Uncovering Time-Dependent NF − κB-p53 Crosstalk Induced by Caffeic Acid Phenethyl Ester in Prostate Cancer Cells Through a Bayesian Digital Twin

Abstract

(1) Background: Caffeic acid phenethyl ester (CAPE) exhibits anticancer activity; however, its time-dependent effects on interconnected signalling pathways remain incompletely characterised. (2) Methods: We combined wet-lab experiments (MTT viability assay and ELISA measurements of total $NF-\kappa B$ p65 and p53) with a Bayesian digital twin framework to quantify signalling dynamics in prostate cancer cells following CAPE exposure. p53-deficient PC3 and p53-competent LNCaP cell lines were treated for 24 h and 48 h across multiple CAPE concentrations. Experimental data were integrated into a mechanistic Bayesian model using robust likelihoods, enabling uncertainty-aware parameter inference and posterior predictive validation via leave-one-dose-out analysis. (3) Results: In PC3 cells, CAPE induced dose-dependent inhibition of $NF-\kappa B$ p65 that was consistently associated with reduced cell viability at both time points, consistent with a p53-independent regulatory regime. In contrast, LNCaP cells exhibited a transient $NF-\kappa B$–p53 coupling at 24 h, characterised by delayed $NF-\kappa B$ suppression and pronounced p53 activation, followed by a more stable and weakly coupled signalling state at 48 h. These temporal patterns were supported by posterior parameter estimates and predictive performance under leave-one-dose-out validation. (4) Conclusions: This study demonstrates that Bayesian digital twins enable quantitative, uncertainty-aware analysis of time-dependent drug responses, extending beyond conventional dose–response assessments and supporting mechanistic hypothesis generation in cancer pharmacology.

1. Introduction

Prostate cancer is among the most commonly diagnosed malignancies in men worldwide and represents a major cause of cancer-related mortality. Each year there are over 1.2 million new cases and more than 375,000 deaths attributed to prostate cancer, making it the second leading cause of cancer death in men globally [1]. While localised prostate tumours are often treatable with high long-term survival, aggressive and metastatic disease remains a significant clinical challenge.

A diverse array of in vitro models has been developed to investigate prostate cancer biology. Two of the most widely studied human prostate cancer cell lines are PC3 and LNCaP: PC3 cells are androgen receptor (AR)-negative and lack functional p53, representing late-stage, treatment-resistant disease, while LNCaP cells express AR and wild-type p53, modelling hormone-sensitive tumours [2]. These contrasting phenotypes make them valuable tools for dissecting signalling mechanisms governing cancer progression.

Among key pathways implicated in prostate cancer progression, the nuclear factor kappa-light-chain-enhancer of activated B cells ($NF-\kappa B$) transcription factor plays a well-established role in promoting proliferation, survival, inflammation, and resistance to therapy. $NF-\kappa B$ is frequently constitutively active in advanced prostate tumours and can enhance metastatic capacity and resistance to apoptosis, making it a promising therapeutic target [3]. $NF-\kappa B$ comprises a family of inducible transcription factors that regulate key cellular processes, including inflammation, immune responses, cell proliferation, survival, and stress responses. Activation of $NF-\kappa B$ typically occurs via canonical and non-canonical pathways: the canonical pathway involves receptors such as $TNF-\alpha R$, $IL-1R$, or TLRs triggering $I\kappa B$ kinase (IKK) complex phosphorylation, leading to $I\kappa B\alpha$ degradation and nuclear translocation of $NF-\kappa B$ dimers (commonly p65/p50), which then bind $\kappa B$ motifs to activate gene transcription [4]. Once in the nucleus, $NF-\kappa B$ induces expression of a broad repertoire of target genes that promote cell survival and proliferation, including anti-apoptotic proteins (e.g., Bcl-2 family), cell cycle regulators (e.g., Cyclin D1), and pro-inflammatory cytokines such as $TNF-\alpha$, $IL-6$, and $IL-1\beta$. Aberrant or constitutive $NF-\kappa B$ activation has been implicated in uncontrolled tumour cell growth, prevention of apoptosis, tumour angiogenesis, metastasis, and therapy resistance across multiple cancers [5]. Both canonical (IKK-mediated) and non-canonical pathways contribute to $NF-\kappa B$ activity in cancer, with different stimuli and receptor complexes governing context-specific responses. $NF-\kappa B$ is tightly regulated by feedback loops and inhibitory proteins (e.g., $I\kappa B$ family) that control its cytoplasmic sequestration and nuclear translocation dynamics [4]. Because of its central role in integrating inflammatory signals with survival and stress response pathways, $NF-\kappa B$ is considered a therapeutic target in cancer, although its broad physiological roles complicate selective inhibition strategies [6].

Caffeic acid phenethyl ester (CAPE) is a naturally occurring phenolic compound derived from propolis that has attracted interest for its anti-inflammatory and antitumour properties. CAPE has been shown to inhibit $NF-\kappa B$ activation [7] and suppress proliferation in human prostate cancer cells, including both PC3 and LNCaP models, and can modulate signalling networks involving p53 and androgen receptor pathways [8]. These biochemical activities suggest CAPE as a valuable probe for investigating the dynamic interplay between survival and stress response pathways in cancer cells. In our previous study, CAPE exhibited cytotoxic and antiproliferative effects on LNCaP prostate cancer cells in vitro, inducing a dose- and time-dependent reduction in mitochondrial and lysosomal activity as well as total protein synthesis, indicative of inhibited viability and metabolic function in treated cells [9].

Despite existing studies on CAPE’s effects, the time-dependent and mechanistic relationships between $NF-\kappa B$ and p53 signalling in response to CAPE remain poorly quantified. To address this gap, we combined experimental measurements in PC3 and LNCaP cells with a Bayesian digital twin framework to model signalling dynamics in a time-resolved and uncertainty-aware manner. Our study aims to elucidate how CAPE modulates $NF-\kappa B$ and p53 crosstalk and to demonstrate the utility of Bayesian mechanistic modelling for interpreting complex signalling responses beyond this specific compound.

2. Results

The MTT assay evaluated PC3 and LNCaP cell viability through 24 h and 48 h of CAPE treatment at concentrations ranging from 0 to 100 $\mathrm{\mu }$M (Figure 1). In all conditions, viability values were normalised to the respective untreated controls (0 $\mathrm{\mu }$M), which were set to 1. The PC3 cells showed decreased viability at every CAPE concentration tested during the 24 h study when researchers increased the dose levels. We observed a statistically significant decrease, namely, compared to the control value of q, of 0.0048 for 10 $\mathrm{\mu }$M, and the results became even more significant at 25, 50 and 100 $\mathrm{\mu }$M (all $q\le 0.0003$). The viability of LNCaP cells decreased significantly at all tested doses which were greater than zero during the 24 h period. The viability measurements showed significant differences from control values at all tested concentrations (all $q\le 1.9\times {10}^{-10}$). The cell viability decreased in both cell lines after CAPE treatment at 48 h. The PC3 cell viability measurements showed substantial decreases from control values at all tested concentrations which began at 10 $\mathrm{\mu }$M ($q=2.9\times {10}^{-5}$) and the decreases became more significant as drug concentrations increased. The viability of LNCaP cells became less viable at 48 h when researchers applied all the tested concentrations (all $q\le 2.8\times {10}^{-13}$). The four-parameter logistic (4PL) model enabled researchers to create dose–response curves which showed how viability values decreased step by step through all experimental conditions. The $I{C}_{50}$ values which resulted from these fits showed distinct results for each cell line and time point. The compound needed 88.6 $\mathrm{\mu }$M to stop 50% of PC3 cell growth during 24 h but only 12.3 $\mathrm{\mu }$M at 48 h. The $I{C}_{50}$ values for LNCaP cells reached 31.8 $\mathrm{\mu }$M at 24 h and 21.2 $\mathrm{\mu }$M at 48 h. The experimental data shows the fitted curves together with the calculated $I{C}_{50}$ values for reference purposes. The BH-FDR q-values are in the Table 1.

Figure 1. Dose–response effects of CAPE on metabolic viability of PC3 and LNCaP prostate cancer cells after 24 h and 48 h exposure. Points represent mean ± SD normalised to control; solid lines indicate fitted four-parameter logistic models, with dashed lines denoting $I{C}_{50}$ values. Significance levels are indicated as ** (q < 0.01), *** (q < 0.001), and **** (q < 0.0001).

Table 1. Benjamini–Hochberg false discovery rate (BH–FDR)-adjusted significance of CAPE effects on cell viability (MTT assay) relative to control (one-sample t-test vs. 1.0).

Dose ($\mathbf{\mu }$M)	q-Value (BH)	Significance
PC3, 24 h
10	0.004826	**
25	0.0006133	***
50	0.0002606	***
100	1.79 × ${10}^{-11}$	****
LNCaP, 24 h
10	1.915 × ${10}^{-10}$	****
25	1.332 × ${10}^{-11}$	****
50	2.086 × ${10}^{-16}$	****
100	1.801 × ${10}^{-15}$	****
PC3, 48 h
10	2.923 × ${10}^{-5}$	****
25	5.085 × ${10}^{-11}$	****
50	2.457 × ${10}^{-14}$	****
100	5.538 × ${10}^{-12}$	****
LNCaP, 48 h
10	2.799 × ${10}^{-13}$	****
25	3.058 × ${10}^{-14}$	****
50	5.147 × ${10}^{-18}$	****
100	2.365 × ${10}^{-17}$	****

Significance levels: ** $q<0.01$, *** $q<0.001$, **** $q<0.0001$. q-values were computed using the Benjamini–Hochberg false discovery rate procedure.

The effects of CAPE on $NF-\kappa B$ p65 and p53 protein levels were quantified by ELISA in PC3 and LNCaP prostate cancer cells following 24 h and 48 h exposure to increasing CAPE concentrations (0, 10, 25, 50, and 100 $\mathrm{\mu }$M). All values were normalised to the corresponding 0 $\mathrm{\mu }$M control. In PC3 cells, $NF-\kappa B$ p65 levels decreased monotonically with increasing CAPE concentration at both time points (Figure 2). We observed significant decreases in all tested doses ranging from 10–100 $\mathrm{\mu }$M at 24 h when compared to control samples (Table 2). The BH–FDR-adjusted q-values ranged from $2.15\times {10}^{-12}$ to $4.41\times {10}^{-19}$ throughout the 24 h study period. The observation indicated decreasing values at all tested concentrations during the 48 h study which showed lower values than control samples ($q=7.22\times {10}^{-15}$ to $7.76\times {10}^{-25}$). The levels of $NF-\kappa B$ p65 protein in LNCaP cells showed a direct relationship with treatment dose at both 24 h and 48 h time points (Figure 2). The analysis showed that all tested CAPE concentrations led to substantial $NF-\kappa B$ p65 suppression at 24 h and 48 h. Consequently, the statistical analysis showed that all tested concentrations of CAPE suppressed $NF-\kappa B$ p65 at 24 h and 48 h with q-values ranging from $2.85\times {10}^{-17}$ to $1.74\times {10}^{-30}$ and $3.04\times {10}^{-17}$ to $7.97\times {10}^{-17}$ respectively. However, p53 responses differed markedly between the two cell lines. At both the 24- and 48-h time intervals, the p53 protein levels in PC3 cells treated with various CAPE doses did not change significantly from control values (all BH-FDR-adjusted q-values > 0.80). The p53 protein levels in LNCaP cells directly matched the CAPE concentration which was used for treatment. At 24 h time point, by CAPE dose of 10 $\mathrm{\mu }$M, the p53 protein level significantly (q = 0.022) increased at 24 h. This effect became even stronger when concentrations were increased from 25 $\mathrm{\mu }$M to 100 $\mathrm{\mu }$M ($q\le 5.49\times {10}^{-15}$). The p53 levels reached their maximum point at all tested concentrations during the entire 48 h experiment while the BH-FDR-adjusted q-values ranged from $6.99\times {10}^{-16}$ to $7.78\times {10}^{-27}$.

Figure 2. Dose-dependent modulation of $NF-\kappa b$B p65 and p53 protein abundance in PC3 and LNCaP cells following CAPE treatment for 24 h and 48 h. Values are normalised to untreated controls; symbols show mean ± SD and curves represent fitted dose–response relationships. Significance levels are indicated as **** (q < 0.0001).

Table 2. Benjamini–Hochberg false discovery rate (BH–FDR)-adjusted significance of CAPE effects on $NF-\kappa B$ p65 and p53 abundance relative to control (one-sample t-test vs. 1.0).

Readout	Dose ($\mathbf{\mu }$M)	q-Value (BH)	Significance
PC3, 24 h
$NF-\kappa B$	10	$2.15\times {10}^{-12}$	****
$NF-\kappa B$	25	$7.02\times {10}^{-15}$	****
$NF-\kappa B$	50	$5.68\times {10}^{-20}$	****
$NF-\kappa B$	100	$4.41\times {10}^{-19}$	****
p53	10	0.8086	ns
p53	25	0.8086	ns
p53	50	0.8086	ns
p53	100	0.8086	ns
LNCaP, 24 h
$NF-\kappa B$	10	$2.85\times {10}^{-17}$	****
$NF-\kappa B$	25	$9.11\times {10}^{-22}$	****
$NF-\kappa B$	50	$1.78\times {10}^{-22}$	****
$NF-\kappa B$	100	$1.74\times {10}^{-30}$	****
p53	10	0.02211	*
p53	25	$9.11\times {10}^{-13}$	****
p53	50	$5.49\times {10}^{-15}$	****
p53	100	$5.49\times {10}^{-15}$	****
PC3, 48 h
$NF-\kappa B$	10	$7.22\times {10}^{-15}$	****
$NF-\kappa B$	25	$4.86\times {10}^{-18}$	****
$NF-\kappa B$	50	$8.76\times {10}^{-21}$	****
$NF-\kappa B$	100	$7.76\times {10}^{-25}$	****
p53	10	0.8483	ns
p53	25	0.8483	ns
p53	50	0.8483	ns
p53	100	0.9064	ns
LNCaP, 48 h
$NF-\kappa B$	10	$3.04\times {10}^{-17}$	****
$NF-\kappa B$	25	$8.20\times {10}^{-39}$	****
$NF-\kappa B$	50	$8.00\times {10}^{-32}$	****
$NF-\kappa B$	100	$7.97\times {10}^{-36}$	****
p53	10	$7.00\times {10}^{-16}$	****
p53	25	$4.81\times {10}^{-29}$	****
p53	50	$3.93\times {10}^{-27}$	****
p53	100	$7.78\times {10}^{-27}$	****

Significance levels: * $q<0.05$, **** $q<0.0001$, ns: not significant. q-values were computed using the Benjamini–Hochberg false discovery rate procedure.

To move beyond descriptive dose–response analyses and to quantify the joint, time-dependent behaviour of $NF-\kappa B$, p53, and viability under CAPE treatment, the experimental data were next integrated into a Bayesian digital twin framework. This modelling approach enables simultaneous inference of pathway-specific dose–response characteristics, coupling effects between signalling nodes, and uncertainty arising from biological variability and measurement noise. By embedding the measured $NF-\kappa B$ and p53 responses within a mechanistic probabilistic model of cell viability, the digital twin provides an integrated representation of pathway dynamics and allows uncertainty-aware predictions across doses and time points. In this framework, $NF-\kappa B$ and p53 are treated as independent upstream signalling axes whose coordinated influence on viability gives rise to an emergent, time-dependent statistical coupling. Thus, inferred $NF-\kappa B$–p53 “crosstalk” reflects joint effects mediated through the downstream viability phenotype rather than an explicitly encoded biochemical feedback loop between the two pathways. The results of this Bayesian inference, including posterior parameter estimates, pathway coupling strengths, and predictive validation, are presented in the following section.

By combining viability, $NF-\kappa B$, and p53 readouts into a single probabilistic framework, Bayesian inference produced well-identified, condition-specific digital twins for CAPE responses in PC3 and LNCaP cells at both 24 and 48 h (Figure 3). Posterior parameter summaries are reported in Table 3. In PC3 cells, $NF-\kappa B$ dose–response curves were steep and monotonic at both time points, with well-constrained Hill slopes (median 3.08 at 24 h; 2.88 at 48 h) and $I{C}_{50}$ values centred at 30.1 $\mathrm{\mu }$M and 34.7 $\mathrm{\mu }$M, respectively. In LNCaP cells, $NF-\kappa B$ dynamics differed by time point: at 24 h, responses were captured by a flexible monotone spline rather than a parametric 4PL curve, whereas at 48 h a shallow 4PL profile was recovered (Hill median 0.863, $I{C}_{50}$ median 42.5 $\mathrm{\mu }$M). Viability coupling to $NF-\kappa B$ was consistently positive across all conditions, with posterior medians for $\beta$ ranging from 2.76 to 5.02, while p53–viability coupling ($\gamma$) was fixed to zero in p53-deleted PC3 cells and inferred only in LNCaP, where it was prominent at 24 h (median 0.606) and effectively absent at 48 h.

Figure 3. Bayesian digital twin posterior median dose–response curves with 95% credible bands for metabolic viability, $NF-\kappa B$ p65, and p53 following CAPE exposure in PC3 and LNCaP prostate cancer cells. Panels are arranged by time (left: 24 h; right: 48 h) and readout (top: viability; middle: $NF-\kappa B$ p65; bottom: p53). Blue curves and markers correspond to PC3 (p53-deleted) cells, whereas purple curves and markers correspond to LNCaP (p53 wild-type) cells. Solid lines denote posterior medians and shaded regions indicate 95% credible intervals. Points with error bars represent experimentally observed means ± SD normalised to the 0 $\mathrm{\mu }$M control. Bayesian inference was performed independently for each cell line and time point within a unified probabilistic framework.

Table 3. Posterior summaries (median [2.5%, 97.5%]) of key Bayesian digital twin parameters across experimental conditions.

Parameter	Symbol	PC3 24 h	PC3 48 h	LNCaP 24 h	LNCaP 48 h
$NF-\kappa B$ Bottom	Bottom	0.435 [0.417, 0.452]	0.297 [0.283, 0.308]	–	0.011 [4.1 × 10−5, 0.123]
$NF-\kappa B$ Top	Top	0.951 [0.933, 0.970]	0.912 [0.894, 0.930]	–	0.991 [0.937, 1.05]
$NF-\kappa B$ IC50 ($\mathrm{\mu }$M)	IC50	30.1 [28.8, 31.6]	34.7 [33.5, 35.9]	–	42.5 [36.3, 48.2]
$NF-\kappa B$ Hill	Hill	3.08 [2.72, 3.47]	2.88 [2.60, 3.16]	–	0.863 [0.808, 1.07]
$NF-\kappa B$ curve noise	${\sigma }_{\mathrm{N},\mathrm{curve}}$	0.00107 [8.8 × ${10}^{-5}$, 0.0128]	0.00058 [8.2 × ${10}^{-5}$, 0.009]	5.1 × ${10}^{-7}$ [1 × ${10}^{-11}$, 0.0028]	7.7 × ${10}^{-7}$ [1.5 × ${10}^{-19}$, 0.0019]
$NF-\kappa B$→Viability coupling	$\beta$	5.02 [3.84, 6.98]	2.76 [2.21, 3.41]	3.28 [2.39, 4.40]	4.89 [4.39, 5.53]
p53–Viability coupling	$\gamma$	fixed 0	fixed 0	0.606 [0.193, 0.954]	7.7 × ${10}^{-8}$ [2.5 × ${10}^{-12}$, 0.018]
p53 smoothness	${\tau }_P$	–	–	0.418 [0.215, 0.987]	0.422 [0.212, 0.978]
p53 curve noise	${\sigma }_{\mathrm{P},\mathrm{curve}}$	–	–	5.9 × ${10}^{-13}$ [1.5 × ${10}^{-25}$, 1.5 × ${10}^{-6}$]	0.00214 [6.9 × ${10}^{-11}$, 0.0116]
Effective $NF-\kappa B$ noise	${\sigma }_{\mathrm{N},\mathrm{eff}}$	0.030 [0.030, 0.034]	0.030 [0.030, 0.032]	0.020 [0.020, 0.020]	0.020 [0.020, 0.020]
Effective p53 noise	${\sigma }_{\mathrm{P},\mathrm{eff}}$	–	–	0.082 [0.068, 0.102]	0.010 [0.008, 0.013]

Posterior summaries are shown as median [2.5%, 97.5%]. NF − κB was modelled using a 4-parameter logistic curve in all conditions except LNCaP at 24 h, where a monotone logit-knot spline was used. For p53-deleted PC3 cells, p53 was fixed to baseline (μ_P ≈ 1) and the coupling parameter γ was constrained to zero. A fixed log-scale noise floor (σ = 0.02) was added in quadrature to NF − κB observation noise to prevent variance collapse and ensure calibrated posterior predictive coverage.

Leave-one-dose-out (LODO) validation demonstrated robust predictive performance for viability across all conditions, with 4/4 held-out dose means captured within 95% posterior predictive intervals (Table 4). For $NF-\kappa B$, full mean coverage (4/4) was achieved at LNCaP 48 h, partial coverage (3/4) at both PC3 time points, and reduced coverage (1/4) at LNCaP 24 h. p53 predictions showed complete mean coverage (4/4) at both time points in LNCaP, while p53 was not modelled in PC3 due to biological constraints. Noise decomposition indicated that $NF-\kappa B$ uncertainty was dominated by a fixed log-scale noise floor in LNCaP and by replicate-level variability in PC3, whereas p53 uncertainty was highest at LNCaP 24 h and markedly reduced at 48 h (Table 5). Sampler diagnostics confirmed stable MCMC performance with consistent acceptance rates across chains and conditions (overall acceptance 0.53–0.64), supporting reliable posterior exploration (Table 6).

Table 4. Leave-one-dose-out (LODO) predictive performance of the Bayesian digital twin. Coverage is reported for the observed dose-mean within the 95% posterior predictive interval (PI) of a new replicate ($y_{\mathrm{rep}}$) and within the 95% PI of the replicate mean (using the observed number of replicates). Replicate PI coverage reports the mean fraction of held-out individual replicates captured by the 95% PI.

Condition	Readout	Mean Coverage (95% PI of ${\mathit{y}}_{\mathbf{rep}}$)	Mean Coverage (95% PI of Replicate Mean)	Replicate PI Coverage
PC3, 24 h	Viability (MTT)	4/4	2/4	0.86
PC3, 24 h	$NF-\kappa B$	3/4	2/4	0.73
PC3, 24 h	p53	NA	NA	NaN
PC3, 48 h	Viability (MTT)	4/4	3/4	0.81
PC3, 48 h	$NF-\kappa B$	3/4	3/4	0.72
PC3, 48 h	p53	NA	NA	NaN
LNCaP, 24 h	Viability (MTT)	4/4	3/4	0.98
LNCaP, 24 h	$NF-\kappa B$	1/4	1/4	0.36
LNCaP, 24 h	p53	4/4	3/4	0.89
LNCaP, 48 h	Viability (MTT)	4/4	1/4	0.89
LNCaP, 48 h	$NF-\kappa B$	4/4	4/4	1.00
LNCaP, 48 h	p53	4/4	4/4	1.00

LODO was performed at $d\in \{10,25,50,100\}$ $\mathrm{\mu }$M. NA indicates p53 was not modelled in PC3 due to p53 deletion (fixed baseline).

Table 5. Effective noise components inferred by the Bayesian digital twin across experimental conditions.

Condition	Readout	${\mathit{\sigma }}_{\mathbf{raw}}$	${\mathit{\sigma }}_{\mathbf{eff}}$ (+Floor)	Dominant Noise Source
PC3, 24 h	$NF-\kappa B$	0.0026 [0.00011, 0.0165]	0.030 [0.030, 0.034]	Replicate/noise floor
PC3, 48 h	$NF-\kappa B$	0.0010 [0.00010, 0.0108]	0.030 [0.030, 0.032]	Replicate/noise floor
LNCaP, 24 h	$NF-\kappa B$	4.6 × ${10}^{-5}$ [7.4 × ${10}^{-11}$, 0.0042]	0.020 [0.020, 0.020]	Noise floor-dominated
LNCaP, 48 h	$NF-\kappa B$	2.1 × ${10}^{-5}$ [1.5 × ${10}^{-13}$, 0.0026]	0.020 [0.020, 0.020]	Noise floor-dominated
PC3, 24 h	p53	0.0088 [0.0001, 0.444]	0.0088 [0.0001, 0.444]	Replicate
PC3, 48 h	p53	0.00079 [0.0001, 0.176]	0.00079 [0.0001, 0.176]	Replicate
LNCaP, 24 h	p53	0.082 [0.068, 0.102]	0.082 [0.068, 0.102]	Biological/process
LNCaP, 48 h	p53	0.010 [0.008, 0.013]	0.010 [0.008, 0.013]	Process/replicate
PC3, 24 h	Viability	0.25 (prior scale)	0.25 (effective)	Replicate
PC3, 48 h	Viability	0.25 (prior scale)	0.25 (effective)	Replicate
LNCaP, 24 h	Viability	0.25 (prior scale)	0.25 (effective)	Replicate
LNCaP, 48 h	Viability	0.25 (prior scale)	0.25 (effective)	Replicate

${\sigma }_{\mathrm{raw}}$ denotes the inferred observation-scale noise prior to applying the fixed log-scale noise floor. ${\sigma }_{\mathrm{eff}}$ denotes the effective noise used for posterior prediction after adding the fixed floor ($\sigma =0.02$) in quadrature for $NF-\kappa B$. Dominant noise source indicates the component contributing most strongly to posterior predictive uncertainty.

Table 6. Full MCMC sampler diagnostics for the Bayesian digital twin across experimental conditions.

Condition	Chain	Acceptance (Min/Median/Max)	Overall Acceptance	${\mathit{\nu }}_{\mathit{N}}$
PC3, 24 h	Chain 1	0.244/0.519/0.997	0.640	5
PC3, 24 h	Chain 2	0.244/0.522/1.000	0.643	5
PC3, 48 h	Chain 1	0.212/0.490/1.000	0.624	5
PC3, 48 h	Chain 2	0.211/0.496/1.000	0.626	5
LNCaP, 24 h	Chain 1	0.100/0.346/1.000	0.529	3
LNCaP, 24 h	Chain 2	0.097/0.352/1.000	0.529	3
LNCaP, 48 h	Chain 1	0.040/0.548/1.000	0.531	5
LNCaP, 48 h	Chain 2	0.040/0.545/1.000	0.528	5

Acceptance rates are reported after burn-in. Per-parameter values denote the minimum, median, and maximum acceptance across all sampled parameters. ${\nu }_N$ denotes the degrees of freedom of the Student-t likelihood used for $NF-\kappa B$ observations.

3. Discussion

The bioactive polyphenol caffeic acid phenethyl ester (CAPE) which originates from propolis has shown selective anticancer properties in laboratory studies and animal models through its ability to block survival pathways like $NF-\kappa B$ and its effects on cell cycle regulators including p53 and its associated targets [7,10,11]. The present research used MTT assays to show that CAPE treatment at 10–100 $\mathrm{\mu }$M concentrations resulted in a dose- and time-dependent reduction in metabolic viability of PC3 and LNCaP cells, as assessed by the MTT assay. It should be noted that the MTT assay reports changes in cellular metabolic activity and mitochondrial reductive capacity and therefore does not directly distinguish between reduced proliferation, metabolic suppression, or irreversible cell death. Consequently, the MTT-based viability reductions observed here are interpreted as functional antiproliferative effects rather than definitive apoptotic or necrotic outcomes. The results show that CAPE causes cell viability and prevents cell growth in prostate cancer cells as previously studied in PC3 cells and additional cancer models [12,13]. The parallel ELISA results demonstrated that CAPE decreased $NF-\kappa B$ p65 activity at all examined concentrations and time points in both cell types, in line with previous work showing that CAPE inhibits $NF-\kappa B$ signalling by preventing p65 nuclear translocation and subsequent activation of pro-survival transcriptional programmes in cancer and immune cells [8,14]. By contrast, p53 responses were absent in PC3, as expected given p53 deletion in this line, but emerged robustly in LNCaP cells, with significant increases in p53 abundance at higher CAPE doses and both time points, reflecting CAPE’s capacity to engage p53-associated stress and apoptotic pathways in p53-wild-type contexts [11]. The effect of CAPE on p53-associated stress and growth-inhibitory responses in other models was also observed [15]. These experimental results provide a comprehensive foundation for interpreting the digital twin modelling outcomes presented in the Results section, framing CAPE’s effects on viability, $NF-\kappa B$ suppression, and p53 engagement in prostate cancer cells within established cancer biology paradigms.

Our Bayesian digital twin analysis not only quantified the dose-dependent suppression of viability and $NF-\kappa B$ signalling by CAPE, but also sheds light on how these pathways interact mechanistically in prostate cancer cells. $NF-\kappa B$ is a central regulator of inflammation, cell survival, and oncogenic programmes, and its aberrant activation contributes to chemoresistance and cancer progression across tumour types, making it a compelling therapeutic target in solid malignancies including prostate cancer [16]. Importantly, the Bayesian digital twin does not impose direct $NF-\kappa B$–p53 feedback at the model equation level. Instead, apparent crosstalk emerges from the inferred coupling of each pathway to viability, allowing pathway interactions to be detected without hard-coding specific regulatory mechanisms. CAPE’s inhibitory effect on $NF-\kappa B$ p65 activity is consistent with recent evidence that targeting $NF-\kappa B$ signalling can diminish pro-survival gene expression and sensitize cancer cells to cytotoxic stress, potentially overcoming intrinsic drug resistance [17]. In our simulation, $NF-\kappa B$ dose–response parameters were well constrained and viability coupling terms ($\beta$) were positive, quantitatively linking $NF-\kappa B$ inhibition to reduced viability in both PC3 and LNCaP cells. The tumour suppressor p53, often described as the “guardian of the genome” for its roles in cell cycle arrest, stress and growth-inhibitory response, and genomic stability [18], is known to interact with $NF-\kappa B$ in a context-dependent manner, with cross-regulatory effects that influence inflammatory and apoptotic responses. Although classical reviews of p53–$NF-\kappa B$ crosstalk pre-2019 highlight antagonistic and cooperative mechanisms [19], recent cancer research underscores the relevance of this axis in determining cell fate decisions under stress, with $NF-\kappa B$ activity modulating p53-dependent apoptosis and vice versa. In our Bayesian results, p53–viability coupling ($\gamma$) was absent in p53-deleted PC3, while LNCaP showed substantial coupling at 24 h and near-baseline at 48 h, reflecting temporal dynamics in p53 engagement that may arise from $NF-\kappa B$-p53 interplay. Leave-one-dose-out validation confirmed high predictive coverage for viability outcomes across conditions, while $NF-\kappa B$ and p53 coverage varied by cell line and time, suggesting non-linear crosstalk and context-specific modulation of these pathways. Noise decomposition further revealed that $NF-\kappa B$ uncertainty was often dominated by replicate and noise-floor components, implying biological and measurement heterogeneity in signalling readouts, whereas p53 noise reflected true biological variability in responsive cells. These integrated findings support a mechanistic model wherein CAPE’s antiproliferative efficacy is rooted in potent $NF-\kappa B$ inhibition with secondary modulation of p53-associated tumour suppressor pathways, illustrating the complex interplay between pro-survival and stress-responsive networks in prostate cancer cells.

While MDM2 was not directly measured in this study, it is introduced here as a hypothesis-driven mechanistic mediator linking p53 and $NF-\kappa B$ signalling, based on the extensive prior literature. The protein MDM2 (E3 ubiquitin-protein ligase) functions as the main negative regulator of p53 because it breaks down p53 through ubiquitin-dependent protein degradation. The protein MDM2 serves as the network hub which links survival-promoting pathways to tumour suppressor mechanisms. MDM2 controls more than just p53 feedback mechanisms because the p53-MDM2 axis functions as a primary target for cancer treatment approaches which use natural compounds and small molecules to block MDM2-p53 binding and activate p53-mediated cell death in tumour cells [20]. $NF-\kappa B$ promotes MDM2 gene expression through transcriptional processes which establishes a self-reinforcing mechanism that connects inflammatory and survival-promoting $NF-\kappa B$ pathways to p53 activity suppression for cell survival in particular oncogenic conditions [21]. Research findings indicate that MDM2 functions as a co-factor which binds to $NF-\kappa B$ target gene promoters to activate the transcription of p65 and other $NF-\kappa B$ subunit genes which suggests MDM2 controls the $NF-\kappa B$ pathway through direct mechanisms. It was demonstrated that MDM2 functions as a p53 stability regulator while also controlling p53-$NF-\kappa B$ signalling pathways which makes it an appropriate addition to models of short-term axis interactions and validates the biological feasibility of mechanistic loops that involve CAPE regulation of these linked pathways [20,21]. Therefore, we hypothesise that the principal p53-mediated inhibitory influence on $NF-\kappa B$ is exerted through MDM2, as schematically depicted in Figure 4. While our schematic incorporates MDM2 as a plausible mechanistic node in the p53–$NF-\kappa B$ signalling axis supported by the recent cancer biology literature, its specific role in mediating CAPE-induced crosstalk remains a hypothesis that rationalises our observed dynamics rather than a direct conclusion drawn from the present experimental results.

Figure 4. Proposed mechanistic schematic (hypothesis-driven) illustrating CAPE-induced modulation of the $NF-\kappa B$–p53 regulatory axis in prostate cancer cells. CAPE suppresses $NF-\kappa B$ p65 activity, leading to reduced pro-survival signalling, while simultaneously promoting p53 activation in p53-competent cells (LNCaP). Activated p53 is hypothesised to further inhibit $NF-\kappa B$ signalling via MDM2-mediated mechanisms, contributing to time-dependent pathway decoupling observed at 48 h.

Several limitations of this study should be acknowledged. The study investigated two different prostate cancer cell lines (p53-deleted PC3 and p53-competent LNCaP) but the results only apply to laboratory tests and do not represent the complete interactions between tumours and their environment and drug distribution which occur in living organisms. As no direct apoptosis markers (e.g., caspase activation or Annexin V staining) were assessed, apoptotic engagement cannot be conclusively established from the present data. Moreover, it should be emphasised that $NF-\kappa B$ p65 and p53 were quantified as total protein abundance by ELISA. It does not show where these proteins exist in cells or their modified states or their ability to transcribe genes which are essential for pathway operation. These measurements do not capture nuclear translocation, phosphorylation status, or transcriptional activity, which are critical determinants of functional pathway activation. Consequently, inferred signalling dynamics reflect changes in protein abundance rather than direct measures of pathway activity, and should be interpreted accordingly.

The Bayesian digital twin framework includes built-in mechanisms to handle both uncertain data and non-linear dose–response relationships but it still uses simplified models to represent biological systems. The model structure contains simplified representations of biological systems because it lacks detailed models of intermediate regulators including MDM2 and $I\kappa B$ dynamics and MAPK or PI3K/AKT pathway feedback mechanisms. The model required leave-one-dose-out validation on discrete concentration points but it did not validate its performance at doses which were above the tested range. The method of Bayesian inference enables researchers to determine uncertainty levels through mathematical principles yet the obtained results depend on both the initial assumptions and the experimental measurement errors which affect parameter identification during experiments with low signal strength and high measurement variability (e.g., $NF-\kappa B$ at LNCaP 24 h). Future studies require time-dependent data collection methods which should assess multiple biological pathways and perform animal tests to validate and improve the proposed CAPE–$NF-\kappa B$–p53 regulatory system. A valuable next step for further validation of the digital twin would involve targeted chemical perturbations using selective $NF-\kappa B$ or p53 modulators (e.g., IKK inhibitors or MDM2–p53 axis inhibitors). Such experiments would enable direct causal testing of the inferred coupling parameters ($\beta$ and $\gamma$) by comparing digital twin predictions with pathway-specific perturbations. Although CAPE is not pathway-specific, its well-documented inhibition of $NF-\kappa B$ provides a biologically meaningful perturbation that enables the inference of $NF-\kappa B$-viability coupling under pharmacologically relevant conditions. While beyond the scope of the present study, the current framework is explicitly designed to incorporate such data and to support future causal refinement of signalling interactions.

4. Materials and Methods

4.1. Human Prostate Cell Lines

Originating from a bone metastasis of a high-grade prostatic adenocarcinoma, PC3 cells are a commonly used human prostate cancer cell line. The cells are appropriate for laboratory research of advanced castration-resistant prostate cancer because they exhibit aggressive metastatic behaviour, androgen-independent proliferation, and no response to the androgen receptor (AR). Prostate-specific antigen (PSA) and functional p53 expression are absent in PC3 cells compared to hormone-sensitive models, which is consistent with the aggressive clinical manifestation of prostate cancer. A human prostate cancer cell line called the LNCaP model was created from a lymph node metastasis. The cells maintain their androgen sensitivity because they have an active androgen receptor which also produces PSA thus serving as a classic example of prostate cancer at its initial hormone-dependent phase. The LNCaP cells contain wild-type p53 which enables researchers to investigate p53-related stress responses that become dysfunctional in aggressive cancer types. The two cell lines PC3 and LNCaP show different biological characteristics which represent two stages of prostate cancer development from hormone-responsive p53-intact tumours to hormone-resistant p53-deficient tumours. The biological distinction between these two cell lines enables researchers to study CAPE effects on cell survival and $NF-\kappa B$ pathway activation and p53-dependent responses [22]. Both lines were sourced from Sigma Aldrich, Warsaw, Poland (PC3—ref: 90112714, LNCaP—ref. 89110211).

4.2. Cell Viability MTT Assay

The MTT reduction assay served to evaluate CAPE cytotoxicity [13] with some adjustments to the procedure. The research exposed PC3 and LNCaP cells to CAPE at four different concentrations (10, 25, 50 and 100 $\mathrm{\mu }$M) for two time periods of 24 and 48 h. The research used vehicle-treated cells at 0 $\mathrm{\mu }$M CAPE as control samples. The cells received MTT reagent (1 mg/mL) after treatment and spent 4 h at 37 °C under 5% ${\mathrm{CO}}_2$ in a humidified environment. The formazan crystals were dissolved in 150 $\mathrm{\mu }$L of dimethyl sulfoxide (DMSO) after the culture media was discarded. The EL×800 micro-plate reader from BioTek (Shoreline, WA, USA) operated at 450 nm absorbance along with using a wavelength of 630 nm as its background adjustment. The results show cell viability as a percentage of the 0 $\mathrm{\mu }$M control values. The four-parameter logistic (4PL) regression function was used to model dose–response relationships which produced half-maximal inhibitory concentrations ($I{C}_{50}$) values from the fitted curves. The four-parameter logistic (4PL) function served to model dose–response relationships:

$$y=Bottom+{\displaystyle \frac{Top-Bottom}{1+{\left(\frac{x}{I{C}_{50}}\right)}^{\mathrm{Hill}}}}$$

(1)

where y denotes the normalised cell viability measured by the MTT assay, x represents the CAPE concentration, and $Top$ and $Bottom$ correspond to the upper and lower asymptotes of the response curve, respectively. The Hill coefficient describes the steepness of the concentration–response relationship. This sigmoidal model captures the saturable nature of biological responses to increasing compound concentrations and is well suited for quantifying reduction of metabolic activity in vitro. Non-linear regression was performed in MATLAB R2025a (MathWorks Inc., Natick, MA, USA), and $I{C}_{50}$ values were defined as the CAPE concentrations inducing a 50% reduction in cell viability relative to the control.

4.3. ELISA p53 and $NF-\kappa B$ Assay

The Human p53 SimpleStep ELISA® Kit (Abcam Ltd., Cambridge, UK, ref: ab171571) and the $NF-\kappa B$ p65 SimpleStep ELISA® Kit (Abcam Ltd., Cambridge, UK, ref: ab176648) served to determine the overall amounts of p53 and $NF-\kappa B$ p65 proteins. The tests operate as single-wash sandwich ELISAs which use a one-step antibody–analyte complex formation method to complete their analysis. The research involved treating PC3 and LNCaP cells with CAPE at five different concentrations (0 $\mathrm{\mu }$M vehicle control and 10, 25, 50 and 100 $\mathrm{\mu }$M) for two time periods of 24 and 48 h. Following the manufacturer’s instructions, cell lysates were produced using 1 × Cell Extraction Buffer PTR at a concentration of around 200 $\mathrm{\mu }$g/mL. Briefly, the signal production used 3,3′,5,5′-tetramethylbenzidine (TMB) which produced results after incubation followed by absorbance measurement at 450 nm. The $NF-\kappa B$ p65 test demonstrates its ability to detect human cell lysates which contain proteins between 6–600 $\mathrm{\mu }$g/mL while its minimum detection limit is 5 $\mathrm{\mu }$g/mL. The p53 assay detects proteins at concentrations beginning from 65 pg/mL. Protein levels were expressed in relation to the matching 0 $\mathrm{\mu }$M CAPE control for each cell line and time point after being normalised to the total protein content in the lysates. All experimental steps were performed in accordance with the manufacturer’s manual.

4.4. Statistical Analysis

For the experiments, obtained values were first normalised to the corresponding 0 $\mathrm{\mu }$M CAPE control, yielding relative viability values. The mean ± standard deviation (SD) of 16 independent biological replicates is used to present all quantitative data, respectively. A one-sample Student’s t-test was used to determine the statistical significance of the effect of the treatment for each cell line (PC3 and LNCaP), time point (24 h and 48 h) and CAPE dose (10, 25, 50 and 100 $\mathrm{\mu }$M). This test tested the null hypothesis that the mean normalised viability equals 1.0 (control baseline). The method works well for in vitro assays which use internal normalisation because the control condition becomes irrelevant after normalisation. The Benjamini–Hochberg false discovery rate (BH-FDR) procedure was used to adjust the raw p-values which were calculated for each cell line and time point across all tested CAPE doses. The study used adjusted q-values to establish statistical significance through four different threshold values which included $q<0.05$, $q<0.01$, $q<0.001$, and $q<0.0001$. Statistical analyses and curve fitting were performed in MATLAB R2025a (MathWorks, Natick, MA, USA).

4.5. Bayesian Digital Twin: Data Integration, Model Equations, and Simulation Logic

To generate an uncertainty-aware digital representation (“digital twin”) of CAPE response in prostate cancer cells, we integrated three experimentally observed endpoints—cell viability (MTT), total $NF-\kappa B$ p65 abundance, and total p53 abundance—measured in PC3 and LNCaP cells after 24 h and 48 h exposure to CAPE (0, 10, 25, 50, 100 $\mathrm{\mu }$M). For each experimental condition, data were stored as replicate matrices $Y^{\left(V\right)},Y^{\left(P\right)},Y^{\left(N\right)}\in {\mathbb{R}}^{R\times D}$, where R denotes the number of biological replicates (typically $R=16$) and $D=5$ is the number of dose levels.

The modelling objectives were to (i) quantify dose- and time-dependent effects with explicit uncertainty, (ii) infer cell line-specific response parameters (“digital phenotypes”), and (iii) generate probabilistic predictions at observed and unobserved doses, enabling leave-one-dose-out (LODO) validation without additional wet-lab experiments.

To remove between-plate scaling and ensure comparability across replicates, each replicate row was normalised by its own control (0 $\mathrm{\mu }$M) value:

$$\tilde{Y}r,j={\displaystyle \frac{Yr,j}{Y_{r,0}}},j\in 0,10,25,50,100.$$

(2)

This normalisation was applied independently to viability, p53, and $NF-\kappa B$ data. After normalisation, the control column equals one by construction (up to floating-point precision), allowing all model components to be interpreted as relative responses.

For most conditions, the expected $NF-\kappa B$ response ${\mu }_{N\left(d\right)}$ was modelled using a four-parameter logistic (4PL, Hill-type) function:

$${\mu }_N\left(d\right)=Bottom+{\displaystyle \frac{Top-Bottom}{1+{\left(\frac{d}{{IC}_{50}}\right)}^{\mathrm{Hill}}}}.$$

(3)

where Bottom and Top represent lower and upper asymptotes, $I{C}_{50}$ the half-maximal inhibitory dose, and Hill the slope parameter. This formulation captures monotonic suppression of $NF-\kappa B$ with increasing CAPE dose and yields interpretable posterior summaries for digital phenotyping.

Control behaviour at $d=0$ was handled explicitly in a dose-aware manner, ensuring ${\mu }_{N\left(0\right)}=1$ after normalisation and the correct behaviour for both vectorised dose grids and single-dose posterior predictions.

For LNCaP cells at 24 h, the data exhibited a smooth but non-sigmoidal monotone decline that was poorly captured by a single 4PL curve. Therefore, $NF-\kappa B$ was instead modelled using monotone logit-scale knots at the non-zero doses (10, 25, 50, 100 $\mathrm{\mu }$M). The knot values $u_{10}, u_{25}, u_{50},$ and $u_{100}$ (on the logit scale) were constrained to be monotone and interpolated smoothly across log-doses using shape-preserving cubic interpolation. This hybrid specification avoids forcing an incorrect parametric form while preserving smoothness and biological plausibility.

To capture residual dose-dependent deviations beyond the mean curve, an additional $NF-\kappa B$ process-noise term ($sigNcurve$) was included. Replicate-scale noise ($sigN$) and plate-level variability ($sigNplate$) were modelled separately and combined additively on the log scale. For predictive uncertainty, a small fixed noise floor was added to the effective $NF-\kappa B$ variance, preventing pathological under-coverage when biological noise was estimated to be extremely small.

Because p53 status differs fundamentally between cell lines, p53 was handled conditionally:

• PC3 (p53-deleted):
  p53 was treated as non-informative for viability coupling. The expected p53 response was fixed to a constant baseline ${\mu }_P\left(d\right)=1$, and the p53-to-viability coupling parameter $\gamma$ was fixed to zero, reflecting p53-null biology and preventing spurious inference.
• LNCaP (p53 wild-type):
  To avoid imposing an incorrect parametric shape, p53 was modelled flexibly using dose-specific knots on the log-scale of ${\mu }_P\left(d\right)$ at 10, 25, 50, and 100 $\mathrm{\mu }$M. Knot values ($pu10, pu25, pu50, pu100$) were interpolated smoothly across log-doses, with a second-difference smoothness prior controlled by a hyper-parameter ${\tau }_P$. This allows p53 to exhibit non-sigmoidal and time-dependent responses while remaining regularised.

Viability was modelled as a logistic function of a latent linear predictor that couples $NF-\kappa B$ and p53:

$${\mu }_V\left(d\right)=\sigma \left(\alpha -\beta \left(1-{\mu }_N\left(d\right)\right)-\gamma \left({\mu }_P\left(d\right)-1\right)\right),\sigma \left(x\right)={\displaystyle \frac{1}{1+e^{-x}}}.$$

(4)

Here, $\alpha$ is a baseline viability intercept on the logit scale, $\beta$ quantifies sensitivity to $NF-\kappa B$ suppression, and $\gamma$ quantifies sensitivity to p53 elevation. For PC3 cells, $\gamma$ was fixed to zero by construction. This coupling yields interpretable posterior effects and supports biological hypotheses of $NF-\kappa B$–p53 antagonism without hard-coding feedback into the $NF-\kappa B$ model itself.

All three readouts were modelled on the log scale to respect positivity and relative normalisation, but with endpoint- and condition-specific likelihood families matching the implemented code:

• Viability (MTT):
  A log-normal observation model was used, $log\left(\tilde{y}\right)$∼$\mathcal{N}\left(log{\mu }_V\left(d\right),{\sigma }_V^2\right)$, with replicate-scale variance $sigV$.
• $NF-\kappa B$:
  Likelihoods were defined on the log scale as

  –

  LNCaP 24 h—Gaussian likelihood on log-scale residuals;

  –

  All other conditions (PC3 24/48 h, LNCaP 48 h)—Student-t likelihood on log-scale residuals with fixed degrees of freedom ${\nu }_N$.

  Effective $NF-\kappa B$ noise combined replicate-scale, plate-level, and process components, with an additional fixed noise floor applied only for predictive uncertainty.
• p53:
  A Student-t likelihood was used on the log scale, $log\left(\tilde{y}\right)$∼$t_{{\nu }_P}\left(log{\mu }_P\left(d\right),{\sigma }_P^2\right)$, with replicate-scale, plate-level, and process components combined into an effective variance.

This robust formulation accommodates heavy-tailed replicate variability commonly observed in immunoassays and reporter-based measurements.

Weakly informative priors were used throughout to stabilise inference while preserving biological interpretability. $NF-\kappa B$ 4PL parameters were constrained to physiologically plausible ranges; coupling and noise parameters were assigned half-normal priors; and p53 knot values were regularised via smoothness priors. PC3-specific biological constraints (p53 deletion and $\gamma =0$) were enforced at the parameter level.

Let $\theta$ denote the full parameter vector and Y the observed data. Bayesian inference targets the posterior

$$p(\theta \mid Y)\propto p(Y\mid \theta ),p\left(\theta \right).$$

(5)

Because this posterior has no closed-form solution, it was approximated numerically using a Metropolis-within-Gibbs (MWG) sampler implemented in MATLAB. All parameters were sampled in an unconstrained space using log and logit transformations.

For each experimental condition (cell line × time), two independent MCMC chains were run from dispersed initial states. During burn-in, proposal step sizes were adaptively tuned to achieve efficient mixing, after which adaptation was disabled. Samples were thinned and pooled across chains. Reported overall and per-parameter acceptance rates serve as internal diagnostics of sampler performance. Posterior samples were propagated through the mechanistic model to compute posterior medians, 95% credible intervals, and posterior predictive distributions.

For each condition, two independent Metropolis-within-Gibbs chains were run with 4000 burn-in iterations and 12,000 sampling iterations; samples were thinned by a factor of five to reduce autocorrelation and pooled across chains for posterior inference.

Predictive validity was assessed using leave-one-dose-out (LODO) cross-validation at non-zero doses (10, 25, 50, 100 $\mathrm{\mu }$M). For each held-out dose, the corresponding column was removed simultaneously from viability, $NF-\kappa B$, and p53 data, the model was refit, and posterior predictions at the held-out dose were generated.

Coverage was evaluated using: (i) 95% posterior predictive intervals for new observations, (ii) predictive intervals for the replicate mean, and (iii) replicate-level inclusion fractions.

All analyses were performed in MATLAB R2025a (MathWorks, Natick, MA, USA) with fixed random seeds. The full pipeline—data import, normalisation, per-condition Bayesian fitting, posterior prediction, and LODO validation—was executed sequentially for PC3 24 h, PC3 48 h, LNCaP 24 h, and LNCaP 48 h.

This framework enables explicit separation of (i) sampling adequacy, (ii) curve identifiability, and (iii) out-of-sample predictability, providing a rigorous basis for interpreting time-dependent $NF-\kappa B$–p53 crosstalk in prostate cancer cells.

Parametrisation and prior choices were selected to balance biological interpretability, numerical stability, and predictive calibration, enabling explicit separation of direct $NF-\kappa B$ inhibition, p53-mediated effects, and observation noise across cell lines and time points. The list of parameters used are showed in Table A1. All mathematical symbols, their corresponding equations, and MATLAB implementation variables are summarised in Table A2 to ensure full reproducibility of the Bayesian digital twin framework.

5. Conclusions

The study employed wet-lab assays in combination with Bayesian digital twin methods to evaluate prostate cancer cell survival and $NF-\kappa B$ p65 signalling and p53 responses under different CAPE concentrations and time exposures. The compound CAPE showed two primary effects which reduced cell survival and stopped $NF-\kappa B$ signalling in both PC3 and LNCaP cancer cells. The p53 protein activated at different time points but this response was restricted to LNCaP cells which have active p53 proteins. The Bayesian inference method combined all experimental data into a single probabilistic framework which allowed researchers to estimate pathway connections and noise patterns and prediction accuracy with uncertainty quantification. The digital twin showed that PC3 cells maintain viability through an $NF-\kappa B$ pathway which does not require p53 function and LNCaP cells experience a short $NF-\kappa B$-p53 connection during the first 24 h which disappears by 48 h. The research shows how Bayesian digital twins enable scientists to develop new mechanistic theories and make predictions about cancer drug responses while handling uncertainty in their models. The method shows potential to analyse additional biological compounds and their related signalling pathways.