To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. This involves defining variables and translating the objective function and constraints into mathematical expressions.

Let's define:
- $x_1$ as the number of first-class seats sold,
- $x_2$ as the number of second-class seats sold.

The objective is to maximize profit, which can be represented by the objective function: $75x_1 + 50x_2$.

The constraints given in the problem are:
1. The train carries at most 500 passengers: $x_1 + x_2 \leq 500$.
2. At least 100 seats are reserved for first-class: $x_1 \geq 100$.
3. At least twice as many passengers prefer second-class over first-class: $x_2 \geq 2x_1$.

Thus, the symbolic representation of the problem can be summarized as follows:

```json
{
    'sym_variables': [('x1', 'number of first-class seats sold'), ('x2', 'number of second-class seats sold')],
    'objective_function': '75*x1 + 50*x2',
    'constraints': ['x1 + x2 <= 500', 'x1 >= 100', 'x2 >= 2*x1']
}
```

Given this representation, we can now implement the solution using Gurobi in Python. The goal is to maximize the profit function under the given constraints.

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(lb=0, vtype=GRB.INTEGER, name="first_class_seats")
x2 = m.addVar(lb=0, vtype=GRB.INTEGER, name="second_class_seats")

# Set the objective function: Maximize profit
m.setObjective(75*x1 + 50*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x1 + x2 <= 500, "total_passengers")
m.addConstr(x1 >= 100, "first_class_minimum")
m.addConstr(x2 >= 2*x1, "second_class_preference")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"First-class seats: {x1.x}")
    print(f"Second-class seats: {x2.x}")
    print(f"Maximum Profit: {m.objVal}")
else:
    print("No optimal solution found")
```