Cellular automata modeling depicts degradation of cellulosic material by a cellulase system with single-molecule resolution

Abstract

Background: Enzymatic hydrolysis of cellulose involves the spatiotemporally correlated action of distinct polysaccharide chain cleaving activities confined to the surface of an insoluble substrate. Because cellulases differ in preference for attacking crystalline compared to amorphous cellulose, the spatial distribution of structural order across the cellulose surface imposes additional constraints on the dynamic interplay between the enzymes. Reconstruction of total system behavior from single-molecule activity parameters is a longstanding key goal in the field.

Results: We have developed a stochastic, cellular automata-based modeling approach to describe degradation of cellulosic material by a cellulase system at single-molecule resolution. Substrate morphology was modeled to represent the amorphous and crystalline phases as well as the different spatial orientations of the polysaccharide chains. The enzyme system model consisted of an internally chain-cleaving endoglucanase (EG) as well as two processively acting, reducing and non-reducing chain end-cleaving cellobiohydrolases (CBHs). Substrate preference (amorphous: EG, CBH II; crystalline: CBH I) and characteristic frequencies for chain cleavage, processive movement, and dissociation were assigned from biochemical data. Once adsorbed, enzymes were allowed to reach surface-exposed substrate sites through "random-walk" lateral diffusion or processive motion. Simulations revealed that slow dissociation of processive enzymes at obstacles obstructing further movement resulted in local jamming of the cellulases, with consequent delay in the degradation of the surface area affected. Exploiting validation against evidence from atomic force microscopy imaging as a unique opportunity opened up by the modeling approach, we show that spatiotemporal characteristics of cellulose surface degradation by the system of synergizing cellulases were reproduced quantitatively at the nanometer resolution of the experimental data. This in turn gave useful prediction of the soluble sugar release rate.

Conclusions: Salient dynamic features of cellulose surface degradation by different cellulases acting in synergy were reproduced in simulations in good agreement with evidence from high-resolution visualization experiments. Due to the single-molecule resolution of the modeling approach, the utility of the presented model lies not only in predicting system behavior but also in elucidating inherently complex (e.g., stochastic) phenomena involved in enzymatic cellulose degradation. Thus, it creates synergy with experiment to advance the mechanistic understanding for improved application.

Keywords: AFM imaging; Cellular automata; Cellulase; Cellulose; Hydrolysis; Surface degradation; System-level modeling.

Scheme 1

A typical set of fungal cellulases acting on mixed amorphous–crystalline cellulose. Chain end-cleaving cellobiohydrolases (CBH I blue; CBH II cyan) and internally chain-cleaving endoglucanases (red) catalyze the hydrolysis of crystalline (yellow) and amorphous (brown) cellulosic material

Fig. 1

CA model of mixed amorphous–crystalline cellulose. Cellulose material is modeled as a large 3D matrix (depicted here in a schematic 2D y–z view) composed of substrate automata that represent cellobiose units. Tokens ‘1’ and ‘2’ are used to indicate cellobiose units in amorphous and crystalline material environment. Token ‘0’ is used to indicate bulk molecules. The corresponding matrix is framed red. Further properties of the cellulose chains such as the orientation in space and the location of the hydrophobic faces are assigned through additional matrices of identical size, as shown in the matrices framed in blue. Each cellobiose unit in the cellulose model is identified unambiguously through the positions in all substrate matrices

Fig. 2

Structure of crystalline (a) and amorphous (b) cellulose as implemented in the CA model. a Cellulose nanocrystals were modeled in analogy to the natural elementary fibril of cellulose allomorph I and were composed of up to 144 cellulose chains (16 nm crystals). Their length was constant at 100 nm equaling 100 cellobiose units per chain, and the width/height varied between 8 and 16 nm. Yellow bars show the cellulose chains with their reducing end indicated in orange. Shown in brown are the so-called hydrophobic faces of the cellulose nanocrystal which is where attack by CBH I takes place. Two orientations of the nanocrystal within amorphous material are considered whereby the crystal’s hydrophobic faces are aligned horizontally (left) or vertically (right). b A random positioning algorithm (Additional file 1) was used to create amorphous cellulose layers with a final mean fragment length of 18 cellobiose units. The figure shows an exemplary amorphous plane. It also shows how amorphous chains with varying orientation are combined in one plane. Amorphous chains can be oriented in x or y direction in one z-plane. Tokens ‘±1’ and ‘±2’ are used within the amorphous chain orientation matrix to distinguish x and y direction, respectively. The internal direction of an amorphous chain is indicated by an algebraic sign (‘+’ or ‘−’) and relative to the origin of the z-plane (0, 0). A chain oriented from the reducing to the non-reducing end with respect to the origin has a positive sign (‘+’) and a vice versa oriented chain is indicated by a negative sign (‘−’). Amorphous cellulose material was obtained by stacking multiple layers of amorphous cellulose on top of each other. c The distribution pattern for the cellulose chain lengths in 30 independent planes making up the amorphous material is shown for x- (left panel) and y-oriented (right panel) cellulose chains demonstrating the essentially random distribution of the chains in all orientations

Fig. 3

The CA model of the cellulases interacting with and degrading the cellulose surface. a The footprint of the enzyme is shown in blue and the center of the enzyme is shown in light green. The center of an enzyme is set within a distance of three water or cellobiose units to a randomly chosen point on the cellulosic surface (one of two dark green squares). Enzymes screen for substrate sites in their immediate surrounding (red box). For EG, any cellobiose (brown) of amorphous cellulose is a substrate, whereas for the cellobiohydrolases, only cellobioses lacking a neighbor cellobiose (indicated with a yellow cross) are substrates. b Top view of a CBH molecule in processive motion is shown. The wall in front of the enzyme in moving direction (red rectangle) is checked for structural obstacles (cellobiose molecules). The enzyme is represented against other enzymes as a sphere (blue circle) to calculate collision events. If neither substrate nor enzyme obstacles are present, the cellobiohydrolase moves along the chain, cleaving one cellobiose per 1/k _cat time interval

Fig. 4

CA model of the action of CBH II on a mixed amorphous–crystalline cellulose substrate. The simulated substrate was a flat amorphous matrix in which a nanocrystal of 16 nm width/height and 100 nm length was embedded in plane. CBH II was placed in saturating amount on the cellulose surface (24 nmol/m²). a Effect of the enzyme’s k _cat on the time course of cellobiose released from 25 mm² of cellulose surface. The same amorphous material was used on all simulations. Note that the rate of cellobiose release depends on the k _cat, whereas the maximum amount of cellobiose does not. b When degradation of amorphous cellulose by CBH II (light blue) was simulated whereby CBH II was modeled as a perfectly processive exo-cellulase (k _off = 0), it was noted that CBH II became gradually trapped at amorphous material (blue circle). Collision between a complexed CBH II molecule and a structural obstacle (nanocrystal or amorphous material) was the origin of the jam, and a thus stuck CBH II presented an obstacle for other CBH molecules acting processively on amorphous cellulose chains nearby (blue circle). The red scale bar shows 5 nm. c Modeled time courses of cellobiose release by CBH II were not consistent with experiment unless the CA model of the enzyme was expanded to include endo-type chain cleavage in amorphous cellulose. Modeled results are shown for a k _endo of 0.03 s⁻¹ and are compared with experimental data. The k _off of CBH II was set to 0.7 × 10⁻²·s⁻¹ in the simulation (Table 1)

Fig. 5

Degradation of amorphous cellulose by EG and CBH II. Different combinations of k _cat for EG and CBH II satisfy the criterion of the experimental V _z in amorphous cellulose, but only the combination k _cat (EG) > k _cat (CBH II) results in a plausible system behavior. a The value of k _off affects V _z strongly when k _cat (EG) < k _cat (CBH II), whereas it does not affect V _z when k _cat (EG) > k _cat (CBH II). b Apparent inactivation of CBH II occurs as result of switch of the enzyme into a resting state on collision with structural obstacles or other enzymes. Green lines indicate the fraction of active CBH II enzymes, while red lines indicate the fraction of resting CBH II enzymes. Solid lines met the criteria k _cat (EG) < k _cat (CBH II) and dotted lines met the criteria k _cat (EG) > k _cat (CBH II). The k _off for CBH II was fixed at 0.7 × 10⁻³·s⁻¹. A relatively low k _off was chosen because the shown effect is particularly pronounced under these conditions (panel a)

Fig. 6

Dynamics of crystalline cellulose degradation depends on the k _off of CBH II. The k _off of CBH II is key to explain the dynamics of crystalline cellulose degradation by cellulases as modeled. Shown are time courses of ∆h_max (vertical height difference between the highest point on a cellulose crystallite and the amorphous material surrounding the crystallite) during enzymatic degradation of large (~16 nm height; panels a, b) and small (~8 nm height; panels c, d) nanocrystals analyzed by experiment and simulation. Nanocrystals were modeled with their hydrophobic faces aligned horizontally and with their top face touching the enzyme-accessible surface. The ∆h_max required to expose the crystal’s hydrophobic faces was therefore half the height, which is indicated by the dashed black line. Experimental time courses are shown in green. Simulated time courses of ∆h_max using low and high boundary values of k _off are shown in red (0.7 × 10⁻³·s⁻¹) and blue (0.7 × 10⁻¹·s⁻¹), respectively. Shown in orange and magenta are two simulations using the iteratively “optimized” value of 0.7 × 10⁻²·s⁻¹ for k _off

Fig. 7

Time-resolved sequences from a simulated degradation of mixed amorphous–crystalline cellulose by the cellulase system. Cellulases are identified by color EG (red), CBH II (light blue), and CBH I (dark blue). a A crystallite is gradually uncovered over time until the horizontally oriented hydrophobic face is revealed. b Upon revealing of the hydrophobic face, the crystallite is attacked in an asymmetric manner. c Finally, the crystallite becomes completely degraded over time. A video of the shown degradation sequence is available (Additional file 2). Please note that occasional visual overlapping of enzymes is caused by their position in different z-planes of the substrate

Fig. 8

Dynamic formation and dissipation of enzyme traffic jams at the amorphous–crystalline interface. Cellulases are identified by color EG (red), CBH II (light blue), and CBH I (dark blue). For an easier viewing, the background is darkened through all panels. a, b An already-trapped CBH II molecule at the interface of crystalline and amorphous cellulose (green rectangle) causes a traffic jam of CBH I molecules. c CBH I molecules move processively from left to right on the upper edge of the crystal and an accumulation of CBH I can be observed (21 s). d, e Degradation starts on the other site of the crystallite too but is soon stopped by another collision with a CBH II molecule (green rectangle, panel f). f After 1/k _off is passed, CBH II dissociates but the amorphous material below the dissociated CBH II is clearly elevated in comparison to the surrounding amorphous material (yellow rectangle). Note: the elevated height is recognized by bright color in the yellow framed area. The lower lying surrounding material is indicated by (dark) brown color. g, h EG molecules attack the (elevated) amorphous part (yellow rectangle) and clearly alter it by reducing its height. i Eventually, CBH I molecules resume hydrolysis (green square, 154 s). However, the next group CBH I enzymes trying to slide along the crystal on a lower plane are trapped again (pink square). Note that, the crystallite shows already degradation (pink square) caused by the first group of enzyme sliding along. A video is available (Additional file 3)

Fig. 9

Modeled time course of cellobiose formation is compared to experiment performed under exactly comparable conditions. The experimentally measured initial production of glucose is expressed as cellobiose released (glucose release/2). The absolute amount of calculated cellobiose is plotted as a function of released cellobiose from 25 mm² cellulose surface with black dashed line and the range of error is shown as dotted line. Experimentally measured points are indicated in red. All experiments and simulations were conducted at least in triplicates

Fig. 10

Exemplary processing of AFM images of enzymatic degradation to determine Δh_max. Crystalline fibrils embedded in amorphous cellulose matrix provide time-resolved quantitative data for comparison of simulation results with the experiment. a Height profiles (red line) of crystals were taken from AFM sequences recorded from enzymatic reactions carried out as described by Ganner et al. [14]. For easier viewing, the amplitude image of the corresponding position is shown. The yellow scale bar shows 10 nm. b The highest point on a cellulose crystallite was found by moving the section profile along the crystallite, and the vertical height difference between the highest point and the amorphous material (indicated by the arrow) surrounding the crystallite was calculated (∆h_max)