To solve the given optimization problem, we first need to translate the natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints based on the information provided.

Let's denote:
- \(x_1\) as the number of premium sandwiches made,
- \(x_2\) as the number of regular sandwiches made.

The objective is to maximize profit. Given that each premium sandwich yields a $5 profit and each regular sandwich yields a $1 profit, the objective function can be represented algebraically as:
\[ \text{Maximize:} \quad 5x_1 + x_2 \]

Now, let's define the constraints based on the given limitations:
1. **Preparation Time Constraint**: The total preparation time for premium and regular sandwiches must not exceed 400 minutes. Since a premium sandwich takes 6 minutes to prepare and a regular sandwich takes 4 minutes, this constraint can be represented as:
\[ 6x_1 + 4x_2 \leq 400 \]
2. **Meat Slices Constraint**: The total slices of meat used for premium and regular sandwiches must not exceed 100 slices. Given that a premium sandwich requires 5 slices of meat and a regular sandwich requires 1 slice, this constraint can be represented as:
\[ 5x_1 + x_2 \leq 100 \]
3. **Non-Negativity Constraints**: The number of sandwiches cannot be negative, so we have:
\[ x_1 \geq 0 \]
\[ x_2 \geq 0 \]

In summary, the symbolic representation of the problem is:

```json
{
  'sym_variables': [('x1', 'number of premium sandwiches'), ('x2', 'number of regular sandwiches')],
  'objective_function': '5*x1 + x2',
  'constraints': [
    '6*x1 + 4*x2 <= 400',
    '5*x1 + x2 <= 100',
    'x1 >= 0',
    'x2 >= 0'
  ]
}
```

To solve this linear programming problem using Gurobi in Python, we can use the following code:

```python
from gurobipy import *

# Create a new model
m = Model("Sandwich_Optimization")

# Define variables
x1 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="premium_sandwiches")
x2 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="regular_sandwiches")

# Set the objective function
m.setObjective(5*x1 + x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(6*x1 + 4*x2 <= 400, "preparation_time")
m.addConstr(5*x1 + x2 <= 100, "meat_slices")

# Optimize the model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of premium sandwiches: {x1.x}")
    print(f"Number of regular sandwiches: {x2.x}")
    print(f"Maximum profit: ${m.objVal:.2f}")
else:
    print("No optimal solution found")
```