Simulation Training

glam2 examines a set of sequences provided by the user, and returns an alignment of segments of these sequences. A typical alignment is shown in Figure 1. Each sequence contributes at most one segment to the alignment. Our approach assumes that a motif is defined by residue preferences at certain positions, which we call key positions. These are analogous to the “turned-on” columns of the second-generation Gibbs sampler, or to the match states of a profile HMM. In a particular motif instance, some key positions may be deleted, and residues may be inserted between key positions (Figure 1).
glam2 defines a scoring scheme for alignments such as that in Figure 1. It rewards alignment of identical or similar residues in the same key position, and penalizes deletions and insertions. However, deletions and insertions are penalized less strongly if they repeatedly occur in the same locations. This is reasonable because some locations in a motif may be more prone to deletions or insertions than others. Having defined a scoring scheme for alignments, it is straightforward to calculate the marginal score of one aligned segment: the score of the alignment including this segment minus the score of the alignment excluding this segment. These marginal scores reflect how well each segment matches the other segments.
Having defined a scoring scheme, glam2 attempts to find a motif alignment with maximum score. Even in the gapless case, the number of possible alignments is too huge to enumerate, and there is no practical algorithm to guarantee finding the optimal alignment. This problem is only exacerbated in the gapped case. Thus glam2 uses a heuristic optimisation method – simulated annealing – highly analogous to the optimisation methods of the gapless Gibbs samplers [10] (link),[27] (link).
Simulated annealing takes an initial, presumably non-optimal, alignment and repeatedly makes changes to it. These changes have an element of randomness: they generally increase the score, but sometimes decrease it, which avoids getting stuck in local optima. The process is analogous to crystallization in a cooling material. Two types of change are performed by glam2, which we call site sampling and column sampling, because they are analogous to similarly-named procedures in the original Gibbs sampler [10] (link),[24] (link). Site sampling adjusts the alignment of one sequence to the motif, using the clever stochastic traceback procedure from hmmer to efficiently sample one from all possible such alignments [22] (link). In column sampling, one key position is moved, added, or deleted. These changes are carefully designed to satisfy the reversibility and detailed balance conditions of simulated annealing (Text S1). Such changes are applied until the score fails to improve for n (e.g. 10000) changes in succession. To check that a reproducible, high-scoring motif has been found, the whole procedure is repeated r (e.g. 10) times from different random starting alignments.
glam2's behaviour can be controlled with numerous adjustable parameters. The allowed alignments can be constrained by specifying a minimum number of key positions (a), a maximum number of key positions (b), and a minimum number of segments in the alignment (z). This z parameter is a useful generalization of the OOPS (one occurrence per sequence) and ZOOPS (zero or one occurrence per sequence) modes of previous motif discovery algorithms [28] . The annealing follows a simple geometric cooling schedule with initial temperature t and cooling rate c per n changes. glam2 can find the optimal number of key positions more quickly if the initial number (w) is set to a near-optimal value. All parameters have sensible default values.
glam2scan takes a motif found by glam2, and scans it against a database of sequences. It performs short-in-long alignments of the motif against the sequences, using position-specific residue scores, deletion scores, and insertion scores, which are derived from the glam2 alignment. The highest-scoring such alignments are reported.

A closed-loop circulatory flow system was designed that included a centrifugal pump (Cole-Parmer, IL, United States), reservoir, flow meter and ultrasonic flow probes, and pressure transducers as shown in Figures 1A,B. The flow loop was connected to the cerebrovascular model, which was placed in the supine orientation. The inlet and outlet flow rates and pressures were monitored using ultrasonic flow probes (Transonic Systems, Inc., Millis, MA) and pressure transducers (Merit Medical, South Jordan, UT), respectively. The pressure transducers were connected to an analog data acquisition module (DAQ, National Instruments, Texas, United States) and recorded using LabVIEW software (National Instruments, Texas, United States).
Similar to Riley et al. (26 (link)), the working fluid consisted of a mixture of water (60% by weight) and glycerol (40% by weight) to obtain a density and dynamic viscosity that is representative of blood (1.09 ± 0.03 g/ml and 3.98 ± 0.14 cP, respectively) at an operating temperature of 22.2°C. Experiments were performed using a steady inlet flow rate of 5.17 ± 0.078 L/min, corresponding to a Reynolds number (Re) in the inlet tube of approximately 3890. This inlet flow rate was chosen to correspond to a representative mean physiological cardiac output of an adult. An extended tube of 900 mm in length was attached to the model inlet such that the flow entering the model inlet was fully developed. To study the effect of a stroke condition on the mean arterial pressure and flow rate in the cerebral arteries, nylon spherical clots of three different sizes (3.15 mm, 4.75 mm, and 6.38 mm in diameter) were manually inserted into the right MCA (RMCA) to completely block the vessel and the corresponding flow through outlet 4, as depicted in Figure 1C.
Three separate experiments were performed for both conditions (normal and stroke) to measure the flow rate and pressure at the inlet and various outlets in order to provide boundary conditions for the CFD simulations and for validation. Because the fluid heats up as it is continuously pumped through the flow loop, we waited for 30 min to allow for the fluid temperature to reach a steady state (22.2°C) before measuring the flow rate and pressure. Importantly, the viscosity of the fluid was tuned to account for the effect of this temperature rise, so as to obtain the desired value of 3.98 cP at the steady-state operating temperature. The regional distribution of the flow rate was tuned to match available literature data (30 (link), 31 (link)) by adjusting clamps downstream of the pressure and flow rate measurement sites. This yielded a regional flow distribution such that 73.2% of the flow passed through the descending aorta and 26.8% of the flow was distributed to the remaining arteries stemming from the aortic arch. The flow rates to individual arteries were also tuned to match that reported in the literature (30 (link), 31 (link)), which are summarized in Table 1.

The NASA LIS system was used in the present study, as it supports DA^{58 (link),59 (link)}, multiple LSMs, and streamflow routing through the HyMAP routing model^{45 (link)}. The LIS LSM and DA system is well suited to resolve this and other extreme events, as it has been successfully demonstrated in several DA scenarios, specifically in the assimilation of surface soil moisture^{39 (link),58 (link),59 (link)}, snow^{24 (link)–26 (link),39 (link)}, leaf area index^{20 (link),57 (link)}, and other surface variables^{18 (link),57 (link),60 (link)}.
Version 4.0.1 of the Noah-MP LSM^{44 (link)} is used with dynamic vegetation, and the runoff scheme is TOPMODEL with groundwater^{44 (link)}. Noah-MP is run at 10-km grid resolution for the full Mississippi River basin domain (28.55°–49.95° N, 77.95°–113.95° W). Noah-MP uses four soil levels with depths (from top to bottom) of 0.1, 0.3, 0.6 and 1.0 m. The OL simulation was spun up from 1 January 1980 and executed through 1 January 2021 with MERRA2 atmospheric forcing and IMERG precipitation (when available starting in 2002). HyMAP streamflow routing is added to the OL simulation starting 1 January 2000 and continued through 1 January 2021. As the SMAP dataset does not begin until early 2015, the AMSR-2 DA, SMAP DA, and full multi-variate DA simulations were all started on 1 April 2015 with initial conditions from the OL simulation. The MODIS-DA simulation was initialized from the OL simulation on 1 July 2002, at the start of the MODIS LAI dataset.
LIS uses a 1-dimensional Ensemble Kalman Filter^{61 (link)} (EnKF) for DA. This DA method is used for all three datasets and the multivariate DA. EnKF has been widely used in prior studies^{61 (link)–66} , and it enables the DA to account for both model errors and non-linear tendencies within the model processes. EnKF additionally allows for the characterization of model errors with an ensemble, and it is capable of resolving non-linear model dynamics and discontinuities in temporal observations^{61 (link)}. EnKF DA has also been demonstrated to add value to model variables in a theoretical case study, where forcing and model biases were introduced, with snow and soil moisture^{65 (link)}. The EnKF process alternates between the model forecast timestep and the analysis update step. EnKF is executed starting with the forecast timestep (i.e., running the LSM forward). In Eq. 1, let the forecast timestep be (k) and ƒsu the prior analysis timestep be (k−1). Subscripts (i) and (j) are the x and y grid points in the model domain, respectively. When the LSM is run, the model state (for any given variable) at a grid point is projected forward from ( ${\widehat{\mathbf{x}}}_{\mathbf{k}-1,\mathbf{i},\mathbf{j}}^{+}$ ) to the LSM state at k ( ${\widehat{\mathbf{x}}}_{\mathbf{k},\mathbf{i},\mathbf{j}}^{-}$ ). The analysis step is then computed as follows: $${\widehat{\mathit{x}}}_{\mathit{k},\mathit{i},\mathit{j}}^{+}={\widehat{\mathit{x}}}_{\mathit{k},\mathit{i},\mathit{j}}^{-}+{\mathit{K}}_{\mathit{k},\mathit{i},\mathit{j}}\left({\widehat{\mathit{y}}}_{\mathit{k},\mathit{i},\mathit{j}},-,{\mathit{H}}_{\mathit{k},\mathit{i},\mathit{j}},{\widehat{\mathit{x}}}_{\mathit{k}}^{-}\right)$$
Here, the posterior state ( ${\widehat{\mathbf{x}}}_{\mathbf{k},\mathbf{i},\mathbf{j}}^{+}$ ) combines the observation state ( ${\widehat{\mathit{y}}}_{\mathit{k},\mathit{i},\mathit{j}}$ ) and the a priori state ( ${\widehat{\mathbf{x}}}_{\mathbf{k},\mathbf{i},\mathbf{j}}^{-}$ ). In this equation, ${\mathit{H}}_{\mathit{k},\mathit{i},\mathit{j}}$ represents the observation operator that relates the model states to the observation. ${\mathit{K}}_{\mathit{k},\mathit{i},\mathit{j}}$ is the Kalman gain, which weights the impact of forecast innovations $\left({\widehat{\mathit{y}}}_{\mathit{k},\mathit{i},\mathit{j}},-,{\mathit{H}}_{\mathit{k},\mathit{i},\mathit{j}},{\widehat{\mathit{x}}}_{\mathit{k},\mathit{i},\mathit{j}}^{-}\right)$ in the analysis update. Kalman gain is based on the model and observation error covariances.
The DA methods and settings used here follow previous work^{57 (link)}, and the settings for forcing perturbations and model variable perturbations are shown in Supplementary Table S1. The forcing perturbations are the same for the snow depth, soil moisture, LAI, and multivariate DA simulations, and these settings follow prior applications of EnKF DA^{59 (link),67 (link)}. Note that the state vectors for snow depth, soil moisture, and LAI, are all independent and executed as separate instances of EnKF DA. All DA cases have 20 ensemble members. The 20-member ensemble configuration is sufficient to avoid sampling errors for snow DA, and has been demonstrated in prior work^{26 (link)}. The forcing state perturbation is 1-h, and the model state perturbation is 3-h. Observation state perturbations depend on the data temporal resolution and are 3-h for soil moisture and snow depth, and 24-h for LAI. Perturbations to atmospheric forcing account for cross-correlation in space and time between variables, which can better simulate the effects of model uncertainty^{25 (link)}.
For snow depth DA, AMSR-2 microwave snow depth retrievals are used, and this is constrained by MODIS snow cover. The Snow Depth and SWE variables of Noah-MP were updated for the AMSR-2 assimilation step. Both of these variables are also perturbed in the DA step (see Table S1 in Supplementary Material). Previous work demonstrated that this method could improve simulated snow states across North America^{24 (link)}. Furthermore, the Noah-MP snow scheme adds value over the earlier Noah LSM, as it uses a 3-layer snow model that accounts for the full energy balance of the snow pack^{44 (link)}. This generally improves the performance of the Noah-MP LSM for resolving snowpack^{68 (link)–72 (link)}.
SMAP soil moisture^{20 (link),41 (link),57 (link)} is used for soil moisture assimilation. SMAP soil moisture is downscaled to 1-km using the Thermal Hydraulic of Soil Moisture Disaggregation (THySM) algorithm^{73 (link)}, which is also available online through the USDA Crop-CASMA system (https://nassgeo.csiss.gmu.edu/CropCASMA/). When available, both ascending (6 PM) and descending (6 AM) paths of data are used for assimilation. As THySM SMAP product is based on surface soil moisture, the shallowest layer (0–10 cm) of Noah-MP is updated. The LIS “anomaly correction” algorithm is used to correct any biases in the SMAP product^{74 (link)}.
The MCD15A2H Version 6 MODIS product⁷⁵ is used for LAI assimilation. While the MODIS MCD15A2H Version 6 product has an 8-day temporal resolution, a smoothing algorithm within the LIS system is used, so that assimilation may be performed daily.
For all the model experiments, means and standard deviations for the model variables (including soil moisture and LAI) are computed from 2016 through 2020. While we acknowledge this is a short analysis period, it is necessary because of the limits of the DA datasets (SMAP is only available from April 2015 to present). For analysis, SWE melt, soil moisture, and LAI are considered in 2019. SWE melt and soil moisture analysis use anomalies (analysis year—mean), and LAI analysis uses standardized anomalies (analysis year–mean)/standard deviation, respectively.
SSIM for SWE melt and LAI is computed using the SSIM function of the python scikit package (available online at: https://scikit-image.org/docs/stable/auto_examples/transform/plot_ssim.html), and gaussian smoothing was performed using the ndimage.filters.gaussian_filter function of the python SciPy package (available online at: https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.ndimage.filters.gaussian_filter.html). Note that this additional step was necessary since EnKF does not include smoothing when DA is performed, and smoothing of model and observation data eliminates the influence local-scale discontinuities and errors in the DA and observation based datasets.

The set of boundary conditions considered in most of our simulations is a physiologically motivated and relatively complex mix of Dirichlet and Neumann type boundary conditions. These boundary conditions are applied in all simulations reported in the Results section of the paper, and all simulations in S1 Appendix, unless otherwise stated.
In the physiologically motivated boundary conditions, we apply a Dirichlet boundary condition fixing the electric potential, ϕ, at zero in an area of the boundary in the y- and z-directions in the extracellular cleft. More specifically, this area, $\partial {\Omega }_D^D$ is defined for the x-values corresponding to the middle third of the extracellular cleft, as illustrated in Fig 3. In $\partial {\Omega }_D^D$ , we apply Dirichlet boundary conditions for the all the ionic concentrations as well. The concentrations in this area are fixed at the initial conditions for the extracellular space (see Table 2). At the leftmost boundary of the pre-junctional cell, marked as $\partial {\Omega }_{N,1}^D$ in Fig 3, we fix the concentrations at the initial conditions for the intracellular space see Table 2), and apply no-flux Neumann boundary conditions for ϕ. At the rightmost boundary of the post-junctional cell, marked as $\partial {\Omega }_{N,2}^D$ in Fig 3, we use the same boundary conditions as for $\partial {\Omega }_{N,1}^D$ in the simulations used to find the resting state of the system (see Section 2.2). However, when the Na⁺ channels of the pre-junctional cell are opened, we apply Dirichlet boundary conditions for ϕ on the right boundary of the post-junctional cell ( $\partial {\Omega }_{N,2}^D$ ), fixing the potential at this boundary at the value of ϕ found at the end of the resting state simulations. On the remaining part of the domain boundary, $\partial {\Omega }_N^N$ , we apply no-flux Neumann boundary conditions for both ϕ and the concentrations.
In summary, the boundary conditions are given by
$$\begin{array}{cc}\hfill \epsilon \nabla \varphi ·\mathbf{n}=0& \text{at}\partial {\Omega }_N^N\cup \partial {\Omega }_{N,1}^D\text{,}\hfill \end{array}$$
$$\begin{array}{ccc}\hfill \epsilon \nabla \varphi ·\mathbf{n}=0& & \text{at}\partial {\Omega }_{N,2}^D\text{for}t\le t^{*}\text{,}\hfill \end{array}$$
$$\begin{array}{ccc}\hfill \varphi =\varphi \left(t^{*}\right)& & \text{at}\partial {\Omega }_{N,2}^D\text{for}t>t^{*}\text{,}\hfill \end{array}$$
$$\begin{array}{ccc}\hfill \varphi =0& & \text{at}\partial {\Omega }_D^D\text{,}\hfill \end{array}$$
$$\begin{array}{ccc}\hfill D_k\nabla c_k·\mathbf{n}=0& & \text{at}\partial {\Omega }_N^N\text{,}\hfill \end{array}$$
$$\begin{array}{ccc}\hfill c_k=c_{k,i}^0& & \text{at}\partial {\Omega }_{N,1}^D\cup \partial {\Omega }_{N,2}^D,\hfill \end{array}$$
$$\begin{array}{ccc}\hfill c_k=c_{k,e}^0& & \text{at}\partial {\Omega }_D^D,\hfill \end{array}$$
for k = {Na⁺, K⁺, Ca²⁺ and Cl⁻}, where $c_{k,e}^0$ and $c_{k,i}^0$ are the extracellular and the intracellular concentrations, respectively, of the ion species k specified in Table 2 and t* is the point in time when the Na⁺ channels of the pre-junctional cell are opened (see Section 2.2). Furthermore, $\partial {\Omega }_{N,1}^D$ , $\partial {\Omega }_{N,2}^D$ and $\partial {\Omega }_D^D$ are illustrated in Fig 3, and $\partial {\Omega }_N^N$ is the remaining part of the domain boundary, defined by $\partial {\Omega }_N^N=\partial \Omega \backslash (\partial {\Omega }_{N,1}^D\cup \partial {\Omega }_{N,2}^D\cup \partial {\Omega }_D^D)$ , where ∂Ω is the entire domain boundary.
Note that in S1 Appendix, we show the results of simulations where Neumann boundary conditions are applied at all parts of the boundary for all ionic concentrations and everywhere except at the right boundary for ϕ. The results of the simulations with this different choice of boundary condition seem to be very close to the results obtained using the boundary conditions described above.