feat(_transcription_dynamics): Added function for multiome dynamics.

Signed-off-by: Alexander Aivazidis <alexander.aivazidis@sanger.ac.uk>

Author: AlexanderAivazidis

src/pyrovelocity/models/_transcription_dynamics.py

@@ -61,6 +61,73 @@ def mrna_dynamics(
     return ut, st
 
 
+@beartype
+def atac_mrna_dynamics(
+    tau_c: Tensor,
+    tau: Tensor,
+    c0: Tensor,
+    u0: Tensor,
+    s0: Tensor,
+    alpha_c: Tensor,
+    alpha: Tensor,
+    beta: Tensor,
+    gamma: Tensor,
+) -> Tuple[Tensor, Tensor]:
+    """
+    Computes the ATAC and mRNA dynamics given temporal coordinate, parameter values, and
+    initial conditions.
+
+    `st_gamma_equals_beta` for the case where the gamma parameter is equal
+    to the beta parameter is taken from Equation 2.12 of
+
+    Args:
+        tau (Tensor): Time points starting at last change in RNA transcription rate.
+        tau_c (Tensor): Time points starting at last change in chromatin opening/closing rate.
+        c0 (Tensor): Initial value of c.
+        u0 (Tensor): Initial value of u.
+        s0 (Tensor): Initial value of s.
+        alpha_c (Tensor): Rate of chromatin opening/closing.
+        alpha (Tensor): Alpha parameter.
+        beta (Tensor): Beta parameter.
+        gamma (Tensor): Gamma parameter.
+
+    Returns:
+        Tuple[Tensor, Tensor]: Tuple containing the final values of c, u and s.
+
+    Examples:
+        >>> import torch
+        >>> tau = torch.tensor(2.0)
+        >>> tau_c = torch.tensor(2.0)
+        >>> c0 = torch.tensor(1.0)
+        >>> u0 = torch.tensor(1.0)
+        >>> s0 = torch.tensor(0.5)
+        >>> alpha_c = torch.tensor(0.45)
+        >>> alpha = torch.tensor(0.5)
+        >>> beta = torch.tensor(0.4)
+        >>> gamma = torch.tensor(0.3)
+        >>> mrna_dynamics(tau_c, tau, c0, u0, s0, alpha_c, alpha, beta, gamma)
+        (tensor(1.1377), tensor(0.9269))
+    """
+
+    A = torch.exp(-alpha_c * tau_c)
+    B = torch.exp(-beta * tau)
+    C = torch.exp(-gamma * tau)
+
+    ct = c0 * A + k_c * (1 - A)
+    ut = (
+        u0 * B
+        + alpha * k_c / beta * (1 - B)
+        + (k_c - c0) * alpha / (beta - alpha_c) * (B - A)
+    )
+    st = s0 * C + alpha * k_c / gamma * (1 - C)
+    +beta / (gamma - beta) * (
+        (alpha * k_c) / beta - u0 - (k_c - c0) * alpha / (beta - alpha_c)
+    ) * (C - B)
+    +beta / (gamma - alpha_c) * (k_c - c0) * alpha / (beta - alpha_c) * (C - A)
+
+    return ct, ut, st
+
+
 @beartype
 def inv(x: Tensor) -> Tensor:
     """