molecular self - assembly , 
 a spontaneous association of disordered 
 components into an ordered supramolecular structure , is responsible 
 for the formation of complex biological systems and is becoming increasingly 
 important in material sciences striving to develop novel biomaterials 
 with a great diversity of biochemical and physical properties .
very 
 little is known about the mechanisms underlying the emergence of ordered 
 structures from disordered components .
recently , a remarkable diversity 
 of polyhedra that self - assemble through excluded volume interactions 
 at high packing fractions into entropic crystals with various degrees 
 of crystalline order was reported . at 
 the opposite spectrum of a high packing fraction
is protein aggregation , 
 which typically occurs in vivo at nanomolar and in vitro at micromolar 
 concentrations and must be consequently facilitated by attractive 
 intermolecular interactions .
aberrant protein aggregation is at the 
 core of many age - triggered diseases , such as alzheimer s , parkinson s , 
 and huntington s disease , amyotrophic lateral sclerosis , type 
 ii diabetes , systemic amyloidoses , and others .
these amyloid proteins do not share any obvious aspects of the primary 
 structure yet they self - assemble into cytotoxic low - molecular weight 
 oligomers and form fibrils with a common cross- structure .
inherent toxicity of amyloid assemblies that 
 cause the disease implies a common assembly mechanism ; however , assembly pathways are not well understood .
here , we introduce a minimal model of self - assembly 
 that unifies common features of protein amyloidogenesis and can be 
 meaningfully mapped onto sequences of amyloid proteins . because the 
 model is not protein specific
, the model predictions extend to self - assembly 
 systems beyond the aggregation of amyloidogenic proteins .
a self - assembly model with a minimal 
 number of beads and interparticle interactions is constructed as following . 

a one - bead molecule does not allow for the implementation of both 
 attractive and repulsive interactions simultaneously . a two- or three - bead 
 molecule is either anisotropic or planar , which imposes a 
 priori geometric restrictions on self - assembling molecules . 

a three - dimensional molecule can be formed by a minimum of four beads , 
 which we place at the four vertices of a tetrahedron ( figure 1a ) .
each tetrahedron molecule thus represents a 
 molecule ( protein monomer ) comprising four beads each of a diameter d connected by four covalent bonds of identical lengths d ( figure 1a , b ) .
each bead within 
 a tetrahedron molecule is assigned either an attractive ( hydrophobic ) 
 or repulsive ( hydrophilic ) character , resulting in three possible 
 model variants : ( i ) one hydrophilic and three hydrophobic beads , ( ii ) 
 three hydrophilic and one hydrophobic bead , and ( iii ) two hydrophobic 
 and two hydrophilic beads .
consistent with discrete molecular dynamics 
 ( dmd ) requirements ( see below ) , the effective intermolecular hydrophobic 
 attraction among hydrophobic beads and effective intermolecular hydrophilic 
 repulsion among hydrophilic beads of different molecules are modeled 
 by square - well potentials with a depth eh and a height ep , respectively , ( figure 1c ) . a hydrophobic and a hydrophilic bead
the ratio of the two potential energies , 
  = ep / eh , is hereafter referred to as the hydropathy ratio . in variant ( i ) , 
 molecules aggregate into a single densely packed quasi - spherical assembly 
 ( data not shown ) . in variant ( ii ) , molecules form stable dimers and 
 trimers that do not assemble further ( data now shown ) . in variant 
 ( iii ) ( figure 1a ) , molecules exhibit complex 
 assembly dynamics , which we examine for several values of the hydropathy 
 ratio . definition of the tetrahedron model .
( a ) a tetrahedron molecule 
 comprises two hydrophobic ( red ) and two hydrophilic ( blue ) beads of 
 a diameter d located at the vertices of a tetrahedron 
 with the edge length d. ( b ) an  infinite  
 square - well intramolecular potential models covalent 
 bonds among the four beads in the tetrahedron molecule .
( c ) effective attractive ( red ) and repulsive ( blue ) intermolecular potentials between pairs of hydrophobic and hydrophilic 
 beads , respectively .
the distance rmin = d corresponds to a sum of the van der waals radii 
 of the two interacting beads and rmax = 
 2.22d is the interaction range distance .
the strengths 
 of the effective hydrophobic attractive and hydrophilic repulsive 
 potential are denoted by eh and ep , respectively .
when interparticle potentials 
 are approximated by a square - well or a combination of square - well 
 potentials , molecular dynamics can be reduced to an efficient , event - driven 
 discrete molecular dynamics ( dmd ) .
more recently , several 
 low , intermediate , and high resolution protein models combined with 
 dmd provided important insights into protein folding and assembly . here
, we apply dmd to examine pathways of self - assembly of 
 the tetrahedron model . in each simulation ,
1000 tetrahedron molecules 
 occupy the simulation box with an edge length of 111d , which corresponds to a volume fraction of 2.80  0.13  
 10 .
we examine self - assembly at five hydropathic 
 ratios  = ep / eh = 0 , 0.5 , 0.75 , 1 , 1.25 .
for each  , 1000 tetrahedron 
 molecules are placed into a cubic lattice in the simulation box , followed 
 by high - temperature ( kbt  eh ) dmd simulations , which 
 produce five distinct initial ensembles of randomly distributed non - interacting 
 tetrahedron molecules .
thus obtained ensembles , five for each of the 
 five  values , resulting in 25 ensembles in total , are then 
 used as initial configurations for the production runs . here , a configuration 
 is defined as an ensemble of coordinates of all tetrahedron molecules 
 in a given trajectory at a given simulation time .
production runs 
 are 10  10 (  = 0 ) or 50  10 (  > 0 ) simulation steps long , and configurations are recorded 
 every 10 simulation steps , resulting in 5  1000 
 = 5000 configurations (  = 0 ) or 5  5000 = 25  000 
 configurations (  > 0 ) . the time scale of one simulation 
 step , t , can be expressed in terms of the 
 spatial resolution , x  d  10 m ( thus approximating a protein 
 by a tetrahedron with a circumsphere 
 radius of 1 nm ) , and thermal energy using the equipartition 
 theorem , where we chose 
 for the mass of one bead mb  10 kg , equivalent 
 to a sequence of 50 amino acids or a  200-residue 
 protein corresponding to each tetrahedron molecule .
we use the thermal 
 energy at a physiological temperature 310 k , kbt  4  10j , resulting in t  
 16  10 s and the total simulation time per 
 trajectory at nonzero  values ( 50  10 simulation 
 steps ) of 8  10 s. note that the 
 unit of length , d , and the mass of each bead , mb , can be adjusted to the size and molecular 
 mass of the protein of interest , which consequently affects the time 
 scale , t  d(mb ) , such that without considering 
 the actual protein sequences , self - assembly of a larger protein occurs 
 on a longer time scale than self - assembly of a smaller protein .
the assembly 
 size is defined as the number of molecules within a connected cluster . 

a recursive algorithm is used to identify tetrahedron molecules that 
 belong to each cluster .
a tetrahedron molecule is identified as a 
 part of the cluster if any of its four beads is at a distance 2.22d from at least one of the beads of the molecules within 
 the cluster . to derive three - dimensional plots displaying the time 
 evolution of assembly sizes for each trajectory ,
assembly size distributions 
 are calculated by using a binning interval of 5  10 simulation steps .
the probability distribution of assembly sizes 
 is calculated by using a 10 simulation steps - wide window , 
 resulting in 100 ensembles , each comprising 1000 tetrahedron molecules 
 in various assembly states .
then , the average over the five probability 
 distributions is taken , and the corresponding standard error of the 
 mean ( sem ) values are calculated .
if an assembly size appears only 
 once in a single trajectory , the corresponding sem value is set to 
 zero . by moving the 10 simulation steps - wide window along 
 the assembly trajectories between 0 and 50  10 simulation 
 steps ,
time evolution of the size distribution for each  is 
 derived . to quantify the degree of orientational 
 order within the assemblies of different sizes acquired for different 
  values ,
each molecule in the assembly is characterized 
 by an axis that connects its two hydrophobic ( red ) beads .
the angle 
  between the axis of each molecule and the long axis of the 
 elongated assembly , determined by the principal component analysis , is then used to calculate s as the average:1over all molecules 
 in the assembly .
note that 
 due to periodic boundary conditions , some assemblies are split in 
 two or more clusters , which are separated by the linear size of the 
 cubic simulation box in either of the three directions .
to accurately 
 calculate the order parameter , these split clusters are merged into 
 a single cluster prior to the order parameter calculation .
the order 
 parameter calculation is performed for each assembly of size larger 
 than 5 within 5000 (  = 0 ) or 25  000 (  > 0 ) 
 recorded 
 configurations , each containing 1000 tetrahedron molecules . for each 
 ensemble of assemblies of a size 6 , the average order parameter 
 and sem values
are calculated by taking into account the order parameter 
 values of all assemblies of a given size . for assemblies comprising 
 100 or more tetrahedron molecules ,
a running window along the assembly 
 axis containing 50 molecules is used to calculate the local order 
 parameter , followed by an average over all local order parameters .
here a minimal model of self - assembling molecules , 
 each characterized 
 by a combination of attractive and repulsive interactions , is examined . 

generally , any molecule prone to aggregation is characterized by attractive 
 and repulsive interactions due to van der waals and electrostatic 
 interactions .
we adopt in the following protein folding and aggregation 
 terminology when discussing the origin of attractive and repulsive 
 interactions . in proteins ,
attractive and repulsive interactions stem 
 from the hydropathic ( polar versus nonpolar ) and electrostatic ( charged 
 versus uncharged ) nature of individual residues .
we posit that in 
 addition to the effective hydrophobic attraction , the effective hydrophilic 
 and/or electrostatic repulsion due to solvation of polar side chains 
 by water molecules plays a critical role in prediction of protein 
 assembly pathways and structures .
we then construct a tetrahedron 
 model with two hydrophobic and two hydrophilic beads as described 
 in the model construction section ( methods ) , 
 in which the hydropathy ratio  = ep / eh between the repulsive ep and attractive eh potential 
 energies is introduced as the only model parameter ( figure 1 ) .
self - assembly of the tetrahedron model 
 is studied in the absence 
 of repulsion among the hydrophilic beads (  = 0 ) and at four 
 different hydropathy ratios (  = 0.5 , 0.75 , 1 , and 1.25 ) . to 
 avoid a comparison of the assembly pathways of proteins with vastly 
 different solubilities , the energy scale eh is adjusted for each  to keep the final monomer concentration 
 within the range of 1016% ( table 1 ) . 
 for each  ,
five trajectories of self - assembly from initially 
 monomeric ensembles of 1000 tetrahedron molecules are acquired and 
 analyzed . for each trajectory ,
time evolution of assembly sizes is 
 monitored and displayed as a three - dimensional plot ( figure 2 ) , in which each assembly is also characterized 
 by an order parameter as described in the order 
 parameter section ( methods ) , measuring a degree of the orientational 
 order of molecules within the assembly ( figure 2 ) .
below , we refer to small , quasi - spherical assemblies as oligomers 
 to distinguish them from elongated , curvilinear assemblies ( protofibrils ) 
 and more ordered assemblies ( fibrils ) .
three - dimensional plots 
 showing probability distributions p(n ) of different assembly sizes n as a function of 
 the simulation time t during the process of assembly 
 starting from an ensemble of spatially separated ( monomeric ) tetrahedron 
 molecules . three 
dimensional plots for each of the five trajectories , 
 labeled as s1 , s2 , s3 , s4 , and s5 , for each of the five  values 
 are displayed .
the average order parameter s that 
 characterizes each assembly is color coded as shown on the color scale .
the potential 
 energy eh , associated with the effective 
 hydrophobic 
 attraction , which represents a unit energy in our model is adjusted 
 for each hydropathy ratio  to yield comparable steady - state 
 monomer concentration ( solubility ) . for each  , steady - state 
 monomer , dimer , and trimer concentrations and their sem values
are 
 calculated by averaging over monomer , dimer , and trimer concentrations 
 for each trajectory within 910  10 steps 
 (  = 0 ) or 4950  10 steps (  
 > 0 ) , followed by averaging over the five trajectories .
figure 3a shows time evolution of assembly 
 sizes for four representative 
 trajectories corresponding to  = 0 , 0.5 , 1 , and 1.25 , respectively . 
 at 
= 0 , the initial hydrophobic collapse driving the assembly 
 occurs rapidly , bringing the monomer concentration from the initial 
 100% to the final value of 14% ( table 1 ) . within the first 10 simulation steps ( figure 4a )
, initial monomers rapidly assemble into an aggregate 
 comprising of 800900 molecules , which remains in a steady - state 
 equilibrium with monomers ( figure 3a , left ) 
 and small populations of dimers and trimers ( table 1 ) .
all assemblies have a relatively low order parameter as 
 color - coded in figure 3a ( left ) .
the assembly 
 proceeds through formation of quasi - spherical oligomers , further elongating 
 into larger curved tubules , which collapse onto themselves , exposing 
 nonhydrophobic ( blue ) and shielding hydrophobic ( red ) beads from the 
  solvent  , and eventually form a large quasi - spherical 
 aggregate with an overall porous morphology ( figure 3a , right ) . the high bending propensity of these aggregates 
 is reflected in their low order parameter for all five trajectories , 
 which display almost identical assembly pathways ( figure 2 ) .
three - dimensional 
 plots showing 
 the assembly size distribution or probability ( z - axis ) 
 of all assembly sizes ( x - axis ) at different times 
 ( y - axis ) for representative trajectories ( left ) and 
 their assembly pathways with characteristic morphologies ( right ) for 
 simulations at ( a )  = 0 ; ( b )  = 0.5 ; ( c )  = 1 ; 
 and ( d )  = 1.25 .
the color code in ( a ) is used to characterize 
 the order parameter of each assembly in the three 
note that the conformations are not displayed on the same scale : 
 the final aggregates along assembly pathways for  = 0 , 0.5 
 ( a , b , right ) are considerably larger than those for  = 1 and 
 1.25 ( c , d , right ) , however , their lateral dimensions are comparable . time evolution of assembly size distributions . 

the average assembly 
 size distributions at  equal to ( a ) 0 , ( b ) 0.5 , ( c ) 1 , and 
 ( d ) 1.25 steadily evolve from the initial to the later assembly stages . 

note that for  = 0 , the size distribution reaches steady state 
 within the initial 10 simulation steps .
the initial assembly stage lasts 10  
 20  10 simulation steps ( figure 3b , left ) , which is an order of magnitude longer than the hydrophobic 
 collapse observed for  = 0 . during this initial stage ,
smaller 
 quasi - spherical oligomers coexist with larger elongated aggregates 
 and a rapid elongation of large aggregates is mostly driven by oligomer 
 addition .
the assembly size distribution averaged over all five trajectories 
 ( figure 2 ) reveals a large number of oligomers 
 comprising 3070 tetrahedron molecules forming within 10 simulation steps ( figure 4b , black 
 curve ) .
these oligomers then coalesce into larger assemblies comprising 
 100140 tetrahedron molecules at 12  10 simulation steps ( figure 4b , blue curve ) . 
 after this initial assembly stage ( at 20  10 simulation steps ) ,
oligomers disappear , marking the onset of the 
 late assembly stage . in the late assembly stage , the elongated aggregates 
 either increase or decrease in length at a rate distinctly lower than 
 the aggregation rate of the initial assembly stage , while persisting 
 in a steady - state equilibrium with the monomer population and minor 
 populations of dimers and trimers ( table 1 ) . 

large aggregates at this late stage increase or decrease in size predominantly 
 by monomer association and dissociation .
in addition to this predominant 
 growth dynamics , two additional albeit less frequent processes are 
 noted : ( a ) merging of two elongated assemblies into a single larger 
 elongated assembly and ( b ) breaking of an elongated aggregate into 
 two smaller aggregates , which continue to either increase or decrease 
 in length . the second process of aggregate breakage results in smaller 
 assemblies of various sizes ( figure 4b , cyan , 
 green , orange , and red curves ) .
the morphology of the assemblies observed 
 in simulation at  = 0.5 differs from those observed for  
 = 0 .
all assemblies at  = 0.5 have a considerably larger order 
 parameter than those observed in simulations at  = 0 .
the order 
 parameter of assemblies increases with their size ( figure 3b , left ) .
elongated aggregates are considerably 
 less prone to bending than those observed for  = 0 , but display 
 kinks and can be slightly bent ( figure 3b , 
 right ) . unlike trajectories at  = 0 , trajectories at  
 = 0.5 display significant variations ( figure 2 ) .
a 
 two - stage assembly process observed for  = 0.5 is absent from 
 simulations at  = 1 .
all trajectories acquired at  = 
 1 reach a steady state rapidly within 10 simulation steps . 

the assembly pathways appear to be characterized by a single assembly 
 growth rate ( figure 2 ) .
this rate is slightly 
 lower than the one characterizing the initial assembly stage but notably 
 larger than the elongation rate observed in the late assembly stage 
 at  = 0.5 .
importantly , the assembly dynamics at  = 
 1 is dominated by assembly breakage , which critically reduces the 
 size of the largest aggregate .
consequently , the largest aggregate 
 in the trajectory shown in figure 3c ( left ) 
 comprises 400 tetrahedron molecules .
time evolution of the 
 assembly size distributions reveals an abundance of quasi - spherical 
 oligomers made of 2030 molecules that form within the first 
 10 simulation steps ( figure 4c , 
 black curve ) .
the corresponding peak in the assembly size distribution 
 decreases and broadens at longer simulation times ( figure 4c , blue , cyan , green , orange , and red curves ) . unlike 
 in simulations at 
= 0.5 , where the oligomer assemblies disappear 
 after the initial assembly stage , the assemblies of 2030 
 molecules persist at all simulation times , and give rise to assemblies 
 of 60  80 and 100  120 tetrahedron 
 molecules , respectively ( figure 4c , red curve ) . 

this multimodal character of the assembly size distribution indicates 
 that larger assemblies form through coalescence of smaller oligomers 
 rather than through monomer addition .
the steady  state pool 
 of assemblies smaller than 120 is a cause of a rapid growth 
 of larger aggregates .
a large breakage rate characteristic for  
 = 1 results in a broad distribution of assembly sizes up to 550 
 ( figure 4c , green , orange , red curves ) .
similar 
 to simulations at  = 0 and  = 0.5 , molecules self - assemble 
 initially into smaller quasi - spherical oligomers , followed by formation 
 of elongated aggregates .
some trajectory - to - trajectory 
 variability is observed at  = 1 , reflected in a variable size 
 of the largest aggregate ( figure 2 ) .
simulations at  = 1.25 are characterized by a steady state 
 dynamics , which is reached within 10 simulation 
 steps ( figure 3d , left ) .
the rate of breakage 
 is significantly higher than at  = 1 , thus abolishing aggregates 
 larger than 120 .
consequently , a steady - state assembly size 
 distribution centered at the assembly size 25 with additional broader 
 and less significant peaks centered at sizes 50 and 90 , 
 respectively , is observed ( figure 4d , red curve ) . 

the assembly proceeds through formation of smaller oligomers , followed 
 by formation of elongated aggregates without any notable kinks or 
 bending ( figure 3d , right ) .
no significant 
 trajectory - to - trajectory variability is observed ( figure 2 ) .
the most variable assembly kinetics is observed 
 for 0.5  
   1 , where the competition between the effective hydrophobic 
 attraction and hydrophilic repulsion combined with thermal fluctuations 
 results in a complex competing dynamics , as demonstrated by simulations 
 at  = 0.75 , which display the largest trajectory - to - trajectory 
 variability ( figure 2 ) .
assembly dynamics shows 
 characteristics of a two - stage aggregation process ( observed for  
 = 0.5 ) as well as a relatively high breakage rate ( observed for  
 = 1 ) .
unique to simulations at  = 0.75 is a simultaneous occurrence 
 of the fast and slow assembly dynamics , where one aggregate displays 
 a fast growth through oligomer addition and the other aggregate displays 
 a slow dynamics through monomer association and dissociation ( figure 5a , black double - sided arrow ) . although the fast 
 aggregation is more frequent earlier in the assembly process , it is 
 sporadically observed at later assembly stages as well , depending 
 on the amount of oligomers available for coalescence with larger elongated 
 aggregates ( figure 2 ) .
figure 6 shows a merging process , during which the smaller assembly 
 attaches itself to the end of the larger aggregate ( c ) , temporarily 
 forms a kink ( d ) , and is subsequently integrated into the larger aggregate , 
 eventually eliminating the kink ( f ) .
( a ) a three - dimensional 
 plot showing the assembly size distribution or probability ( z - axis ) of all assembly sizes ( x - axis ) 
 at different times ( y - axis ) of the assembly process 
 at  = 0.75 .
( b ) assembly size distributions at several early 
 ( top ) and later ( bottom ) assembly stages are displayed .
( c  e ) 
 the assembly pathway and corresponding morphologies observed in simulations 
 at  = 0.75 . ( c )
initially , tetrahedron molecules form small 
 quasi - spherical oligomers , which further assemble into elongated protofibril - like 
 aggregates .
finally , a large elongated assembly with distinct structural 
 domains ( i  iv ) , including a sandwich - like ordered structure 
 ( ii ) , emerges .
( d ) a closeup of the four distinct domains with the 
 respective cross sections , displaying distinct degrees of lateral 
 ordering .
( e ) time evolution of the sandwich - like ordered structure 
 ( ii ) .
( a  c ) 
 large elongated multi - domain assembly and a small aggregate ( inside 
 a green circle ) that are initially spatially separated , ( d  f ) 
 merge into a single aggregate .
the time frames correspond to ( a ) 42.33 
  10 ; ( b ) 42.34  10 ; ( c ) 42.35 
  10 ; ( d ) 42.36  10 ; ( e ) 42.38 
  10 ; and ( f ) 44.37  10 simulation 
 steps .
note that within the 2  10 simulation steps 
 between the last two time frames ( e and f ) , the initially formed kink 
 between the two merged assemblies smooths out as the smaller assembly 
 integrates into the structure of the larger aggregate . in the early assembly stage , we observe oligomers 
 comprising 30 
 molecules ( figure 5b , top graph ) , which gradually 
 decrease in abundance at later assembly stages at the expense of assemblies 
 comprising up to 150 molecules as well as considerably larger aggregates 
 comprising up to 750 molecules ( figure 5b , bottom graph ) .
this broad distribution of assembly sizes is caused 
 by breakage , which is more frequent than for  = 0.5 but less 
 frequent than for  = 1 .
assembly into smaller quasi - spherical 
 oligomers is followed by formation of elongated curvilinear protofibrils 
 that eventually evolve into large multi - domain aggregates , separated 
 by kinks ( figure 5c ) .
figure 5d shows four structurally distinct domains of a polymorphic 
 aggregate that forms following the pathway described above .
domains 
 i , iii , and iv are cylindrically symmetric yet display variable degrees 
 of lateral order from the most disordered ( iv ) to the most ordered 
 ( i ) , as shown in the corresponding profiles ( figure 5d , bottom ) .
domain ii , which is not cylindrically symmetric , 
 appears to be the most ordered of the four domains and displays a 
 sandwich - like profile .
molecules in domain ii are arranged into a 
 double layer of 68 molecules in width .
this highly structured 
 domain emerges from a less ordered domain and spreads to neighboring 
 regions in the course of the simulation ( figure 5e ) .
the assembly kinetics in our model is driven by a competition between 
 spontaneous aggregation and disaggregation , controlled by the hydropathic 
 ratio  , which affects the assembly size distribution , breakage 
 rate , and assembly morphologies .
the orientational order parameter 
 of assemblies increases with the assembly size for each  ( figure 7a ) .
in addition , pentacosamer ( 25-mer ) conformations 
 are more ordered and less prone to bending as  increases ( figure 7b ) .
( a ) the average order parameter as a function of the assembly size 
 for the five different hydropathy ratios . for each assembly 
 size , the order parameter is calculated for a window containing a 
 connected cluster of 50 tetrahedron molecules and then averaged over 
 all windows along the assembly axis .
( b ) characteristic pentacosamer ( 25-mer ) conformations 
 at different values of . complex 
 and diverse self - assembly dynamics observed for 0.5 <  < 
 1 results in multi - domain aggregates , consistent with in vitro observed 
 molecular - level polymorphism of amyloid fibrils , with bends and kinks ( figure 8a ) 
 that strongly resemble the morphology of in vitro amyloid fibers ( figure 8b ) .
amyloid proteins assemble into cross- 
 fibrils , stabilized by intermolecular hydrogen bonding along the fibril 
 axis . yet
, protein aggregation can occur in the absence 
 of hydrogen bonding . in sickle cell anemia ,
sickle hemoglobin aggregates 
 into polymers , causing a characteristic sickle - cell shape of red blood 
 cells .
whereas free energy calculations 
 need to account for hydrogen bonding , our model reveals that a multi - stage 
 assembly process that proceeds through formation of quasi - spherical 
 and elongated mesoscopic - size aggregates , which eventually undergo 
 a structural conversion into polymorphic aggregates , occurs also in 
 the absence of hydrogen bonding .
( a ) a long in 
 silico aggregate before ( upper ) and after ( lower ) the breakage occurring 
 at the position marked by black arrows ( dmd simulations at  
 = 0.75 ) .
cross - sections to the left and to the right of the breakage 
 with mismatched morphologies are displayed at the bottom .
( b ) transmission 
 electron micrograph of a42 fibers characterized by several 
 kinks ( green arrows ) and breakages ( red arrows ) .
the multi - stage assembly mechanism 
 observed in our model is consistent 
 with previously reported theoretical and experimental findings on 
 amyloidogenic proteins .
a recent theoretical study postulated both 
 association and breakage processes to predict a variety of distinct 
 aggregation pathways in agreement with experimental data .
experimental studies on amyloid -protein 
 ( a ) , one of the most studied proteins due to its relevance 
 to alzheimer s disease , which is the leading cause of dementia 
 in elderly , showed that its two predominant alloforms , a40 
 and a42 , form oligomers , which further assemble into protofibrils 
 and finally convert into fibrils .
the existence 
 of a nucleated conversion from prefibrillar into fibrillar assemblies 
 was recently confirmed through flash monitoring of cys - cys - a 
 aggregation .
a structural conversion 
 from within a molten collapsed intermediates into a structurally ordered 
 fibril was also shown to be consistent with experimental data on yeast 
 sup35 prion protein .
a tetrahedron molecule in our model represents 
 a monomer that retains its tertiary structure throughout the assembly 
 process . as such ,
it can be identified as an aggregation - prone monomeric 
 state of a natively unfolded protein . whereas the internal degrees 
 of freedom associated with intrinsically disordered nature of natively 
 unfolded protein monomers are not incorporated in our model
, the existence 
 of such disorder implies that most amino acids in the sequence will 
 be for some time exposed to the solvent , analogous to the four beads 
 in the tetrahedron model .
consequently , it is reasonable to map the 
 ratio of hydrophilic to hydrophobic amino acids , np / nh , in the sequence of natively 
 unfolded amyloidogenic proteins to the hydropathy ratio  of 
 our model . in this mapping , we implicitly assume that hydrophobic 
 collapse dominates protein self - assembly and that hydrogen bonds , 
 which are strongly directional , form among peptide regions only after 
 these regions have been driven into proximity by effective hydrophobic 
 interactions
. this mapping also does not distinguish between charged 
 and noncharged hydrophilic amino acids and thus all charged hydrophilic 
 amino acids effectively contribute to repulsive interactions .
in aqueous 
 solutions , electrostatic interactions among charged amino acids depend 
 on ph and ionic concentrations that may under specific conditions 
 modify the overall repulsive to attractive interaction ratio . however , 
 most protein sequences are dominated by noncharged amino acids and 
 salt bridges between oppositely charged amino acids tend to be exposed 
 to the solvent and rarely stabilize protein structures in water . thus , attractive electrostatic interactions are mostly not expected 
 to significantly contribute to inter - residue interactions during protein 
 self - assembly . by employing several hydropathy scales previously 
 reported by chothia , janin , kyte and 
 doolittle , eisenberg et al . , and engelman et al .
, we identified the hydrophilic and hydrophobic amino acids within 
 each scale , counted the number of hydrophobic and hydrophilic amino 
 acids in the sequence , and calculated np / nh for several natively unfolded proteins , 
 known to form amyloid fibrils ( table 2 ) .
a 
 histogram of the np / nh values from table 2 demonstrates 
 that the np / nh values are distributed in the range 0.5 < np / nh < 2 ( figure 9 ) .
the highest np / nh values are found for the human prion , prion - like yeast - sup35 , 
 and tau ( table 2 ) .
histogram of the ratio of hydrophilic to hydrophobic number 
 of residues , np / nh , in the sequence of natively unfolded amyloidogenic proteins .
the ratio of hydrophilic to hydrophobic 
 amino acids ( np / nh ) for each protein is calculated within each of the five distinct 
 hydropathy scales .
proteins are identified by their names and/or i d 
 codes from protein data bank or universal 
 protein resource .
for the three proteins 
 without the i d codes , the source references are cited next to the 
 protein name in the first column .
the length of proteins is given 
 in terms of the number of amino acids in the sequence ( third column ) . 

the averages over the five np / nh values corresponding to each of the five hydropathy 
 scales for each protein are shown in the last column .
the average 
 and standard deviation of the average np / nh values per protein of the last column 
 are displayed at the bottom right ( bold font ) .
high values of  > 1 in 
 our model result in a high breakage 
 rate and relatively short and ordered aggregates .
prion proteins are 
 known to form brittle aggregates , prone to breakage , which is associated 
 with their infectious nature .
substantial 
 evidence indicates that tau protein displays prion - like characteristics 
 by initially aggregating in a few nerve cells in discrete brain areas 
 and then self  propagating and spreading to distant brain regions .
most np / nh values from table 2 fall into the range 0.5 < np / nh < 1 , for which our model predicts complex 
 and diverse assembly dynamics with a broad assembly size distribution . 

these proteins include a40 and a42 associated with alzheimer s 
 disease , amylin associated with diabetes mellitus type 2 , transthyretin 
 associated with systemic amyloidosis , -synuclein associated 
 with parkinson s disease , and huntingtin associated with huntington s 
 disease .
interestingly , a42 , which aggregates faster and forms 
 larger assemblies than a40 under equivalent in vitro conditions , 
 has a lower np / nh value of 0.59 than a40 with np / nh = 0.66 , consistent with our 
 model s maximal assembly size decreasing with  in the 
 range 0.5 <  < 1
. for   0.75 , assembly 
 breakage in our model creates a steady - state pool of oligomers , which 
 form from smaller and merge into larger assemblies . assuming that 
 oligomers are the key toxic species responsible for pathogenesis , 
 even if lifetimes of oligomers are short , the assembly dynamics that 
 produces a constant pool of oligomers is a persistent source of toxicity , 
 which provides a plausible mechanism through which a metastable assembly 
 with a short lifetime might exert toxicity over a longer time period .
in summary , our tetrahedron model 
 predicts complex self - assembly 
 into quasi - spherical oliogmers , curvilinear protofibrils , and multi - domain 
 aggregates , characteristic of in vitro observed amyloid formation 
 by natively unfolded amyloidogenic proteins .
spontaneous assembly 
 breakage and merging , assembly size distributions , and a diversity 
 of assembly morphologies in our model are controlled by a single parameter , 
 the hydropathic ratio  , which can be meaningfully mapped onto 
 the hydrophilic to hydrophobic ratio of the sequence of amyloidogenic 
 proteins .
we show that most natively unfolded amyloidogenic proteins 
 are characterized by 0.5 <  < 1 , for which our model 
 predicts the most complex and diverse assembly dynamics .
there 
 is no consensus on the size and structure of toxic oligomers . 
 in alzheimer
s disease , different a pre - fibrillar species 
 were reported in vitro and in vivo , depending on the methodologies 
 and experimental settings . moreover , 

relatively minor changes in the sequence between a40 and a42 
 result in distinct assembly pathways and 
 oligomer structures that likely underlie 
 distinct cytotoxic properties of the two peptides , with a42 
 that is genetically more strongly associated with alzheimer s 
 disease than a40 . similarly , a single amino acid mutation of 
 a can cause earlier onset of the disease and/or altered pathology .
a seeming contradiction between two sides of 
 the protein aggregation puzzle : ( a ) a large class of proteins with 
 variable sequences that aggregate into amyloid fibrils and ( b ) small 
 changes in the sequence and/or solvent conditions that significantly 
 alter the assembly pathways and structures , can be reconciled within 
 our model by noting that both sequence and solvent modifications alter 
 an overall hydropathic nature of the protein , thus  , in the 
 range , where the assembly dynamics is the most sensitive .
short 
 hydrophobic peptides incubated with a42 were recently 
 reported to inhibit a42-induced toxicity in cell cultures and subsequent computer simulations of a42 
 assembly in the presence of these inhibitors elucidated the structure 
 of the resulting large amorphous hetero - oliogmers . by binding hydrophobic peptides to a42 , the overall 
 hydropathic ratio of hetero - assemblies ,  , is lowered to  
  0.5 , thereby biasing aggregation toward presumably less toxic 
 amorphous aggregates .
the hydropathy parameter might thus be critical 
 for a deeper understanding of protein aggregation dynamics and developing 
 drugs that aim to alter aggregation pathways in a way to minimize 
 the damage caused by cytotoxic oligomeric assemblies .
while the discussion 
 described above is specific to protein aggregation , our model and 
 its predictions are applicable to a wide range of self  assembling 
 systems and elucidate a critical role of the repulsive to attractive 
 intermolecular interaction ratio in supramolecular structure prediction .
 molecular 
 self - assembly is ubiquitous in nature , yet prediction 
 of assembly pathways from fundamental interparticle interactions has 
 yet to be achieved . here 
 , we introduce a minimal self - assembly model 
 with two attractive and two repulsive beads bound into a tetrahedron . 
 
 the model is associated with a single parameter  defined as 
 the repulsive to attractive interaction ratio . 
 we explore self - assembly 
 pathways and resulting assembly morphologies for different  
 values by discrete molecular dynamics . 
 our results demonstrate that 
  governs the assembly dynamics and resulting assembly morphologies , 
 revealing an unexpected diversity and complexity for 0.5  
  < 1 . one of the key processes that governs the assembly 
 dynamics is assembly breakage , which emerges spontaneously at  
 > 0 with the breakage rate increasing with . the observed 
 assembly 
 pathways display a broad variety of assembly structures characteristic 
 of aggregation of amyloidogenic proteins , including quasi - spherical 
 oligomers that coassemble into elongated protofibrils , followed by 
 a conversion into ordered polymorphic fibril - like aggregates . we further 
 demonstrate that  can be meaningfully mapped onto amyloidogenic 
 protein sequences , with the majority of amyloidogenic proteins characterized 
 by 0.5   < 1 . 
 prion proteins , which are known to 
 form highly breakage  
 prone fibrils , are characterized by  
 > 
 1 , consistent with the model predictions . 
 our model thus provides 
 a theoretical basis for understanding the universal aspects of aggregation 
 pathways of amyloidogenic proteins relevant to human disease . as the 
 model is not specific to proteins 
 , these findings represent an important 
 step toward understanding and predicting assembly dynamics of not 
 only proteins but also viruses , colloids , and nanoparticles . 
Introduction
Methods
Results
Conclusions and Discussion