a pressure of 38 N/cm².

Knee protector

A sample group of 71 protectors was selected to cover a broad range of force distribution and of forms (shaped or otherwise) available on the market. They are made of fabrics, foam or rubber with or without shells (cf. Figures 2 and3). 7

In this study, we considered two main families of knee protectors:

plane homogeneous foams of constant thickness; and shaped knee protectors.

Of the 71 protectors, 10 were selected to be evaluated by subjects. They had to be representative of the forms available on the market and of the various fastening means. They also had to make it possible to propose a force distribution, as measured using the protocol defined by the standard, that was very wide: the maximum pressures lay in the range 15 N/cm² to 145 N/cm².

Relevance of the 3 measurement points

The objective of this phase was to determine the maximum pressure on a sample group of knee protectors, so as to verify whether or not it corresponds to the pressure recorded at one of the positions of the three measurement points defined in the standard.

It was necessary to find means for measuring the force or pressure distribution over the entire bearing contact area of the knee, so that it was possible to establish mapping of the distribution, and to determine the maximum of the force and the position of that maximum. Furthermore, those means needed to be usable not only with an artificial test knee, but also during evaluations with the test subjects.

The means chosen were constituted by a 16 cm x 16 cm grid of pressure sensors that was equipped with 1024 pressure sensors whose sides were 5 mm in length. Those means offered the advantage of operating with the Nowell system that is present at INRS, and that had already been evaluated through other studies.

Measurement with the grid as compared with the measurement method recommended by Standard EN14404 was validated with knee protectors of different designs (insert pads, and shaped knee protectors), by superposing the grid of pressure sensors on the standardised knee as equipped with the three force sensors.

During the comparison of that grid with the 3 force sensors, the maximum pressure was recorded.

Representativeness of the knee

The objective of this phase was to evaluate the influence of the size and of the shape of the knee on the distribution of the bearing force, and thus on the maximum pressure. For this purpose, the following were defined and fabricated:

three knees of different sizes and having shapes representative of human knees; and a simplified knee made up of simple geometric shapes that were easy to represent.

Then, the measurements of the force distribution were taken on knee pads with the 4 knees in succession.

Producing the knees representative of different sizes and the simplified knee

The objective was to define and to make 3 knees that were representative of different sizes: large, medium, and small. For this purpose, the right knees of 40 subjects of varied morphologies were digitized by being scanned. Their heights varied from 1.57 m to 1.90 m, and their weights varied from 51 kg to 126 kg. A theoretical mean knee was computed by taking the mean of the 40 scans. Each scanned knee was then compared with the theoretical mean knee, in order to find the one that was closest to the mean: that knee was then taken as being the representative mean knee and was fabricated.

In the same way, a knee representative of the small sizes was defined on the basis of the 20 smallest scanned knees, and a knee representative of the large sizes was defined on the basis of the 20 largest knees. The dimensions of the simplified knee were defined on the basis of the representative mean knee.

Measurements of the force distribution with the different knees

The force distribution was measured with the 4 knees (small, medium, and large size knees, and the simplified knee) on various foams and knee protectors, with the defined grid of pressure sensors. The other test parameters remained unchanged and were those of Standard EN 14404.

The following were recorded:

the pressure values at the 3 measurement positions defined in the standard: these positions were determined on the new knees in the same manner as on the standardised knee: the position of the 2 nd force sensor, in a central point, is the first surface that comes into contact with the floor; the other 2 positions are on the axis of the leg, on either side of the central point and placed at 2.5 cm therefrom; -the value of the maximum pressure.

Measurements of the force distribution on plane foams

In a first stage, the measurements were taken on the plane and homogeneous foams by applying the force of 750 N regardless of the knee. The pressures recorded with the different knees on the same foam were compared. In a second stage, considering that the bearing force is a function of subject morphology, measurements were taken while modulating the bearing force as a function of knee size: it was kept at 750 N with the large knee and it was adjusted with the other knees until it reached the maximum pressure obtained when measuring with the large knee. The bearing forces obtained with the small and medium sizes and with the simplified knee were compared with theoretical forces reduced in proportion to the mean weights of the subjects. Considering that the force of 750 N applied to the large knee was proportional to the mean weight of the 20 heaviest test subjects, the theoretical forces for the other knees were defined as follows:

for the medium size and for the simplified knee, it was proportional to the mean weight of all of the subjects; -for the small size, it was proportional to the mean weight of the 20 lightest subjects.

Measurements of the force distribution on shaped knee protectors

The measurements were taken on a series of shaped knee protectors, by applying the standardised force of 750 N, in order to compare, for the same protector, the pressures obtained with the different knees.

Validation of the maximum force specified by the standard

The object of this phase was to validate the limit of 30 N specified in the standard or to redefine it on the basis of evaluation by test subjects. The principle consisted in evaluating the degree of comfort afforded by the protectors as felt by the subjects, and, in parallel, in measuring the maximum pressure at the knee. The limit value selected had to be positioned in such a manner that only those knee protectors that were deemed to be comfortable could be deemed to conform.

The protocol and the measurement means had to be defined:

choice and duration of the kneeling positions; -sample group of test subjects; -comfort scale; -number and period of the measurements and notations; and means for measuring the pressure.

Choice of the test position

A kneeling position, representative of a working situation and inducing a maximum weight on the knees, was defined. It had to be stable and balanced (from front to back and from right to left) in order to limit the variations in bearing pressure on each of the knees during the tests, so as to avoid differences during the measurements.

Various postures, shown diagrammatically in Table 1, were evaluated with 10 test subjects. Their heights varied from 1.72 m to 1.85 m, and their weights varied from 64 kg to 96 kg. The distribution of the weight over the knees was measured using the following method:

the subjects were weighed; and they successively took up the positions indicated in Table 1, and each knee of the test subjects was placed on scales (cf. Figure 4) in order to record the distribution of the weight over each of the knees.

Evaluation of the knee protectors

A test protocol was established: it simulated the working conditions in the defined kneeling position. Half of the 10 test subjects were students, and therefore did not have any experience with knee protectors, and the other half were trade professionals using the protectors regularly.

The maximum pressure on the knee was measured and the degree of comfort felt was recorded at 3 different times:

P1 P2 P3 P4 P5 P6
Scales NS 318, April 2014 10 -T1: on kneeling down; -T3: after 3 minutes of kneeling on both knees without moving; -T6: after 6 minutes, with, between T3 and T6 a series of movements defined in the protocol having been made by the test subjects. The degree of comfort felt by the test subjects was evaluated on a scale from 0 (the least comfortable) to 10 (the most comfortable). For measuring the maximum pressure, the above-defined grid of pressure sensors was positioned between the knee of the subject and the knee protector.

Main Results

Validation of the grid of pressure sensors

For each of the three measurement positions defined by the standard (in blue: position 1; in red: position 2; and in green: position 3), Figure 5 shows the correlation between the pressure measurements obtained by means of the three standardised force sensors and the pressure measurements obtained with the grid of pressure sensors, on the entire sample group of the 71 protectors. We can observe that this correlation does not depend on the type of knee protector, or on the position of the force sensor 1, 2, or 3. All three of the coefficients of determination (R²) are greater than 0.9 and all three of the slopes of the linear regression lines are very close to 1 with ordinates at the origin lying in the range 0.9 to -0.5. The correlation between the set of measurements obtained with the three force sensors and those obtained with the grid of pressure sensors is high. The coefficient of determination R² is equal to 0.96. The slope of the line is 0.91 with an ordinate at the origin equal to 1.6. The grid of pressure sensors was thus used for the remainder of the study, i.e. with the various artificial test knees studied and with the human subjects.

Comparison of the maximum pressure

For a sample group of knee protectors, Figure 6 gives the pressure values obtained with the grid of sensors at the three measurement positions defined by the standard, and the maximum value. The maximum value of the pressure was very often higher than the values measured at the three positions defined by the standard.

The protectors numbered from 1 to 17 were plane homogeneous foams of constant thickness, and the maximum pressure was, in general, located in the vicinity of sensor No. 2. On the other knee protectors numbered from 18 to 71, the surfaces had, for example, deep grooves, projections, or holes, and the maximum pressure could be located at positions different from the positions of the measurement points defined by the standard. These curves confirm that the measurement positions specified by the standard do not make it possible to guarantee that the maximum value will be found regardless of the protector. For the majority of the products, their conformity was not called into question: of the 72 protectors tested, 13% were found not to conform when the test protocol defined by the standard was applied, and 18% appeared not to conform when maximum pressure was taken into account. The three protectors for which the conformity is called into question are samples Nos. 12, 69, and 72: they are provided with large grooves for improving their flexibility. If, during the measurement, the knee protector finds itself positioned in such a manner that the groove faces the force sensors, the measured pressure is very low and underestimates the real pressure on the knee. The force recorded at sensor 1 (cf. Figure 1) was systematically lower than the pressure at sensor 2, or than the pressure at sensor 3. The zero pressure obtained in the zone of sensor No. 1 with protector No. 43 is due to its particular shape that eliminates the bearing pressure at the front of the knee and shifts it to the tibia.

Representativeness of the knees

Artificial knees representative of different sizes

The dimensions of the scanned knees varied considerably. On the basis of the scans of 40 real knees, three mean knees representative of large, medium, and small sizes were fabricated. The choice of the 20 smallest and the 20 largest knees was made on the basis of the intercondylar distance. It lay in the range 8 cm to 12 cm. After the three knees had been fabricated, a knee of simplified shapes and that was easy to make was defined for a low cost (less than €150). Figure 7 gives an example of a modelled knee.

Comparative measurements with the 3 knees and with the simplified knee

The comparative maximum pressure measurements as a function of the various knees can be broken down into two main families of protectors. We can observe that the foams conforming to the standard remain conforming regardless of the artificial knee used, and foam No. 1 that did not conform with the large knee remained non-conforming with the other three knees.

Plane foams of constant thickness

For the same bearing force, the maximum pressure was a function of the size of the knee: the smaller the knee, the higher it was. Regardless of the knees, including the simplified knee, the correlation between the maximum pressure values recorded on the various foams with the various knees was high: the coefficient of determination R² was greater than 0.95. The pressures recorded with the simplified knee lay between the pressures recorded with the small knee and the pressures recorded with the medium-size knee.

Measurements were taken while modulating the bearing force as a function of knee size. Table 3 gives the force to be applied to the various different knees in order to obtain the maximum pressure obtained with the large knee with a bearing force of 750 N. firstly, the theoretical forces that should be applied to the knees of different sizes in order to take account of the mean weight of the subjects corresponding to that size; they were computed by taking as a reference the value of 750 N of the standard that corresponds to the large knee (mean weight of 82 kg), and then for the other two knees by applying the rule of three: F = 750 N x Mean Weight / 82; -secondly, the mean force to be applied to the various knees in order to obtain the same maximum pressure. These values are indicated in Table 3. These tables show that if the size of the knee decreases, and even if the bearing force is reduced so as to be proportional to the weights of the corresponding subjects, the reduction in the bearing contact area is such that the maximum pressure recorded is greater than the maximum pressure recorded with a knee of larger size.

Shaped knee protectors a) Maximum pressure

Figures 8 and9 give examples of lines of correlation between the maximum pressures obtained with the various different knees and with the sample group of knee protectors. The correlation between the knees of large and medium sizes and the simplified knee is relatively good: the R² coefficient is approximately in the range 0.8 to 0.9. The slopes of the linear regression lines are close to 1. Conversely, the correlation of the small knee with the other three is poor, the values for R² decrease to 0.6 and the slopes of the linear regression lines can reach 0.4. The different distribution of the bearing force can be explained by the bonier shape of the small knee.

b) Pressures recorded at the positions defined in the standard

In the same way, analyses made it possible to compare the pressures recorded at the 3 measurement positions of the standard using the various different knees.

At the level of sensor 2, the correlations are correct: the R² coefficients vary in the range 0.8 to 0.9. In particular, the slopes a and b of the line of linear regression between the medium knee and the simplified knee are close to 1: the simplified knee gave values comparable to those obtained with the medium-size knee. This good correlation can be explained by the fact that, in general, the position of this sensor corresponds to the first point of bearing contact of the knee.

At the sensors 1 and 3, the correlations are lower: the R2 coefficients vary in the range 0.4 to 0.9. These results can be explained by a distribution of the pressure as function of the respective shapes of the knee protector and of the knee. That was observed in particular on certain knee protectors having a particularly large reinforcement at the tibia so as to limit the bearing pressure at the patella. That leads to the bearing points being shifted, that shift varying from one knee to another, especially with the small knee, which has a more bony and more knobbly shape.

Validation of the specification

The position for performing the subjective tests was chosen on the basis of 3 criteria:

the percentage of the weight on the knee should be as high as possible;

the position should be as stable as possible; and the position should be as repeatable as possible by the various subjects. Position 2 shown in Table No. 1 was selected. It gives the lowest standard deviation, with a weight on the knees of close to 50% of the weight of the individual. It corresponds to the torso and the legs being perpendicular to the floor, to the arms being down along the body, and to the head being up. It should be noted that the value for the bearing force of 750 N that is specified by the standard corresponds to the pressure on the knee of an individual weighing about 120 kg in the down-on-one-knee position as in position No. 5.

Figure 10 shows the comparison of the degree of comfort felt by the test subjects relative to the maximum pressure recorded on their knee for each subject and for the 10 protectors chosen. The vertical lines show the maximum pressure measured on the protector with the test bench specified by the standard. The limit as from which the evaluations of the degree of comfort felt by the subjects were considered as acceptable degrees of comfort was set at 5. That limit was defined arbitrarily. This information was not given to the test subjects.

In order to give them some reference points, the first protector they evaluated gave very good force distribution values using the standardised test and the second they evaluated gave very poor ones. The subjects were free to evaluate them on a scale of 0 to 10. The protectors P1, P5, and P7 that were considered to be poor by the test subjects gave, with the knees of the subjects, pressure values that were high, lying in the range 19 N/cm² to 58 N/cm² (58 N/cm² being the maximum pressure it is possible to measure with the grid of pressure sensors) and means that lay in the range 38 N/cm² to 53 N/cm². The measured pressures were thus consistent with the degree of comfort felt by the subjects, who evaluated them as being uncomfortable. Furthermore, when those three protectors were measured on the test bench using the force sensors defined by the standard, they gave maximum pressures lying in the range 85 N/cm² to 145 N/cm² and thus they do not conform to Standard EN 14404.

The 6 knee protectors P2, P3, P4, P6, P8 and P9 were considered to be comfortable by the subjects. The pressures measured with the knees of the subjects were less than 38 N/cm² except for one measurement with a subject that was 50 N/cm². The values obtained on the standardised test bench were also less than 38 N/cm². The protector P10 does not conform to Standard EN 14404 even though it was evaluated as comfortable by the test subjects. The maximum pressure measured during the subjective tests was 37 N/cm², i.e. a value less than the limit of 38 N/cm². The pressure measured during the standardised test was considerably higher than the pressure recorded on the knees of the subjects. The standardised force of 750 N corresponds to the force on the knee of an individual weighing about 120 kg in the kneeling down on one knee position, which is an extreme situation that was not reached during the tests on the test subjects.

To recapitulate:

-6 knee protectors conforming to the standard were deemed comfortable by the subjects; -3 knee protectors deemed uncomfortable by the subjects did not conform to the standard; and -1 knee protector deemed comfortable did not comply with the specifications of the standard.

Discussion and conclusions

The objective of this study was to analyse the relevance of the parameters for testing force distribution that are defined in Standard EN 14404, in particular:

the positions of the three points at which the bearing force is measured; -the artificial test knee defined in the standard; and a limit value of 30 N, for force sensors whose measurement area is constituted by a disk having a diameter of 1 cm.

This study shows that, for the plane and homogeneous foams, the maximum pressure is often situated at sensor No. 2. Conversely, whenever the protectors have shapes that, for example, include grooves or projections in order to shift the bearing points, the maximum pressure can be situated outside the 3 measurement positions defined by the standard.

The difference between the maximum pressure and the measurements taken at the standardised positions may be considerable depending on the shapes of the knee protectors: it is therefore advantageous to have the possibility of measuring that pressure at any point over the knee. The grid of pressure sensors used for this study makes it possible to procure immediate mapping of the force distribution.

Three knees representative of large, medium, and small sizes were defined on the basis of scans of 40 real knees. After the three knees had been fabricated, a knee of simplified shapes (cf. Appendix 1) and that was easy to make was defined for a low cost.

Comparing the maximum pressures obtained with the three representative knees and with the simplified knee gave a good correlation when the measurements were taken on plane foams, whereas the correlation became relatively poor with protectors having a variety of shapes. Generally, when the same bearing force is applied to knees of different sizes, the smaller the knee, the higher the pressure is. This can be explained by the smaller bearing contact area when the size of knee is smaller. When the size of the knee decreases, and even if the bearing force is reduced so as to be proportional to the weights of the corresponding subjects, the reduction in the bearing contact area is such that the maximum pressure recorded is greater than the maximum pressure recorded with a knee of larger size. Relatively speaking, the measurements with a smaller knee and a lower bearing force give pressures higher than measurements with a larger knee and a higher bearing force. We can thus note that it was not relevant, in the normalised method, to take a knee of very large size that distributes the pressures better than a smaller knee does, but, conversely, the specified bearing force of 750 N would appear to be higher than the real weight bearing on the knee. The pressures obtained with the simplified knee are within the range of those recorded with the other knees, and in particular with the medium size knee. Use of the simplified knee may therefore be considered in place of an artificial knee that is difficult to fabricate and that is not representative of morphological diversity.

A comfort evaluation on a sample group of 10 knee protectors was conducted by 10 test subjects in order to validate the limit specified by the standard. That evaluation made it possible to arrive at the general conclusion that the specified limit of 30 N for a force sensor whose bearing contact area is constituted by a disk having a diameter of 1 cm is consistent: it made it possible to distinguish between the protectors recognised as comfortable and appropriate by the test subjects.

Figure 1 :

 1 Figure 1: diagrammatic view of the bench for measuring the force distribution.

Figure 2 :

 2 Figure 2: insert pad.Figure3: shaped protector.

Figure 4 :

 4 Figure 4: kneeling position for measuring the distribution over each of the 2 knees.

Figure 5 :

 5 Figure 5: comparison of the grid of pressure sensors with the force sensors defined in the standard.

Figure 6 :

 6 Figure 6: comparison of the maximum pressure with the pressures at the positions defined by the standard.

Figure 7 :

 7 Figure 7: example of a modelled knee.

Figure 8 :

 8 Figure 8: correlation between the maximum pressures with the knees of large and medium sizes.

Figure 9 :

 9 Figure 9: correlation between the maximum pressures with the knees of medium and small sizes.

Figure 10 :

 10 Figure 10: comparison between the degree of comfort felt and the maximum pressure.

Table 1 : diagrammatic representation of the kneeling positions.

 1

Table 2

 2 gives the maximum pressure recorded during the measurements on the different foams with the knees of different sizes and with the simplified knee while applying the force of 750 N.

	13

Table 2 : maximum pressures obtained with the different knees for a bearing force of 750 N.

 2

Table 3 : comparison of the bearing forces on the different knees for an equal maximum pressure

 3

Table 4 gives :

 4:

Table 4 : comparison between the theoretical forces to be applied to the different knees and the force giving the same maximum pressure.

 4

			Mean weight	Theoretical	Mean force applied
			(kg)	force	(N)
				(N)
	Small knee	20 smallest subjects	61	550	433
	Medium knee	All of the subjects	71	650	616
	Large knee	20 subjects largest	82	750	750

Appendix 1

Figure 15: drawing of the simplified knee.


Fanny Ruhland

Violette Chiara

Marie Trabalon

Age and egg-sac loss determine maternal behaviour and locomotor activity of wolf spiders (Araneae, Lycosidae)

Age and egg-sac loss determine maternal behaviour and locomotor activity of wolf spiders (Araneae, Lycosidae)

Introduction

Pardosa saltans spiders' behaviour (in frequencies) during exploration: (A) virgins (n = 14) and mothers carrying a 0-, 10-or 15-day old egg-sac (n = 57); (B) virgins and mothers after egg-sac loss; (C) virgins and mothers after contact and hanging of egg-sac. Frequencies were compared using χ 2 tests: different letters indicate a significant difference at p < 0.05. Latencies to egg-sac retrieval (sec), duration of egg-sac manipulation (sec), and latency of egg-sac hanging (sec) (n = 57). Latencies were compared using ANOVAs: different letters indicate a significant difference at p < 0.05.

FiguresFig. 2

 2 Figures

  Fig. 2.

Fig. 3 .

 3 Fig. 3. Locomotor activity of virgins (n = 14) and mothers (n = 57) carrying their egg-sac (Test 1), after lost their egg-sac (Test 2) and after egg-sac hanging (Test 3): (A)

Fig

  Fig. 4.