Many diseases are correlated with the presence of ordered fibril-like structures, so called amyloids, that are formed by mis-folded proteins. For instance, in the case of Alzheimer's disease are these amyloids, here made of Ab peptides and tau-proteins, extensively studied [1] [2] [3] [4] and found in the brain. However, amyloid deposits are not always localized. For example, in Serum Amyloid A (AA) amyloidosis is the damage to tissues and organs spread all over the human body. 5, 6 This systemic amyloidosis, which besides humans also appears in more than 50 other species, 7, 8 is connected with overexpression of serum amyloid A (SAA), as sometimes occurring in inflammatory bowel disease, rheumatoid arthritis, or certain cancers. 9, 10 As a consequence of the primary disease can the concentration of acute-phase SAA be 1000 times higher than seen under healthy conditions. 11 The resulting elevated risk for misfolding and aggregation leads in some cases to AA amyloidosis as a secondary disease that adds to the pathology of the primary disease and complicates its treatment.
In its active form, the 104-residue long SAA assembles into a hexamer and is involved in cholesterol transport and the regulation of inflammation. 12 On the other hand, SAA fibrils are not formed by the full-sized protein, but are associations of shorter fragments that are derived after enzymatic cleavage. The most commonly found fragment is SAA . We have studied in previous work 13 the conditions that lead to decay of the hexamer, cleavage of the monomers, and their subsequent misfolding. We could propose a mechanism for downregulating SAA activity in case of overexpression that relies on the cleavage of the protein as a way to shift the equilibrium from hexamer to the monomer. 13 At neutral pH is a structure (coined by us helix-weakened) dominates that can be easily degraded lowering quickly the SAA concentration. However, the N-terminal helix has in helix-weakened configurations a higher risk to unravel, and to release the first eleven residues. These N-terminal residues are known to be amyloidogenic [14] [15] [16] can then form a strandlike segment that in turn may nucleate amyloid growth. If acidic conditions raise this risk, the equilibrium therefore shifts toward the second motif (termed by us helix-broken) where the Nterminal residues stay cached in a helix, making the SAA 1-76 fragment more difficult to degrade, but also less aggregation-prone.
The above described mechanism assumes a competition between fast degradation of SAA fragments and their tendency to form amyloids. Amyloidogenic conditions are countered by a shift to a more stable and less easily degradable form, often avoiding amyloidosis in cases where the primary disease causes overexpression of SAA. However, the equilibrium between amyloids and free SAA fragments depends not only on the relative frequency of the two fragment motifs but also on the stability of the final fibrils. Hence, possible treatment options could involve either stabilization of the N-terminal segment in the SAA 1-76 fragments (decreasing the risk to nucleate amyloid formation), or destabilizing the fibrils. The present work is motivated by the second avenue and explores the factors that drive fibril formation and stabilize the fibril geometry.
The kinetics of fibril formation is difficult to probe in either experiments or in silico.
Computationally more accessible 17, 18 is the stability analysis of fibrils which, however, requires a structural model of the fibril. The human SAA fibril model was only recently resolved by electron cryo-microscopy, 19 and is deposited in the Protein Data Bank (PDB) under identifier 6MST. Because of experimental difficulties only the fragment SAA 2-69 (shown in Figure 1 ) was resolved, and here only residues 2-55 have a defined structure while residues 56-69 are disordered. The ordered part of the human SAA fibril adopts a right-hand twist, with each chain forming seven β-sheets, β1-β7. The N-terminal region is made of about 20 residues and is stabilized by hydrophobic interaction involving six hydrophobic residues (F3, F5, F10, M16, Y17, and A19).
Unlike the N-terminal region, the C-terminal region of human SAA fibril is stabilized by the ionic interactions among the charged residues (D23, R25, E26, K34, D43, and R47) buried inside a Cterminal cavity. Two proto-fibrils are packed together, with the packing interface stabilized by ionic, hydrophobic, and polar cross-stack interactions. The center of the interface is a so-called "steric zipper", formed by the packing of β3-strands (residues Y29 and I30) of chains on opposite proto-fibrils. A salt-bridge connecting the N-terminus of a chain in one layer of a protofibril with residue D33 of a chain of the same layer in the opposite proto-fibril is also stabilizing the packing interface, 19 see Figure 1 . The tight packing of the protofibrils makes us wonder whether packing interactions between two chains may already lead to a stable nucleus, i.e., whether fibril formation starts with nucleation and elongation of single protofibrils, or whether both protofibrils grow out of such dimers kept together by packing interactions. Hence, the first question that we pose is that for the smallest stable fibril fragment, i.e., the nucleus which starts fibril growth, and whether this fragment is stabilized by stacking or packing interactions. As it is known that acid conditions further SAA fibril formation, we ask as a second question how acidic conditions affect the fibril stability. Finally, in our third line of research we study the role of the first eleven residues (an amyloidogenic region) and of the residues 56-76 ( a disordered region ) for fibril stability. We study the later region because SAA 1-76 (and not SAA 1-55 ) is the most commonly found fragment in human SAA fibrils. Hence, it appears to be likely that the disordered region, which we assume to extend to residue 76, plays also a role in fibril formation and fibril stability. 19 Analyzing our molecular dynamics simulations we conclude that the critical size for fibril stability is a two-fold two-layer tetramer. We conjecture that stacking within the same fold precedes packing between two folds. Acidic conditions are most important at early stages of fibril formation.
By altering the salt-bridge network for the C-terminal cavity, they encourage the stacking of chains, initiated by N-terminal regions of different chains forming anchors. As shown in our previous work, 13 these N-terminal residues are more easily released from the N-terminal helix in the native structure once the C-terminal residues 65-76 become disordered. However, the disordered C-terminal tail does not affect by itself the stability of the SAA fibril geometry.
Models used in our simulations are generated from the cryo-EM structure as deposited in the Protein Data Bank (PDB) under identifier 6MST. The fibril structure is made of SAA 2-55 fragments, i.e., without the first residue which we also do not consider in our investigation. The model is a two-fold six-layer system, from which we have successively deleted chains in the bottom and top end layer to obtain a sequence of models with two-fold three-layers (F2L3), twofold two-layers (F2L2), and two-fold one-layer (F2L1). By deleting one of the folds, corresponding single-fold models are generated, i.e., one-fold three-layers (F1L3), one-fold twolayers (F1L2) , and one-fold one-layer (F1L1) systems. In order to test the role of the first eleven N-terminal residues we generated also a series of truncated models (F1L1-,F1L2-,F1L3-,F2L1and F2L2-) where these residues where deleted from the chains. In another series of models (F1L1A, F1L2A, F1L3A, F2L1A, F2L2A, F2L3A) we accounted for the effect of acid conditions in an approximated way by altering the protonation states of residues E9, E26, H37 to the one expected at pH=4 while these protonation states correspond in the parent models to pH=7. Each of the single-fold models is placed in a cubic box of edge size ~ 9.6 nm and filled with ~ 28000 water molecules, while for two-fold models the box has an edge size of ~ 10.4 nm and is filled with ~ 37000 water molecules. Since most fibrils taken from patients are made from the larger SAA(1-76) fragments, we have in addition generated one model where the residues 56-76 were added. These residues are not resolved in the cryo-EM structure indicating that that this segment is unstructured. We generated this specific model only after we had determined first the two-fold two-layer F2L2 as the smallest stable system. Hence, our extended model F2L2+ is also a twofold two-layer system, with the added additional residues 56-76 assuming a coil configuration.
The extended model is again placed in a cubic box of edge size of ~ 17.5 nm that is filled with ~ 74000 water molecules. All simulated models are listed in Table 1 . Our simulations rely on the GROMACS 2018.2 software package. 20 We use the CHARMM36m force field to describe interactions between atoms, and TIP3P for modeling water. 21, 22 The LINCS and SETTLE algorithms are used to constrain bond length and water geometry. 23 using Amber 12 package and its MMGBSA tools. 29, 30 Results and Discussions
In order to study the stability of the recently resolved SAA fibril model, we start by determining the critical size of the fibril, that is the size above which the fibril is stable over extended periods of time while below this size the fibril will decay on the time scale of our simulations.
Representative structures from simulations of various fibril fragments built from SAA 2-55 -chains are shown in Figure 2 . The simulations were done at neutral pH and the configuration taken at the as nucleus for fibril growth. However, it may serve as a metastable state on the way to forming the two-fold two layers system F2L2, the smallest assembly seen in our simulation that is stable enough to nucleate fibril elongation. Figure 3a and 3b. The differences between the observed stabilities are small, and systems larger than the critical size preserve their fibril structure on the time scale of our simulations, i.e., can be considered to be stable, see also the representative final configurations in Figure 2d and 2g. The critical size determined by our simulation is also consistent with previous measurements of the Gibbs free energy needed for dissociation of two protofibrils. Using PDBePISA it was found that a protofibril has to have at least two layers in order for this dissociation energy to be positive. 19 
The situation is different at pH=4,. We have simulated such acidic conditions in a simplistic way by protonating the sidechains of Glutamic acid and Histidine. While the fibril stability changes similarly with number of folds and layers as in the simulations done at neutral pH, we observe differences. For instance, the root-mean-square-fluctuations (RMSF) of the residues in Figure 5a is smaller for the one-fold two layers system F1L2A than the one for the corresponding system at pH=7, namely F1L2. Differences are most prominent for residues 12 -37 and for the Cterminal cavity formed by residues 45 -55. The lower root-mean-square fluctuations imply a higher stability of this assembly than seen under neutral condition. These differences in RMSF and stability are not seen when comparing the two-fold two-layer F2L2A tetramer with the corresponding system at neutral pH, F2L2, see Figure 5b . Hence, while acidic conditions do not raise the stability of fully-formed fibrils, see Figure 5c and 5d, we conjecture that they enhance fibril formation by making it easier to form a nucleus. This is because at pH=4 leads the stacking of two chains to a dimer that is more stable than it is at pH=7. While at neutral pH only a few long-range side-chain contacts (between F3 and M24) contribute to stability, under acidic conditions are additional contacts found in the C-terminal cavity region.
Example are the contacts between residue E26 and the group of residues K46, R47 and G48; and the contacts between the group of residues R25, E26 and K34 with D43. Note especially the role of the glutamic acid E26 which becomes protonated at pH=4. This allows the residue to form now multiple sidechain hydrogen bonds and salt-bridges in the C-terminal cavity (residues 23 -51) that further stabilize the meta-stable dimer. For instance, at neutral pH salt-bridge interactions between E26 and K34 (8 % occupancy) or E26 and K46 (7 % occupancy) connect the glutamic acid weakly with the two groups of residues 47-K46-D43 and residues Y35-K34-D33-Y29, see Figure 6g ). However, at pH=4 interacts E26 preferentially with D43 enabling now also an sidechain hydrogen bond interaction between R25 and D43 (8 % occupancy), and a salt-bridge interaction between K34 and D33 (7 % occupancy). As R25 forms also a contact with D23 (with occupancy of 50 %), and D43 with R47 (with 27 % occupancy), the two groups of residues D23-R25-E26 and R47-K46-D43 become connected, and form now together with the residue group Y35-K34-D33-Y29 a network of intra-chain side chain hydrogen bonds and salt bridges, see Figure 6h . The stronger intrachain contact network in the C-terminal cavity change with lowering the pH, increasing the structural stability of the chain folds.
On the other hand, the pH-dependence of fibril stability disappears once the critical nucleus size is reached or exceeded. This can be seen, for instance, in Figure 6d and 6e, where the contact map for two-fold two-layer tetramer F2L2 (pH=7) and F2L2A (pH=4) are shown. At pH=7 exist in the tetramer already a multitude of contacts within both the N-terminal and C-terminal regions of SAA chains, and as a consequence, there is little difference in the overall stability when going from neutral to acidic conditions. This picture is similar for larger assemblies, which indicates that once a critical size is reached the already at pH=7 existing intra-chain and inter-chain contacts suffice (unlike for the meta-stable F1L2 dimer) to maintain the stability of the fibril. Hence, acidic conditions do not change the stability of fibrils but encourage their formation by stabilizing the early-stage on-pathway meta-stable F2L2 dimers. 
The above discussion has already pointed out the importance of inter-and intra-molecular interactions involving the N-terminal residues for maintaining SAA fibril stability. Our observation is consistent with previous experimental studies, which showed that the truncation of the first eleven residues prevents SAA fibril formation. Hence, in order to understand the role of these residues in more detail, we have also considered truncated fragments where the N-terminal residues are missing, and compared their stability with that of the parent fragments. These 6c) and its rigidity implies a higher stability of these fibril models. This is also seen in a binding energy (as approximated by MMGBSA calculation) that with -70.8 (20.5) kcal/mol has a larger magnitude than the non-truncated system (59.0 (20.9) kcal/mol). These results appear to contradict experimental observations that SAA cannot form fibrils when the first eleven residues are truncated. However, our results can be reconciled with the experiments if one assumes that the SAA fibril form by a process where the N-terminal regions serve as anchors, that ease the stacking of the rest of chains. Once the stacking of two chains is completed, they will be stabilized not only by interactions between N-terminal residues, but in addition also by such involving the C-terminal residues, see Figure 6c and 6f. Our hypothesis implies that existing fibrils will not dissolve if the first eleven residues are cleaved, but that fragments derived from such fibrils with truncated chains cannot nucleate fibril growths and elongation. While going beyond the scope of this work, this could in principle be tested in suitable seeding experiments. We remark that in ~ 15 % of conformations of the single-fold double-layer N-terminal-truncated F1L2-dimer simulations, the region spanned by residues 12-21 in a chain folds back into a helix, see Figure 8a . This refolding is not observed for the full-sized F1L2 dimer.
Hence, while the more rigid C-terminal region is stabilizing the fibril geometry, stacking contacts between layers involving residues 12-21 get lost without the N-terminal eleven residues, leading to an overall loss of stability and degradation of the SAA fibril. This can be seen in Figure 3c and 3d for the N-terminal truncated two-fold models F2L1-and F2L2-, which despite the additional intra-chain contacts in the C-terminal cavity (see Figure 5f ) are less stable than the corresponding models F2L1 and F2L2 built from full-sized chains. The loss of stability is also seen in the MMGBSA-derived binding energies which with -7.7 (8.6) kcal/mol for F2L1-and -8.9 (7.0) kcal/mol are much smaller in magnitude than the corresponding energies for the nontruncated F2L1 (-40.7 (11.2) kcal/mol) and F2L2 (-55.2(28.2) kcal/mol). Hence, the truncation of the N-terminal residues encourages break-up of the two protofibrils which when isolated lack the stabilizing packing contacts and easily decay, see the representative structure in Figure 8b . surprisingly a stabilization of metastable structure. This stabilization is also seen in the MMGBSA-derived binding energy approximations which with -123.5 (28.6) kcal/mol exceed the value of -59.0 (20.9) kcal/mol seen for the SAA 2-55 fragment. This stabilization is consistent with a role of the C-terminal tail in the early stages of fibril formation but not for the stability of the fully formed fibril. We speculate that the elongated disordered C-terminal tail decreases the chances that the amyloidogenic first eleven N-terminal residues refold into the N-terminal helix, therefore stabilizing the fibril geometry at early stages. 13 Figure 8 . Representative structure of largest cluster seen in simulations under neutral conditions of the truncated models F1L2-(a) and F2L2-(b). Configurations from all three trajectories where clustered with a cut-off of 3 Å of systems at pH 7, and the dominant cluster contains 35% N-and C-terminus are marked by blue and red spheres, respectively. The β-strands are colored in yellow, helices in purple, turns in cyan and coil regions in gray.
SAA in its native structure is stabilized by intra-chain interactions, especially the salt-bridge between residues 1R and 9E, which stabilizes the N-terminal amyloidogenic region and prevents misfolding and aggregation. On the other hand, both packing of chains by the N-terminals and a network of ionic interactions and hydrogen bond inside the C-terminal cavity stabilize the fibril structure. We have identified as the critical size of fibrils a tetramer with two folds and two layers, which we conjecture is formed by first stacking two chains on each other, before two such dimers pack together into a two-fold structure anchored by their N-terminal regions. The critical role of the N-terminus explains the experimental observation that without the first eleven residues SAA does not aggregate into fibrils. We speculate that amyloid formation requires these residues to unravel from the N-terminal helix seen in the native structure. In earlier work 13 we have shown that this process is aided by a disordered C-terminal region, which however, itself is not contributing to fibril stability. Once the N-terminal residues are no longer part of a helix, the now exposed segment can interact with a segment from another chain, forming a β-anchor that will be the starting point of fibril formation. While our stability analysis therefore gives already some hints on the underlying dynamics, more elaborated simulations are needed to establish in detail the mechanism of SAA fibrilization.