Intracellular Payload Release Through Endosomal and Lysosomal Steps

Why Separate Endosomal and Lysosomal Processing?

After an ADC is internalized, payload liberation need not occur immediately. The ADC may first reside in an endosomal state and then transfer to a lysosomal state, where processing liberates payload. Representing these stages separately makes the timing of liberation explicit: a slow endosomal transfer can delay lysosomal arrival, while slow lysosomal processing can delay liberation even after arrival.

The model below begins with intracellular ADC influx, not administered dose or plasma concentration. The earlier material on ADC species states and internalization provides the context for defining that influx. Linker stability outside the cell is a separate process; here, an ADC entering the model carries a fixed amount of payload until lysosomal processing liberates it.

The model assumes well-mixed states, constant positive rate constants, and first-order transfer and processing. It also assumes that ADC is not lost through competing pathways after entry, that each ADC carries the same \(q\), and that liberated payload is counted cumulatively rather than removed from a free-payload pool. The standard starting condition is \(E(0)=L(0)=P(0)=0\). These assumptions make the model a transparent description of sequential delay, not a complete account of every intracellular fate.

Deriving the Time Course of Liberation

Consider first a single cohort of ADC molecules that enters the endosomal state at time zero. Its endosomal residence time is exponentially distributed with rate \(k_E\), and its lysosomal processing time is exponentially distributed with rate \(k_L\). The total time until processing is complete is the sum of these two residence times. For \(k_E\ne k_L\), the probability that a cohort molecule is still unprocessed at age \(a\) is

Thus, the fraction processed by age \(a\) is \(F(a)=1-S(a)\). One justification is to integrate over the possible time \(s\) spent in the endosomal state: completion by age \(a\) requires \(0\le s\le a\), with endosomal transfer density \(k_Ee^{-k_Es}\) and lysosomal completion probability \(1-e^{-k_L(a-s)}\). Therefore,

If the rates are equal, \(k_E=k_L=k\), the limiting expression is \(S(a)=e^{-ka}(1+ka)\), so \(F(a)=1-e^{-ka}(1+ka)\). This equal-rate form avoids division by zero and is useful when the two estimated rates are close.

For a time-varying influx, each input cohort contributes according to its age. The cumulative number of processed ADC molecules per cell by time \(t\) is \(\int_0^t u(s)F(t-s)\,ds\), and cumulative payload liberation is \(q\) times that amount. This convolution makes the delay explicit: recently internalized ADC contributes little to liberation when the processing steps take time.

Constant Input and the Effect of Rate Changes

For constant input \(u\) from time zero through time \(t\), the ages of the entered ADC molecules range from zero to \(t\). Integrating the cohort completion fraction gives the number processed. Equivalently, integrate the unprocessed fraction \(S(a)\) and subtract it from the total input \(ut\):

Here \(R(t)\) is processed ADC in ADC molecules per cell, and \(P(t)\) is cumulative liberated payload in payload molecules per cell. The expression follows because \(\int_0^t S(a)\,da=\{\frac{k_L}{k_E}(1-e^{-k_Et})-\frac{k_E}{k_L}(1-e^{-k_Lt})\}/(k_L-k_E)\). This formula applies when the input has been constant throughout the interval being evaluated; for an input that stops earlier, the cohort-convolution form remains applicable.

What Happens When Input Stops?

Stopping input does not stop the processing of ADC already in \(E\) or \(L\). For example, if constant input ends at time \(T\), later liberation comes from the cohorts that entered before \(T\); it is not calculated by setting \(P(t)\) to its value at \(T\). In the no-loss model, the eventual processed amount equals the total ADC that entered. The time course differs with \(k_E\) and \(k_L\), even though the eventual yield does not.

Diagnostics, Alternatives, and Limitations

The model provides direct checks on calculations and on fitted trajectories. With zero initial amounts, the ADC mass balance is

This balance follows by adding the three state equations after dividing the \(P\) equation by \(q\): transfer terms cancel, leaving the input. If a computed solution violates this identity, check units, signs, initial conditions, and numerical implementation. Also check that \(E\), \(L\), and \(P\) remain nonnegative, that \(P(t)\) does not decrease, and that liberation cannot exceed \(q\) times cumulative ADC input.

A poor fit can indicate that one or more assumptions are inadequate. For example, a persistent mismatch in the observed timing of lysosomal arrival may indicate that a single first-order endosomal state is too simple. A measured loss of ADC-derived material would violate the no-competing-loss assumption and alter eventual yield. Measurements must also distinguish ADC still carrying payload from liberated payload; otherwise the two modeled processes may not be separately identifiable from the data.

A one-step model can be useful when only the overall delay matters, but it cannot attribute that delay to endosomal transfer versus lysosomal processing. The two-step model preserves that distinction at the cost of additional parameters and data requirements. If observations support a broad distribution of transit times rather than exponential residence times, a more detailed transit representation may be considered; adding states without informative measurements, however, can make rate estimates difficult to interpret. This model also does not describe the concentration of free payload after liberation, its removal, or its effects on cell viability.