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Design the conditions
around the chemistry.

A catalyst, a layer and a route for reactants.
Explore the mechanisms that make an architecture worth testing.

Enter the models

Two questions.
Two defined models.

How much substrate changes a well-mixed reaction? How much of a reactive layer remains supplied?

A material-associated catalytic system introduces an architectural question alongside the enzyme question. The retained activity may have little practical value if the intended reactant cannot reach the active region under the chosen conditions. The useful next comparison can therefore concern layer depth, access or the operating window.

The first model explores Michaelis–Menten saturation. The second solves a steady first-order planar reaction–diffusion problem. Their assumptions are deliberately different; a first-order constant is not silently derived from the nonlinear enzyme model. Neither view substitutes for characterising the actual preparation.

The kinetic context follows the published analysis of NanoLuc systems (Nemergut et al., 2023). The planar model follows the worked reaction-engineering derivation in Green and Wittrup (2007), with the coordinate measured from the exposed surface towards the backing.

Read the relationship.
Then change the conditions.

Well-mixed saturation kinetics
v=Vmax⁡SKm+S,vVmax⁡=S/Km1+S/Kmv=\frac{V_{\max}S}{K_m+S},\qquad \frac{v}{V_{\max}}=\frac{S/K_m}{1+S/K_m}

The concentration ratio determines the marginal rate gain in this model. At S = Km, the rate is half its limiting value; doubling an already high concentration offers a smaller relative gain.

vv
Reaction rateConcentration per unit time
Vmax⁡V_{\max}
Limiting reaction rateSame unit as v
SS
Substrate concentrationSame concentration unit as Km
KmK_m
Michaelis constantConcentration

Model conditions. Steady-state Michaelis–Menten conditions in a well-mixed system with specified active enzyme. No transport resistance, product inhibition, substrate depletion trajectory or photon-conversion model is added.

Kinetic model | Defined assumptions

More substrate.
A different marginal gain.

Explore a well-mixed Michaelis–Menten relationship. Increasing substrate changes rate nonlinearly; it does not automatically solve access inside a material.

50.0%

Model rate relative to Vmax

A parameterised explanatory curve. It is neither a fit to a Bio experiment nor a measurement of photon output.

10.500510Substrate | S/KmRate | v/Vmax
The marker follows the selected concentration ratio. The curve is calculated from the stated equation.

Reaction–diffusion model | One-sided planar layer

The inside has
its own conditions.

Change layer depth, diffusivity and a first-order consumption constant. Examine how the assumed concentration at an impermeable backing changes when consumption competes with transport.

These are editable model inputs. They are not measured properties of a Bio preparation.

10.50SurfaceHalf depthBackingPosition | x/LConcentration | C/Cs
Maintained concentration at the exposed surface; no flux at the backing. The shaded area is the calculated concentration profile.
Thiele modulus | φ
1.58
Internal effectiveness | η
58.1%
Backing concentration | C/Cs
0.395
Diffusion time scale | L²/D
250 s

What changes the development question?

Consumption and transport are comparable in this selected model. Characterising both would help identify whether the proposed layer is using its catalyst throughout its depth.

What needs an experiment?

Actual diffusivity, partitioning, reaction order, preparation state and boundary conditions. Swelling, oxygen co-limitation, catalyst loss and external transfer resistance would require a different or expanded model.

Download this model brief

The brief records both models, your current inputs, calculated outputs, assumptions and source links. It is prepared locally.

η describes internal reaction effectiveness for this idealised first-order model. It is not photon yield, material performance or a commercial efficiency claim. L²/D is a characteristic scale, rather than an exact equilibration time.

Inside the planar model.

Steady planar balance and boundary conditions
Dd2Cdx2−kC=0,C(0)=Cs,dCdx∣x=L=0D\frac{d^2 C}{dx^2}-kC=0,\qquad C(0)=C_s,\qquad \left.\frac{dC}{dx}\right|_{x=L}=0

The exposed surface has a maintained concentration. The impermeable backing has zero reactant flux. First-order consumption competes with diffusion through the layer.

CC
Reactant concentration at position xA consistent concentration unit
DD
Effective diffusivity in the homogeneous layerm²/s
kk
First-order consumption constants⁻¹
LL
Depth from exposed surface to backingm

Model conditions. Steady state, constant D and k, planar geometry, isothermal conditions, maintained surface concentration and no flux through the backing. This is a defined model for a hypothetical layer.

Internal access and effectiveness
ϕ=LkD,C(x)Cs=cosh⁡[ϕ(1−x/L)]cosh⁡ϕ,η=tanh⁡ϕϕ\phi=L\sqrt{\frac{k}{D}},\qquad \frac{C(x)}{C_s}=\frac{\cosh[\phi(1-x/L)]}{\cosh\phi},\qquad \eta=\frac{\tanh\phi}{\phi}

The Thiele modulus compares reaction and transport within the chosen geometry. The profile describes internal concentration, while η compares the layer’s integrated first-order rate with the ideal rate if the entire layer experienced Cs.

φφ
Thiele modulusDimensionless
ηη
Internal reaction effectivenessDimensionless
x/Lx/L
Normalised depth0 at surface; 1 at backing
CsC_s
Maintained surface concentrationSame unit as C

Model conditions. For this one-sided layer, L is its full depth. A symmetric two-sided slab would use half-thickness for the corresponding half-domain. η is neither optical yield nor observed product performance.

Characteristic transport scale
tD∼L2Dt_D\sim\frac{L^2}{D}

With diffusivity held fixed, doubling layer depth increases this characteristic time scale fourfold. It is a useful question about architecture, rather than an exact equilibration-time prediction.

tDt_D
Characteristic diffusion time scales
LL
Depth of the model layerm
DD
Effective diffusivitym²/s

Model conditions. A diffusion scale under a specified D and geometry. Transient boundary conditions, swelling, partitioning and reaction can change the actual time-dependent response.

The model sharpens the experiment.

Measure the preparation and the relevant conditions. Use the calculation to identify what would change the development decision.

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The useful next step
belongs to the matrix.

For a proposed retained catalyst, useful measurements include the effective transport of each limiting reactant, partitioning at the material interface, activity after incorporation and the time evolution of the preparation. Oxygen co-limitation, enzyme deactivation or swelling could dominate a result that a simple steady profile cannot explain.

An assay developer may care about detector response, background and integrated signal over a short operating window. A materials developer may care about repeated activation, handling and a consistent layer. Both can use a defined model to identify an experiment, while their success criteria remain distinct.

The proposed programme therefore connects kinetic selection, material immobilisation, reactant access and optical measurement. A development discussion should establish the application, the architecture and the observation that would justify the next stage.

Explore the biochemical-light programmeDiscuss a reaction or material question

Sources and model scope.

Nemergut et al. (2023). Illuminating the mechanism and allosteric behavior of NanoLuc luciferase. External kinetic context; the interactive saturation curve is calculated from the displayed general relationship.

Green, W. and Wittrup, K.D. (2007). MIT 10.37, Problem Set 10 Solutions, planar catalyst derivation. The coordinate here is reversed relative to the source’s backing-origin convention. Equations are implemented for a defined hypothetical planar layer, not a fitted Bio system.