Computational modeling of the relationship between amyloid and disease

Abstract

Amyloid is the name given to a special type of linear protein aggregate that exhibits a common set of structural features and dye binding capabilities. The formation of amyloid is associated with over 27 distinct human diseases which are collectively referred to as the amyloidoses. Although there is great diversity amongst the amyloidoses with regard to the polypeptide monomeric precursor, targeted tissues, and the nature and time course of disease development, the common underlying link of a structurally similar amyloid aggregate has prompted the search for a unified theory of disease progression in which amyloid production is the central element. Computational modeling has allowed the formulation and testing of scientific hypotheses for exploring this relationship. However, the majority of computational studies on amyloid aggregation are pitched at the atomistic level of description, in simple ideal solution environments, with simulation time scales of the order of microseconds and system sizes limited to 100 monomers (or fewer). The experimental reality is that disease-related amyloid aggregation processes occur in extremely complex reaction environments (i.e. the human body), over time scales of months to years with monitoring of the reaction achieved using extremely coarse or indirect experimental markers that yield little or no atomistic insight. Clearly, a substantial gap exists between computational and experimental communities with a deficit of ‘useful’ computational methodology that can be directly related to available markers of disease progression. This review will place its focus on the development of these latter types of computational models and discuss them in relation to disease onset and progression.

Introduction

Our aim is to discuss recent progress in the computational modeling of a diverse range of amyloid related diseases collectively known as the amyloidoses (Pepys 2006; Norrby 2011; Jucker and Walker 2011; Merlini and Bellotti 2003). Although this topic has recently been reviewed from a number of different perspectives (Friedman 2011; Straub and Thirumalai 2012; Hall 2008; Caflisch 2006; Cohen et al. 2011), we have focused our review on heuristic computational models that describe a specific link between the time course of amyloid formation and the progression of disease (Masel et al. 1999; Hall and Edskes 2009). With this bias raised and noted, we have structured our review in the following four parts. In the first section, “Background on amyloidosis”, we examine the amyloidoses, describing their history, classification and modes of pathogenesis. In the next, “Background on amyloid”, we provide an introduction to what is known of the chemical and physical properties of amyloid along with its involvement with disease. In the third section, “Computational modeling of amyloid’s relationship to disease”, we describe a general class of heuristic models that discuss amyloidosis in relation to the time course of amyloid formation placing a special focus on two of our computational modeling papers. In the fourth and final section, “The way forward”, we identify gaps in the literature and suggest methods for using computation for advancing our understanding of the role amyloid plays in disease.

Background on amyloidosis

Amyloidosis is the name given to a range of diseases characterized by the deposition of a class of aggregated protein, known as amyloid, into various tissues and organs (Sipe et al. 2010; Pepys 2006). The phenomenon of amyloidosis was scientifically recognized in the early 1800s (Rokkitansky 1842); however, first usage of the term amyloid was made by Virchow in 1854 (Virchow 1854) when he falsely attributed a positive iodine staining test exhibited by amyloid laden liver to an overabundance of starch (amylum—Latin for starch). In the early 1900s, it was demonstrated by Benhold (Bennhold 1922) that the dye, Congo Red, could be used as a universal identifier of amyloid under autopsy. Using such histopathological stain-based approaches, amyloid deposits were subsequently found in heart, intestines, tongue, spleen, liver, brain, kidneys, lungs, adrenal glands, lymphatic system, and skeletal muscle (Symmers 1956). Electron microscopy studies of ex vivo amyloid samples conducted in the 1950s (Cohen and Calkins 1959) revealed the fundamental structural unit of the amyloid deposit to be the amyloid fibril—a one-dimensional form of protein aggregate (Fig. 1). X-ray fiber diffraction studies conducted on patient-derived fibers revealed them to have a cross-β-sheet structure, in which the individual strands of each β-sheet run perpendicular to the fibril axis while the β-sheets run parallel to the fiber axis (Eanes and Glenner 1968). Molecular analysis of the contents of ex vivo amyloid deposits (Glenner 1980; Glenner and Wong 1984; Prusiner et al. 1984) revealed that the dominant component was a single type of protein, unique for that disease sub-type (Pepys 2006; Sipe et al. 2010). A recent report from the International Society for Amyloidosis describes 27 different proteins capable of forming disease-related amyloid in humans and a further 6 types of protein that can form intracellular ‘amyloid-like’ deposits associated with disease (Sipe et al. 2010). Nearly all amyloid deposits associated with disease have also been found to contain minor amounts of accessory proteins such as SAP (serum amyloid P protein1). In certain instances, amyloid deposits may also become mineralized (Quintana et al. 2007). Modern imaging methods for detecting amyloid in the living patient range from whole body scintigraphy (Hawkins et al. 1988; Gillmore et al. 2010), dual energy X-ray analysis (Howard et al. 2012), ultrasound methods (Koyama et al. 2002; Onur et al. 2012), and amyloid probe-based MRI (Higuchi et al. 2005; Onur et al. 2012) /PET scanning methods (Klunk 2011). Development of monoclonal antibodies directed against various forms of amyloid and amyloid-associated components (such as SAP) (Leliveld and Korth 2007; Pul et al. 2011) have significantly expanded the range and scope of sero- and histopathological analysis of samples derived from human biopsy/autopsy. Taken together, these methods have improved our ability to detect amyloid formation, in some cases prior to display of disease-related symptoms (Klunk 2011).

Fig. 1.

Insight into amyloid structure: a Typical Transmission Electron Micrograph (TEM) image of amyloid fibers (in this case formed from insulin) visualized at ×60,000 magnification (scale bar 100 nm). Fibers are typically long and thin with a length distribution in the range of nm AND μm and width distribution in the range 4–20 nm. b Single amyloid proto-filaments are formed as a result of polypeptide units forming intermolecular β-sheets along the long axis of the fiber. The example diagram shows a rectangular box representation of a proto-filament which is formed by a polypeptide with two stacks of β-sheet. c Proto-filaments can undergo self-association to form ‘mature fibrils’ which are typically helical or lateral arrangements of multiple proto-filaments

Classification of amyloidosis disease is made according to a number of different measures (Rostagno et al. 2010; Pepys 2006; Sipe et al. 2010). Traditionally amyloidosis was (somewhat vaguely) grouped as being either primary or secondary in nature based on whether or not an alternative predisposing cause could be found; i.e., if the initiation of amyloidosis was not induced (primary) or alternatively was induced (secondary) by another illness (Symmers 1956; Kyle and Bayrd 1975). Modern molecular approaches towards disease nomenclature pin their classification on the identity of the protein monomer. In this schema, each particular amyloid and associated disease is designated by use of the capital letter A (for amyloid) followed by a ∼1- to 4-letter acronym of the protein monomer, e.g., APro for amyloidosis derived from formation of amyloid from the Prolactin protein (Pro) (Sipe et al. 2010) (see Table 1). The amyloidoses may be further classified as either localized (for example, IAPP in type II diabetes) or systemic (for example, transthyretin amyloidosis) dependent on whether or not the amyloid deposits are restricted to a single tissue/organ type or whether these deposits are dispersed throughout the body (Biewend et al. 2006; Pepys 2001). They may also be profitably grouped as either spontaneous (acquired) or hereditary based upon whether the disease has a heritable genetic origin defined by a mutant allele (Pepys 2006; Rostagno et al. 2010). The final classification of the amyloidoses we will discuss is their division into transmissible or non-transmissible diseases (Moreno-Gonzalez and Soto 2010), whereby some types of amyloidosis (prion diseases) are capable of being passed from one organism to another via ingestion of tainted food product (Gibbs et al. 1980) or by introduction through surgical procedure (iatrogenic) (Norrby 2011). The designation of non-transmissibility obviously relates to a transmission route, so ruling out transmissibility is often less straightforward than the converse case (Westermark and Westermark 2009; Polymenidou and Cleveland 2011); however, with this said, most of the amyloidosis diseases exhibit a noted lack of infectious character. In closing this section, we note that the above classifications are not mutually exclusive with some amyloidosis sharing multiple designations.

Table 1.

Amyloid and amyloid-like proteins related to human amyloidosis

Protein abbreviation	Protein precursor	Localized/systemica	Hereditary	Disease
AL	Immunoglobulin light chain	S,L	N	Primary systemic amyloidosis, AL amyloidosis
AH	Immunoglobulin heavy chain	S,L	N	AH amyloidosis
Aβ2M	β2-microglobulin	S	N	Hemodialysis-related amyloidosis
ATTR	Transthyretin	S	Y,N	Familial amyloid polyneuropathy, Senile systemic amyloidosis
AA	Serum amyloid A	S	N	Secondary systemic amyloidosis, Familial Mediterranean fever
AApoAI	Apolipoprotein AI	S	Y	Familial amyloid polyneuropathy III
AApoAII	Apolipoprotein AII	S	Y	Hereditary renal amyloidosis
AApoAIV	ApolipoproteinAIV	S	Y	Renal medullary amyloidosis
AGel	Gelsolin	S	Y	Familial Finnish amyloidosis
ALys	Lysozyme	S	Y	Hereditary systemic amyloidosis
AFib	Fibrinogen α-chain	S	Y	Heriditary renal amyloidosis
ACys	Cystatin C	S	Y	Hereditary cerebral amyloid angiopathy
ABri	ABriPP	S	Y	Familial British dementia
ALect2	Leukocyte-chemotactic factor	S	Y	Renal amyloidosis
ADan	ADanPP	S	Y	Familial Danish dementia
Aβ	Aβ protein precursor (AβPP)	L	Y,N	Alzheimer’s disease
APrP	Prion protein	L	Y,N	Transmissible spongiform encephalopathies
ACal	Procalcitonin	L	Y	Medullary thyroid carcinoma
AIAPP	Islet amyloid polypeptide	L	Y	Type II diabetes
AANF	Atrial natriuretic factor	L	Y	Atrial amyloidosis
APro	Prolactin	L	Y	Aging pituitary prolactinomas
AIns	Insulin	L	Y	Insulin-related amyloidosis
AMed	Lactadherin	L	N	Senile thoracic aortic amyloidosis
AKer	Kerato-epithelin	L	Y	Corneal amyloidosis
ALac	Lactoferrin	L	Unknown	Familial subepithelial corneal amyloidosis
AOaap	Odontogenic amelobast-associated protein	L	Y	Calcifying epithelial odontogenic tumors
ASemI	Semenogelin I	L	Y	Senile seminal vesicle amyloidosis
b, c	α-Synuclein	L	Y,N	Parkinson’s disease
b, c	Huntingtin	L	Y	Huntington’s disease
ATauc	Tau	L	Y,N	Alzheimer’s disease

Table adapted from the following references (Sipe et al. 2010; Norrby 2011; Eisenberg and Jucker 2012; Rostagno et al. 2010)

^aSystemic or organ restricted

^bNo systematic name

^cAmyloid-like protein aggregation diseases, all others being known disease-related amyloid

Background on amyloid

Present-day usage of the term amyloid or ‘amyloid-like’ extends to the description of an aggregated state of a protein that exhibits the following phenomenological properties (Baxa 2008; Sipe et al. 2010; Tycko 2011; Shewmaker et al. 2011)—a gross ultra-structure of the minimal amyloid unit that falls within a set limit of nano- to microscale structural constraints (unbranched with width of ∼2–10 nm, length ranging from nm to μm) (Chamberlain et al. 2000; Hall 2012), loss of original structure (Esler et al. 2000; Uversky 2008), adoption of a high extent of β-sheet content present in the aggregated state (as exhibited by FTIR and CD spectroscopy studies) (Nilsson 2004), arrangement of the β-sheets into a cross-β arrangement (as indicated by X-ray fiber diffraction) (Eanes and Glenner 1968; Jahn et al. 2010) and the capability for binding a range of intercalating amyloidophilic dyes (with two prominent members being Congo Red and Thioflavin T) (Naiki et al. 1989; Levine and Walker 2010). As amyloid formation (Fig. 2) involves loss of the original protein structure, the first general step in the process requires either partial or complete denaturation, or alternatively partial proteolysis, to expose amyloidogenic peptide regions2 (see Rostagno et al. 2010). Alternatively, natively unstructured domains can interact and form β-sheets. The mechanism of amyloid formation, both in vitro and in vivo, is thought to be a nucleated growth phenomenon (Naiki et al. 1991; Jarrett and Lansbury 1993; Lomakin et al. 1996; Carulla et al. 2005), displaying kinetic and thermodynamic phase transition behavior3 similar in nature to that obtained and predicted for one-dimensional crystal growth (Oosawa and Asakura 1975; Hall and Minton 2002, 2004). As with any nucleated process, the exact mechanistic progression of amyloid nucleation is not necessarily determinate. As such nucleation of amyloid may be best viewed through the prism of a generalized process having multiple pathways (Nguyen and Hall 2004; Patil et al. 2011); however, under some conditions, a single pathway may be dominant. Roles in enhancing/inhibiting the various modes of amyloid nucleation have been suggested for non-specific surfaces (Linse et al. 2007; Morinaga et al. 2010), various cellular and membrane components (Narindrasorasak et al. 1991), ligand bound and/or covalently modified monomer (Kuner et al. 2000; Liang et al. 2006), and non-specific aggregated forms of the monomer (Nguyen and Hall 2004). Growth of amyloid is necessarily a template-dependent process in which the joining molecule adopts the particular cross-β-sheet structure of the amyloid fiber to which it is being added (Tycko 2011; Wickner et al. 2010), and, as such, amyloid can be regarded as a self-perpetuating structural state capable of transferring information in an epigenetic fashion. Slight differences in the template result in different perpetuated structural states. These structural differences may confer unique properties to the amyloid and are thought to be the physical basis behind the existence of different amyloid strains (Tanaka et al. 2004; King and Diaz-Avalos 2004; Ross et al. 2005; Petkova et al. 2005). A number of three-dimensional structures for amyloid fibers produced in vitro, from disease and non-disease peptide, have been determined by solid-state NMR and high resolution cryo-electron microscopy measurements (Iwata et al. 2006; Fändrich 2007; Baxa 2008; Tycko 2011). Such measurements have confirmed the cross-β-sheet structure and have shown that individual fibril units, termed protofibrils, will typically undergo higher order self-association to form mature fibers, in which multiple protofibrils form various kinds of complex helical structures (Jimenez et al. 2001; Fändrich 2007). Even higher order arrangement may occur resulting in the formation of large amyloid spherulites (micron-length scale) (when fibers self-associate in an ordered fashion) (Krebs et al. 2009) or alternatively non-specific aggregated deposits of fibers similar to those observed in disease (when fibers overlap in a disordered fashion) (Ban et al. 2003; Mishra et al. 2011).

Fig. 2.

Concepts in the progression of amyloid formation: i To form amyloid a protein must be either partially or completely unfolded (transparent blue arrow) or alternatively the amyloidogenic fragment must be released via proteolytic cleavage (red arrow). ii Nucleation of amyloid involves formation of a structural template which is consolidated and replicated by polymerization. Such nucleus formation may occur by a number of pathways, e.g., from covalent or reversible modification of the peptide, non-specific aggregation of the polypeptide, via a defined kinetic pathway or via reversible hetero-association with other components, e.g., a surface. iii Once formed, the nucleus is extended to form amyloid proto-fibrils which result from intermolecular β-sheet formation along the fibril axis. Different amyloid proto-filament structures (formed via different nucleation mechanisms) are perpetuated as distinct amyloid ‘strains’. iv Mature amyloid fibrils are formed by the self-association of proto-fibrils to form thicker fibrils. v Proto-filaments and mature fibrils may undergo higher-order self-association to form clumped deposits. Most disease-related amyloidosis-related deposits have additional factors associated with the deposits such as serum amyloid component P (SAP) and proteoglycans such as heparin sulfate

The material properties associated with a limited number of amyloid fiber types have been characterized in relation to their cellular biocompatibility (Pike et al. 1991; Hope et al. 1996; Kayed and Lasagna-Reeves 2012). Various kinds of amyloids have been observed to display a cellular toxicity that is dependent upon both the specific type (monomer/strain) and dimension of the aggregate (Xue et al. 2009; Kayed and Lasagna-Reeves 2012). A somewhat general finding over the last 20 years has been that smaller oligomeric forms of amyloid seem to display more cytotoxicity than larger mature forms of amyloid, prompting establishment of the toxic oligomer hypothesis (Glabe 2006; Stefani 2010; Kayed and Lasagna-Reeves 2012) [also known as the ADL (Amyloid Diffusable Ligand) theory; Klein 2006]. This theory states that amyloid oligomers are cytotoxic whereas mature amyloid fibrils represent a non-cytotoxic relatively ‘safe’ form of stored amyloid.4 A number of mechanisms accounting for amyloid cytotoxicity have been proposed (Stefani 2010; Kayed and Lasagna-Reeves 2012): these include the action of amyloid to puncture/disrupt cell membranes (Lashuel and Lansbury 2006), action as an adsorbing surface (Olzscha et al. 2011; Treusch and Lindquist 2012), a negative effector of normal chaperone and proteasome functions (Csermely 2001), and a general disruptor of cellular cytostatic mechanisms for maintaining redox and ion balance within the cell (Smith et al. 2007). Due to the noted diversity of the amyloid product, both in polypeptide sequence and three-dimensional structure, it is possible/probable that multiple avenues of cytotoxicity may exist (Stefani 2010; Kayed and Lasagna-Reeves 2012).

A noted feature of deposited amyloid is its extreme resistance to breakdown by proteolysis (Soto and Castaño 1996), thereby making it difficult (though not impossible; Nalivaeva et al. 2012) for the body to remove. Within the body, amyloid clearance may result from breakdown by proteolysis and autophagy (Nalivaeva et al. 2012; Harris and Rubinsztein 2012), dispersal by chaperones (Wilson et al. 2008), or ingestion by phagocytic cells such as macrophages or astrocytes (Wyss-Coray et al. 2003) (followed by subsequent intracellular breakdown). Due to the differential rates of its production and clearance amyloid levels are highly variable across the range of the amyloidosis diseases. In some cases, ‘literally kilograms’ of amyloid may be deposited (Pepys 2006), with the surrounding cells remaining largely healthy before eventual tissue dysfunction results from either physical obstruction of normal cellular metabolic function or physical damage caused by the presence of an unyielding solid mass (note remarks by Symmers 1956; Glenner 1980; Pepys 2006). Alternatively, in other forms of amyloidosis, significant extents of deposited amyloid may not be apparent, with cell loss suspected to occur either from, or in part due to, amyloid’s cytotoxic effects (Ng et al. 2005; Liberski et al. 2005; Shankar et al. 2008; Perrin et al. 2009). In the following sections, we will examine the possible generalities linking amyloid with disease progression across the full domain of the amyloidoses.

Computational modeling of amyloid’s relationship to disease

In seeking to use computational modeling to relate amyloid growth to disease, it is worthwhile clarifying what such a set of models are specifically trying to achieve. For the purposes of this review, we are primarily interested in modeling approaches which can, across the span of the amyloidoses, (1) describe the time course of amyloid formation, (2) incorporate a level of amyloid structural detail (3) provide a causal link between amyloid formation and the onset of disease (and hence describe the time course of symptomatic display and disease progression) (4) both incorporate and compare against experimental data over the characteristic time scales of the disease, and (5) offer testable predictions for use in guiding patient therapy and treatment. Heuristic or phenomenological rate models constructed on the basis of experimentally derived data (or alternately postulates derived from experimental observation) satisfy all of these requirements (e.g., see Masel et al. 1999; Craft et al. 2002; Simmons et al. 2005) to various extents and are the type that we will explore in this review. As most5 models of this type share, as their central pillar, a chemical rate model of amyloid formation, we first examine some of the alternative formulations of such chemical rate models before reviewing some recent efforts.

Many have utilized sets of kinetic rate equations to model experimental measurements of amyloid growth under in vitro conditions (Sluzky et al. 1991; Lomakin et al. 1996; Pallitto and Murphy 2001; Lee et al. 2007; Xue et al. 2008; Binger et al. 2008; Bernacki and Murphy 2009; Ghosh et al. 2010; Foderà and Donald 2010). Rather than detail all the different modeling approaches that have been adopted, here we initially take a very general view of the aggregation process and list some useful limiting cases that can be (and have been) derived therefrom. In the statistical limit, aggregation kinetics can be described using a generalized Rate Model (RM) approach first developed by von Smoluchowski for the description of vapor condensation (von Smoluchowski 1916, 1917). On the condition of a single isomer per species6 this model can be expressed by (Fig. 3, equation set 1) and, in some cases, the individual rate parameters can be analytically formulated. In this scheme, C_i represents the concentration of species of size i. The rate constants k_A(y,z) and k_S°(y,z), respectively, represent a second order association rate constant (units M⁻¹ s⁻¹) and a first order dissociation rate constant (units s⁻¹) governing formation/dissociation of species i = y + z from/into sub-unit species of size y and z. Nucleated growth phenomena, such as amyloid formation, may be productively viewed through one of two general limiting cases of equation set 1 (Fig. 3a), a kinetic limit in which the behavior is considered simply as a slow step followed by a fast step or a thermodynamic limit, in which a thermodynamically rare species (read low relative population at equilibrium) termed the critical nucleus, mechanistically precedes the formation of a thermodynamically favored species (read high relative population at equilibrium). Mathematical realization of either case can be effected by judicious selection of rate constants in equation set 1 (Fig. 3a) and as such it offers a particularly powerful conceptual view of the problem. Traditionally because of either limits in computing power, incompatibility with experimental observation, or a desire for conceptual transparency a number of simplifications of equation set 1 (Fig. 3a) have been made (Edelstein-Keshet 2006). For example, when analytical formulation of rate constants is not possible (e.g., Pallitto and Murphy 2001; Lee et al. 2007; Hall and Hirota 2008), parameter space may be reduced by making all values of k_A and k_S above and below the size of the critical nucleus equal to a set of arbitrarily constant values. Other important simplifications worthy of note include discounting end-to-end fiber annealing (Fig. 3a, equation set 2), the no end-to-end fiber annealing/no internal fiber breakage approximation (Fig. 3a, equation set 3) and the no end-to-end fiber annealing/irreversible fiber growth approximation (Fig. 3a, equation set 4). For some of the classes of equations, a simpler relationship, only capable of describing lumped-system parameters, can be made by summing the sets of rate equations derived from the generalized rate model (Fig. 3a, equation set 5). A further set of model equations worthy of mention (although not listed) are the class of ultra-simplified mechanistic or empirically based descriptive equations (Hall and Minton 2004; Powers and Paolucci 2006, 2008; Morris et al. 2009; Bernacki and Murphy 2009). This class of equations, sometimes analytical in nature, display the same general form as more complex mechanism-based relationships and therefore have proven to be useful in data analysis and data reduction (Philo and Arakawa 2009; Bernacki and Murphy 2009).

Fig. 3.

Minimal kinetic and thermodynamic description of amyloid formation in vitro. a Aggregation processes can be modeled using a generalized Rate Model (RM). A generalized RM encompasses all possible association and dissociation possibilities of the aggregated species. Neglecting isomeric differences Equation set 1 describes such a description for a generalized one-dimensional growth processes with terms reflecting fiber growth via monomer addition, growth via annealing, fiber shrinkage via monomer loss and fiber shrinkage via internal fragmentation. Equation set 2 describes a useful truncation of the full RM in which end-to-end fiber annealing steps have been discounted as being of secondary importance. Additionally an arbitrary nucleus size, n, is set such that all association rate constants prior and post the n stage are designated as k_N and k_G respectively. Fiber fragmentation between all forms of aggregated monomers is set at k_S°. Equation set 3 is a further simplification in which growth and loss occur via monomer addition or loss with no internal breakage. Equation set 4 represents a final simplification in which growth is considered irreversible. Equation set 5 represents a process whereby equations can be summed amongst the pre and post nucleation steps to reduce the number of equations and express the system in terms of lumped parameters reflecting total amyloid mass, the number concentration of fibers and the mass and number concentrations of the pre-nuclear species. The equation set at the bottom of the panel shows the result of such a summation for a system corresponding to Equation set 2, when the nucleus size is equal to 2. Righthand panels effect of parameter variation within the confines of the general phenomenological model represented by Equation set 2 expressed in lumped parameter form. Simulations were made using the following base set of parameters: n = 2, k_G = 100 M⁻¹ s⁻¹, k_S° = 1 × 10⁻⁶ s⁻¹, (C_M)_TOT = 100 nM and k_N = 0.01 M⁻¹ s⁻¹. b. Effect of variation of the nucleation rate k_N varied over 0.001, 0.01, 0.1, 1, and 10 M⁻¹ s⁻¹. c Effect of variation in the scission rate k_S° varied over 0,0.1 × 10⁻⁶, 0.5 × 10⁻⁶, 1 × 10⁻⁶, 2 × 10⁻⁶, 5 × 10⁻⁶, and 10 × 10⁻⁶ s⁻¹. d Effect of variation of the growth rate k_G varied over 1, 10, 25, 50, 100, and 200 M⁻¹ s⁻¹. e Effect of variation in the total concentration of monomer, (C_M)_TOT varied over 10, 15, 20, 25, 50, 75, 100, and 200 nM. f For cases where K_N < K_G, the equilibrium concentrations of total amyloid and free monomer follow a sharp phase transition centered around a critical concentration, C_C, that is proportional to ∼1/K_G. The red and blue arrows, respectively, demarcate the phase transition boundaries (demarcated by the C_C) for the total concentration of amyloid and free monomer concentration. In our implementation, we consider unidirectional growth and bi-directional dissociation; therefore, there is a factor of 2 the calculation of C_C

The mathematical formalization of amyloid growth in vitro, using such nucleated growth rate models (along with their approximations), has given a basic physical understanding of amyloid’s behavior, which, in turn, has provided predictive power in assessing its response to an imposed change. In Fig. 3, we describe some key general observations to have emerged which, described in the shorthand notation used in the figure, include the following.

• (i.)Reversible nucleated growth behavior (where K_N << K_G) displays a critical concentration, C_C, which may be effectively interpreted as the solubility limit of the monomer.7
• (ii.)Critical concentration is approximately equal to the reciprocal of the growth association equilibrium constant, i.e. C_C ∼ 1/K_G with K_G = k_G/k_S°.
• (iii.)Time evolution of the shape/extent of the distribution of amyloid and pre-nuclear species will be sensitive to nucleus size, k_N, k_G, k_S° and C_M.

Such general physico-chemical concepts have been used to great effect by many (Maggio and Mantyh 1996; Dobson 2001; Lansbury and Lashuel 2006; Hall 2008) in making worded argument about the progression of ‘mis-folding diseases’ involving deposition of insoluble protein. Based on the assumption that such phenomenological behavior is transferable from the test tube to the human body, some (Masel et al. 1999; Hall and Edskes 2004, 2009) have used rate model descriptions to underpin the construction of heuristic models charting amyloid’s relationship to disease. In the following sections, we will review two such heuristic computational models using the shorthand titles ‘Two Hit Model’ and ‘Subtle Model’8 to, respectively, refer to the following two references (Hall and Edskes 2004, 2009).

A two hit model of amyloid’s relationship to disease

In 2004, we utilized a chemical rate model of amyloid formation to develop a model that linked amyloid growth with certain aspects of amyloidosis disease progression (Hall and Edskes 2004). This model was built upon four postulates

1. Rate of amyloid formation followed a kinetic scheme comporting with nucleated-growth and fibril breakage of a linear polymerization reaction (with no end-to-end fiber annealing) occurring in a well-mixed reaction vessel (Fig. 4a) (corresponding to Fig. 3a, equation set 2).

2. Causal relationship between amyloid levels and disease development was set linking the onset of disease to either a total level of amyloid (Fig. 4b, thick yellow line) or a set extent of monomer loss (Fig. 4b, dotted yellow line).

3. Monomeric protein concentration9 in the human body was bounded by two extrema corresponding to a fixed unchanging free monomer concentration [(C_M)(t) = constant] (Fig. 4b, thick blue line) and a set total monomer level [(C_M)(t) = (C_M)_TOT − C_AMYLOID(t)] (Fig. 4b, thin blue line).

4. Rate parameters of the kinetic model could be considered phenomenological functions of amyloid type (structure/strain), host environment, and amyloid size.

Fig. 4.

Model postulates for relating amyloid formation to disease. a Amyloid growth is modeled via a heuristic model of a nucleated polymerization reaction featuring fibril fracture. In this treatment nucleus formation is phenomenologically modeled as a bimolecular process. The rate constants governing nucleation, k_N, growth, k_G, and breakage, k_S°, are considered functions of the amyloid size, host environmental status and amyloid type/strain. Amyloid fiber breakage is considered to happen via fracture between any two monomers in the amyloid fiber at an intrinsic rate constant k_s°. Fragmentation species the size of monomer are considered to rejoin the monomer pool whereas fragmentation species of size greater than monomer are considered as short amyloid fragments. b In the ‘Two Hit Model’ (Hall and Edskes 2004), we considered the negative effects of amyloidosis disease in relation to a chemical progress curve describing the total extent of amyloid (red lines) and the concentration of free monomer (blue lines). Progress curves were simulated using an arbitrary set of parameter values (k_N = 0.01 M⁻¹ s⁻¹, k_G = 100 M⁻¹ s⁻¹, k_S° = 1 × 10⁻⁶ s⁻¹) with Fig. 3a, equation set 2. The behavior of monomer was considered to follow one of two limiting kinetic cases: i Monomer synthesis occurring quickly relative to its rate of depletion, C_M(t) ∼ C_M(t = 0) (thick blue line), ii Monomer synthesis occurring slowly relative to its rate of depletion, C_M(t) ∼ [C_M(t = 0) − C_AMYLOID] (thin blue line). The thick and thin red lines correspond to the extent of amyloid formed by the corresponding limiting case monomer conditions. Disease onset was postulated to occur in relation to either the concentration of amyloid exceeding a certain critical threshold (thick yellow line) or the free monomer concentration dipping below a set limit (dotted yellow line). The time corresponding to violation of limits represented the time associated with disease symptomatic display (green arrows). c In the ‘Subtle Model’ (Hall and Edskes 2009), we expanded our starting postulate of amyloid’s relationship to disease by considering only a certain region of the amyloid distribution as having toxic qualities (green area) whilst the rest of the amyloid distribution was considered relatively inert. d The Subtle Model postulated an additional mechanism for amyloidosis disease occurrence as a result of toxic amyloid injury when the concentration of toxic species, denoted C_TS, exceeded a designated critical threshold (green dotted line)

Although following in the footsteps of Masel and coworkers (Masel et al. 1999) who described the prion amyloidosis in terms of an unbounded set of coupled differential equations, we approached the problem somewhat differently in a number of respects. By eliminating steps corresponding to monomer formation and degradation, we eliminated two rate parameters thereby making our model slightly more transparent. We also considered the process of breakdown by phagocytosis as too undefined10 and we chose to lump this aspect into a more general consideration of monomer production and fiber growth and breakage. Another major difference was our approach to solving the equations—rather than reducing the set of equations (to a smaller set describing lumped system parameters), we attempted the explicit solution of the complete unbounded set. Explicit solution of the amyloid distribution relied upon numerical integration of all the equations describing the rate of formation of the number concentration of each chemical species. The number of species was allowed to grow as required throughout the simulation so as to make solution as rapid as possible at early times (Hall 2003). As the number of computational steps required at each time step in the numerical integration of this set of coupled differential rate equations would normally scale in relation to (N_max)², we utilized an approximation that made the solution effort scale with just N_max instead—hence making these calculations achievable on a desktop computer (the nature of the approximation is outlined in Hall and Edskes (2009). Using this numerical strategy, the model was solved for a number of cases relating to the spontaneous development of amyloidosis and infectious transmissible forms of the disease. We noted that by considering variation in rate parameters occurring both (1) along the lifetime of the individual and (2) within members of a population, we could rationalize many previously puzzling aspects of the amyloidosis diseases and especially of the infectious amyloidoses. These included such features as resistance/susceptibility to infection (as evinced by the concept of prion strains) (Marsh and Bessen 1994; Derkatch et al. 1996), the species barrier (the inefficient infection of an organism by a prion propagated in another organism) (Collinge and Clarke 2007; Cuille and Chelle 1939), and the possible role of outside agents inflicting multiple injuries on the host thereby promoting the onset of disease (Roberts et al. 1994; Bush 2003). A startling feature of the model was that, due to the phase transition behavior dictated by the underlying mathematics of the nucleated growth model, extremely small undetectable amounts of amyloid could persist in a stable equilibrium state waiting for a change in the host’s parameters that would enable it to rapidly amplify and exceed the postulated safe level. This observation was paraphrased (somewhat lyrically) within the paper as a potentially silent prion lying in wait.

Within the context of the starting postulates, the major finding of the paper was the striking conceptual similarity between disease progression in the amyloidoses and the ‘two hit model’ of disease progression developed by Knudson to explain the genetic basis of retinoblastoma (cancer) (Knudson 1971). The physical basis of the original two hit model developed by Knudson11 involved the concept of Mendelian allelic variation, i.e. each gene has a dominant, A, or recessive, a, allele.12 In applying the concepts of the two hit,model to the progression of disease in the amyloidoses, we divorced the model from its original genetic association identified by Knudson and applied it in the context of amyloid’s kinetic mechanism and thermodynamic behavior (Fig. 5). We considered defects relating to the receipt of an amyloid template (via nucleation or infection) and in the ability to manage amyloid (via changes in the growth/breakage rate constants or monomer concentration i.e. those factors directly affecting the relationship between C_M and the C_C). In this context, a ‘no hit’ individual might be represented by a person who could not spontaneously nucleate the disease (k_N = 0) or was not infected by amyloid of external origin (C_AMYLOID = 0) and had an efficient amyloid management system (k_G, k_S° and C_M) capable of dealing with infection/nucleation if encountered. A person suffering ‘one hit’ could be considered as someone with defects in either their amyloid management or nucleation/infection status. In the latter case, an exceedingly small but non-zero quantity of amyloid may be present, but kept at low levels due to efficient amyloid management dependent upon kinetic (due to slow progression manifested by low values of all/some of k_G, k_S° and C_M) or thermodynamic effects (due to a set of k_G, k_S° and C_M that effected conditions such that C_M < C_C). A patient suffering two hits was one who developed amyloid in excess of safe levels due to defects in the remaining normal functioning aspect of either their nucleation/infection status or their amyloid management capability. Figure 5 describes some of the basic concepts relating to the two hit computational model formulation explored in Hall and Edskes (2004). The lefthand panels refer to different modes of spontaneous development of amyloidosis for patients comporting with the ‘no hit’ (Fig. 5a), ‘one hit’ (Fig. 5b) and ‘two hit’ (Fig. 5c) designations. The righthand panels describe simulations reflecting disease resulting from the transmissible amyloidoses with a case of a host resistant to infection but capable of maintaining a silent prion infection (Fig. 5d), a host who although initially resistant to infection, sustains the silent prion and then suffers a second hit later in life (Fig. 5e) and finally a host susceptible to infection, who initially had a first hit derived from defective amyloid management capabilities but sustained no prion by virtue of the fact that they could not self-nucleate amyloid, but who upon infection, rapidly developed disease (Fig. 5f).

Fig. 5.

Major Concepts Associated with the Two Hit Model of Amyloidosis. Top panel Application of the two hit model concept to amyloidosis. The two hit model was employed in a heuristic sense to describe accumulation of defects in the nucleation and management of amyloid based on the mathematical relations governing its kinetic and thermodynamic behavior. White sections (No Hits) refer to patients who cannot incipiently nucleate amyloid\ have not been infected with transmissible amyloid and who have an amyloid management system capable of preventing significant amyloid formation if such amyloid was introduced. Blue sections (First Hit)—The first hit may represent either a change in incipient nucleation\infection status or a change in amyloid management status. Pink sections (Second Hit) the occurrence of the alternate possibility not enacted during the first hit. Lower panels Simulations describing condensed examples of how the two hit theory works in practice. Panels a to c describe cases relating to the development of spontaneous/hereditary amyloidosis disease. d–f describe cases relating to the development of transmissible amyloidosis disease. The yellow lines refer to the maximum permissible amyloid levels above which occurs disease related symptomatic display as outlined in the explanation of postulate 2 (Fig. 4b). Parameter values used: (a) white section (C_M)_TOT = 100 nM, k_N = 0 M⁻¹ s⁻¹, k_G = 100 M⁻¹ s⁻¹, k_S° = 7 × 10⁻⁶ s⁻¹; blue section k_N = 0.01 M⁻¹ s⁻¹; pink section (C_M)_TOT = 200 nM (lines represent Case A and Case B monomer behavior). (b) Blue section (C_M)_TOT = 100 nM, k_N = 0.01 M⁻¹ s⁻¹, k_G = 100 M⁻¹ s⁻¹, k_S° = 7 × 10⁻⁶ s⁻¹; pink section k_S° = 1 × 10⁻⁶ s⁻¹ (dotted line) or (C_M)_TOT = 200 nM (solid line). (c) Pink section (C_M)_TOT = 200 nM, k_N = 0.01 M⁻¹ s⁻¹, k_G = 100 M⁻¹ s⁻¹, k_S° = 7 × 10⁻⁶ s⁻¹. (d) White section (C_M)_TOT = 100 nM, k_N = 0 M⁻¹ s⁻¹, k_G = 100 M⁻¹ s⁻¹, k_S° = 7 × 10⁻⁶ s⁻¹; blue section C_AMYLOID(t = 3.8 × 10⁶) = 50 nM, C_N(t = 3.8 × 10⁶) = 0.5 nM. (Pe) White section (C_M)_TOT = 100 nM, k_N = 0 M⁻¹ s⁻¹, k_G = 100 M⁻¹ s⁻¹, k_S° = 7 × 10⁻⁶ s⁻¹; blue section C_AMYLOID(t = 3.8 × 10⁶) = 50 nM, C_N(t = 3.8 × 10⁶) = 0.5 nM; pink section (C_M)_TOT = 200 nM. (f) Blue section (C_M)_TOT = 100 nM, k_N = 0 M⁻¹ s⁻¹, k_G = 100 M⁻¹ s⁻¹, k_S° = 1 × 10⁻⁶ s⁻¹; pink section C_AMYLOID(t = 3.8 × 10⁶) = 50 nM, C_N(t = 3.8 × 10⁶) = 0.5 nM. (Adapted from Hall and Edskes (2004)

A subtle model of relationship between amyloid and disease

Although the Two Hit Model was successful in accounting for a number of challenging aspects of amyloidosis disease (such as transmission barriers in prion infection; for an interesting discussion, see Jones et al. 2009), a major defect of the model was that it provided no insight into the exact mode of tissue injury effected by amyloid when exceeding the postulated critical level. In this sense, our model made no distinction between tissue injury resulting from the amyloid related toxicity or solid mass effects discussed in the introductory sections. In our development of the ‘Subtle Model of Amyloid’s Relationship to Disease’ (Hall and Edskes 2009), we refined the postulate linking amyloid to disease to reflect the toxic oligomer hypothesis (Glabe 2006; Klein 2006; Kayed and Lasagna-Reeves 2012), i.e. the idea that small regions of the amyloid distribution possessed inherent cytotoxicity as compared to larger mature fibers. Postulate 2 of the Two Hit Model was effectively revised as follows,

• (2)Causal relationship between amyloid levels and disease development was set linking the onset of disease to either a total level of amyloid (Fig. 4b, thick yellow line), a set extent of monomer loss (Fig. 4b, dotted yellow line) or the summed extent of a certain toxic region of the amyloid distribution (Fig. 4c, d).

Similar to the Two Hit Model, postulates 1, 3 and 4 were taken to hold true. The computational realization of the modified second postulate was carried out by integrating over an arbitrary region of the amyloid distribution assigned ‘toxic’ properties and monitoring the evolution of this toxic integral in comparison to the total mass of polymer product (Fig. 4d). In solving the model an arbitrary set of parameters were utilized to examine the effect of varying the intrinsic fracture rate in relation to the general time course of the amyloid formation reaction. In this context the actual values of the parameters were less important than their general scaled relationships. Figure 6a, b, respectively, describe results obtained from Hall and Edskes (2009) for the time evolution of toxic species and total amyloid for the two types of governing monomer behavior. Three general cases of breakage rate were considered, fast scission (relative to the rate of polymerization), relatively slow scission and a zero scission rate.13 Five major findings were concluded from the results of the Subtle Model simulations:

• (i.)The amount of amyloid formed was sensitive to the total concentration of monomer and its relation to the critical concentration defined by k_G and k_S°.
• (ii.)Relatively fast scission occurring under conditions above the C_C, results in the production of significant extents of small toxic species with very little long amyloid fibrils produced.
• (iii.)Intermediate scission rates could effectively decouple the emergence of toxicity from the appearance of amyloid fibers by orders of magnitudes in time.
• (iv.)Intermediate scission rates in closed systems (total conservation of monomer) may exhibit a critical time window to the emergence of toxicity (Fig. 6d)
• (v.)Unbreakable fibers (zero scission rate) did not yield a significant production of toxic species but did produce a significant extent of amyloid fibers.

Fig. 6.

Major concepts associated with the Subtle Model of amyloidosis. According to the postulates described in Fig. 4c, d, a region of the amyloid size distribution is imbued with a differential cytotoxicity. Explicit solution of the rate of formation of all amyloid species using Fig. 3a, equation set 2 allows for the grouping and summation of an arbitrary region of the amyloid size distribution enabling simulation of the time evolution of this region separately from the total mass of amyloid. Lefthand panels Case A monomer conditions whereby the free concentration of monomer is at all times constant. Righthand panels Case B monomer conditions whereby the total concentration of monomer is a conserved quantity. a, b The time evolution of the summed toxic species (solid lines) and the total concentration of amyloid (broken lines) for the two different monomer concentration regimes. Red unbreakable fibrils (k_S° = 0). Blue fibrils that break quickly in relation to the time course of formation of total amyloid. Black fibrils that break slowly in relation to the time course of formation of total amyloid. Green line the maximum permissible levels of all toxic species above which occurs disease related symptomatic display as outlined in the explanation of revised postulate 2. Parameter values used: (C_M)_TOT = 100 nM, k_N = 0.01 M⁻¹ s⁻¹, k_G = 100 M⁻¹ s⁻¹, Slow breaking fibrils k_S° = 1 × 10⁻⁹ s⁻¹ (black lines); fast breaking fibrils k_S° = 1 × 10⁻⁶ s⁻¹ (blue lines); unbreakable fibrils k_S° = 0 s⁻¹. c, d The evolution of the average degree of polymerization <dop> of the amyloid for the two different cases of monomer presentation and the three different types of fibril breakage behavior slow breaking fibrils (black lines), fast breaking fibrils (blue lines) and unbreakable fibrils (red lines). Note the incredibly slow relaxation of the <dop> for the slow breaking fibers occurring orders of magnitude more slowly than the original polymer formation reaction. (Adapted from Hall and Edskes (2009)

These findings, taken together with results generated by the Two Hit Model, allowed for the construction of a general paradigm of amyloid’s relationship to disease based on the relative fracture rate of the fibrils occurring under conditions above the C_C. The general take home message of this model was that the different modes (solid mass vs. cytotoxic effects) and timing of tissue injury observed across the amyloidosis disease spectrum could well represent different limiting cases of a nucleated growth model of amyloid formation featuring fibril fracture in which different regions of the amyloid size distribution exhibited differential cytotoxicity. Interestingly two models derived from experimental observations (Xue et al. 2009; Sandberg et al. 2011) were subsequently published14 which add striking support to the Subtle Model’s general potential as a theoretical mechanism for relating the time course of amyloid development to disease (Fig. 7).

Fig. 7.

Summary of the Amyloidosis Disease Fibril Fracture Model. On the condition that disease occurs, fibril fracture provides a convenient switch between the different causative modes of amyloidosis disease due to i toxic oligomer effects or ii solid mass effects. Comments within the circle make reference to the mode of cell and tissue injury in relation to the type of monomer regulation. i.e. Case A or Case B type systems. The Subtle Model thus represents an extension to the original Two Hit Model (adapted from reference Hall and Edskes (2009)

The way forward

We have reviewed, in detail, two heuristic computational models of amyloidosis that provide a rational explanation and offer qualitative predictive power for many observed aspects of the amyloidosis diseases, such as onset, heterogeneity, resistance/susceptibility, and hyper-variation, in the association between the appearance of amyloid fibers and tissue injury (Pepys 2006; Selkoe 2011). However it would be fair to say that, despite being somewhat successful, these models are not complete and here we suggest general strategies for model improvement.

At present, the model approach described offers only heuristic, but not molecular, insight into possible differences in amyloid nature due to involvement of different polypeptide building blocks. One general way of providing such insight is through incorporation of the results of molecular level simulation procedures such as Molecular Dynamics (MD) or Monte Carlo (MC) (Hall 2008; Straub and Thirumalai 2012) into the construction of the heuristic model. Such studies have provided substantial insight into amyloid structure (Standley et al. 2006), possible pathways followed in the aggregation of peptides (Nguyen and Hall 2004; Nasica-Labouze et al. 2011), and the interaction of amyloid with other biological components (such as membranes) (Connelly et al. 2012) at various levels of realism. Incorporation of such methods has been archieved using various forms of multi-scaling (Hall and Hirota 2008). In the multi-scaling approach, a combination of models (e.g., MD, BD, RM), each with different levels of detail in particle representation and each operating over a set of time/distance/complexity scales, are used such that fine detail is conserved when required by nesting the higher order models within the coarser ones and only calling them into action when higher order detail is required. An example of such a modeling approach is that provided by Hall and Hirota (2008) who incorporated a polymer model15 of a protein capable of predicting microscopic rate constants, into a kinetic rate model of amyloid formation (Hall et al. 2005; Hall and Hirota 2008). Another related deficiency in the heuristic modeling approach is its lack of detail in how amyloid exerts its specific effects. In this sense, molecular level detail obtained from higher-order models could be similarly introduced.

A major weakness of the discussed modeling strategy is its lack of a spatial aspect. Such a feature would require incorporation in either the form of generalized compartment modeling in the ODE set (Craft et al. 2002; Simmons et al. 2005) or recasting of the model in terms of reaction/diffusion/convection system based on a set of partial differential equations (Matthäus 2006). The inclusion of such spatial effects will be of importance when addressing such questions as why some amyloidosis are systemic whereas others are predominantly localized in nature (Pepys 2001; Biewend et al. 2006). Accounting for the spatial domain will also allow for the correction of a second major weakness in the model which is the lack of designation of specific tissue susceptibility. Other major steps forward would involve both resurrecting and clarifying the amyloid clearance term used by Masel et al. (1999) and explicitly including the possible effects of fiber annealing (Ghosh et al. 2010) as well as non-specific aggregation. Towards the goal of modeling amyloid formation in a whole body context, these improvements would signal a major step forward, both in our perception and treatment of the amyloidoses. Indeed, potential comparison of such time-dependent simulations (based on the complete size and spatial distribution of amyloid in a whole body environment) against in vivo measurements of amyloid based on the use of sensitive imaging probes (Simmons et al. 2005) would provide significant insight, both to clinicians making decisions on patient treatment and companies developing more effective drugs against the range of diseases constituting the phenomenon of amyloidosis.

Conclusions

Some of the amyloidosis diseases, notably Alzheimer’s disease and type 2 diabetes are major causes of mortality in modern society. Nearly all vastly affect the quality of life for both the afflicted and their caregivers over a long period. Whether by dealing with the amyloidoses as a potentially common group or as individual diseases, computational approaches have a very important role to play in helping to explore and untangle some of the complexity associated with their onset, progression, and possible treatment. It is easy to get lost in exploring interesting computational avenues in which the information benefit return, as a fraction of expenditure of our time, effort, and resources, is of questionable value. In this review, we have concentrated on coarser modeling methods that we believe jump this prospective effort/benefit hurdle. While some might view the heuristic rate equation approach as a scientific retreat, we believe that this is the right tool for the job for one wishing to come to grips with the underlying medical problem as it inherently deals in the macroscopic quantities directly observable in treatment and diagnosis of disease. In closing, we would venture that categorical linking of the amyloidoses by a consistent unified mechanism of disease would open the doorway (a little wider) in the development of possible treatments—one of the major aims in the application of computational biophysics to disease.